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tendsto_sum_indicator_at_top_iff [is_finite_measure μ]
(hfmono : ∀ᵐ ω ∂μ, ∀ n, f n ω ≤ f (n + 1) ω)
(hf : adapted ℱ f) (hint : ∀ n, integrable (f n) μ)
(hbdd : ∀ᵐ ω ∂μ, ∀ n, |f (n + 1) ω - f n ω| ≤ R) :
∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top at_top ↔
tendsto (λ n, predictable_part f ℱ μ n ω) at_top at_top | begin
have h₁ := (martingale_martingale_part hf hint).ae_not_tendsto_at_top_at_top
(martingale_part_bdd_difference ℱ hbdd),
have h₂ := (martingale_martingale_part hf hint).ae_not_tendsto_at_top_at_bot
(martingale_part_bdd_difference ℱ hbdd),
have h₃ : ∀ᵐ ω ∂μ, ∀ n, 0 ≤ μ[f (n + 1) - f n | ℱ n] ω,
{ refi... | lemma | measure_theory.tendsto_sum_indicator_at_top_iff | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range_mono",
"monotone_nat_of_le_succ"
] | An a.e. monotone adapted process `f` with uniformly bounded differences converges to `+∞` if
and only if its predictable part also converges to `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_sum_indicator_at_top_iff' [is_finite_measure μ]
{s : ℕ → set Ω} (hs : ∀ n, measurable_set[ℱ n] (s n)) :
∀ᵐ ω ∂μ,
tendsto (λ n, ∑ k in finset.range n, (s (k + 1)).indicator (1 : Ω → ℝ) ω) at_top at_top ↔
tendsto (λ n, ∑ k in finset.range n, μ[(s (k + 1)).indicator (1 : Ω → ℝ) | ℱ k] ω)
at_top a... | begin
have := tendsto_sum_indicator_at_top_iff (eventually_of_forall $ λ ω n, _) (adapted_process hs)
(integrable_process μ hs) (eventually_of_forall $ process_difference_le s),
swap,
{ rw [process, process, ← sub_nonneg, finset.sum_apply, finset.sum_apply,
finset.sum_range_succ_sub_sum],
exact set.... | lemma | measure_theory.tendsto_sum_indicator_at_top_iff' | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range",
"measurable_set",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_mem_limsup_at_top_iff (μ : measure Ω) [is_finite_measure μ]
{s : ℕ → set Ω} (hs : ∀ n, measurable_set[ℱ n] (s n)) :
∀ᵐ ω ∂μ, ω ∈ limsup s at_top ↔
tendsto (λ n, ∑ k in finset.range n, μ[(s (k + 1)).indicator (1 : Ω → ℝ) | ℱ k] ω)
at_top at_top | (limsup_eq_tendsto_sum_indicator_at_top ℝ s).symm ▸ tendsto_sum_indicator_at_top_iff' hs | theorem | measure_theory.ae_mem_limsup_at_top_iff | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range",
"limsup_eq_tendsto_sum_indicator_at_top",
"measurable_set"
] | **Lévy's generalization of the Borel-Cantelli lemma**: given a sequence of sets `s` and a
filtration `ℱ` such that for all `n`, `s n` is `ℱ n`-measurable, `at_top.limsup s` is almost
everywhere equal to the set for which `∑ k, ℙ(s (k + 1) | ℱ k) = ∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
predictable_part {m0 : measurable_space Ω}
(f : ℕ → Ω → E) (ℱ : filtration ℕ m0) (μ : measure Ω . volume_tac) : ℕ → Ω → E | λ n, ∑ i in finset.range n, μ[f (i+1) - f i | ℱ i] | def | measure_theory.predictable_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"finset.range",
"measurable_space"
] | Any `ℕ`-indexed stochastic process can be written as the sum of a martingale and a predictable
process. This is the predictable process. See `martingale_part` for the martingale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
predictable_part_zero : predictable_part f ℱ μ 0 = 0 | by simp_rw [predictable_part, finset.range_zero, finset.sum_empty] | lemma | measure_theory.predictable_part_zero | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"finset.range_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted_predictable_part : adapted ℱ (λ n, predictable_part f ℱ μ (n+1)) | λ n, finset.strongly_measurable_sum' _
(λ i hin, strongly_measurable_condexp.mono (ℱ.mono (finset.mem_range_succ_iff.mp hin))) | lemma | measure_theory.adapted_predictable_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted_predictable_part' : adapted ℱ (λ n, predictable_part f ℱ μ n) | λ n, finset.strongly_measurable_sum' _
(λ i hin, strongly_measurable_condexp.mono (ℱ.mono (finset.mem_range_le hin))) | lemma | measure_theory.adapted_predictable_part' | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"finset.mem_range_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_part {m0 : measurable_space Ω}
(f : ℕ → Ω → E) (ℱ : filtration ℕ m0) (μ : measure Ω . volume_tac) : ℕ → Ω → E | λ n, f n - predictable_part f ℱ μ n | def | measure_theory.martingale_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"measurable_space"
] | Any `ℕ`-indexed stochastic process can be written as the sum of a martingale and a predictable
process. This is the martingale. See `predictable_part` for the predictable process. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
martingale_part_add_predictable_part (ℱ : filtration ℕ m0) (μ : measure Ω) (f : ℕ → Ω → E) :
martingale_part f ℱ μ + predictable_part f ℱ μ = f | sub_add_cancel _ _ | lemma | measure_theory.martingale_part_add_predictable_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_part_eq_sum :
martingale_part f ℱ μ =
λ n, f 0 + ∑ i in finset.range n, (f (i+1) - f i - μ[f (i+1) - f i | ℱ i]) | begin
rw [martingale_part, predictable_part],
ext1 n,
rw [finset.eq_sum_range_sub f n, ← add_sub, ← finset.sum_sub_distrib],
end | lemma | measure_theory.martingale_part_eq_sum | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted_martingale_part (hf : adapted ℱ f) :
adapted ℱ (martingale_part f ℱ μ) | adapted.sub hf adapted_predictable_part' | lemma | measure_theory.adapted_martingale_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_martingale_part (hf_int : ∀ n, integrable (f n) μ) (n : ℕ) :
integrable (martingale_part f ℱ μ n) μ | begin
rw martingale_part_eq_sum,
exact (hf_int 0).add
(integrable_finset_sum' _ (λ i hi, ((hf_int _).sub (hf_int _)).sub integrable_condexp)),
end | lemma | measure_theory.integrable_martingale_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_martingale_part (hf : adapted ℱ f) (hf_int : ∀ n, integrable (f n) μ)
[sigma_finite_filtration μ ℱ] :
martingale (martingale_part f ℱ μ) ℱ μ | begin
refine ⟨adapted_martingale_part hf, λ i j hij, _⟩,
-- ⊢ μ[martingale_part f ℱ μ j | ℱ i] =ᵐ[μ] martingale_part f ℱ μ i
have h_eq_sum : μ[martingale_part f ℱ μ j | ℱ i]
=ᵐ[μ] f 0 + ∑ k in finset.range j, (μ[f (k+1) - f k | ℱ i] - μ[μ[f (k+1) - f k | ℱ k] | ℱ i]),
{ rw martingale_part_eq_sum,
refine... | lemma | measure_theory.martingale_martingale_part | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_part_add_ae_eq [sigma_finite_filtration μ ℱ] {f g : ℕ → Ω → E}
(hf : martingale f ℱ μ) (hg : adapted ℱ (λ n, g (n + 1))) (hg0 : g 0 = 0)
(hgint : ∀ n, integrable (g n) μ) (n : ℕ) :
martingale_part (f + g) ℱ μ n =ᵐ[μ] f n | begin
set h := f - martingale_part (f + g) ℱ μ with hhdef,
have hh : h = predictable_part (f + g) ℱ μ - g,
{ rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictable_part (f + g) ℱ μ),
martingale_part_add_predictable_part] },
have hhpred : adapted ℱ (λ n, h (n + 1)),
{ rw hh,
exact adapted_predic... | lemma | measure_theory.martingale_part_add_ae_eq | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
predictable_part_add_ae_eq [sigma_finite_filtration μ ℱ] {f g : ℕ → Ω → E}
(hf : martingale f ℱ μ) (hg : adapted ℱ (λ n, g (n + 1))) (hg0 : g 0 = 0)
(hgint : ∀ n, integrable (g n) μ) (n : ℕ) :
predictable_part (f + g) ℱ μ n =ᵐ[μ] g n | begin
filter_upwards [martingale_part_add_ae_eq hf hg hg0 hgint n] with ω hω,
rw ← add_right_inj (f n ω),
conv_rhs { rw [← pi.add_apply, ← pi.add_apply,
← martingale_part_add_predictable_part ℱ μ (f + g)] },
rw [pi.add_apply, pi.add_apply, hω],
end | lemma | measure_theory.predictable_part_add_ae_eq | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
predictable_part_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ}
(ℱ : filtration ℕ m0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ∀ i, |predictable_part f ℱ μ (i + 1) ω - predictable_part f ℱ μ i ω| ≤ R | begin
simp_rw [predictable_part, finset.sum_apply, finset.sum_range_succ_sub_sum],
exact ae_all_iff.2 (λ i, ae_bdd_condexp_of_ae_bdd $ ae_all_iff.1 hbdd i),
end | lemma | measure_theory.predictable_part_bdd_difference | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_part_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ}
(ℱ : filtration ℕ m0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ∀ i, |martingale_part f ℱ μ (i + 1) ω - martingale_part f ℱ μ i ω| ≤ ↑(2 * R) | begin
filter_upwards [hbdd, predictable_part_bdd_difference ℱ hbdd] with ω hω₁ hω₂ i,
simp only [two_mul, martingale_part, pi.sub_apply],
have : |f (i + 1) ω - predictable_part f ℱ μ (i + 1) ω - (f i ω - predictable_part f ℱ μ i ω)| =
|(f (i + 1) ω - f i ω) - (predictable_part f ℱ μ (i + 1) ω - predictable_pa... | lemma | measure_theory.martingale_part_bdd_difference | probability.martingale | src/probability/martingale/centering.lean | [
"probability.martingale.basic"
] | [
"abs_sub",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_frequently_of_upcrossings_lt_top (hab : a < b) (hω : upcrossings a b f ω ≠ ∞) :
¬((∃ᶠ n in at_top, f n ω < a) ∧ (∃ᶠ n in at_top, b < f n ω)) | begin
rw [← lt_top_iff_ne_top, upcrossings_lt_top_iff] at hω,
replace hω : ∃ k, ∀ N, upcrossings_before a b f N ω < k,
{ obtain ⟨k, hk⟩ := hω,
exact ⟨k + 1, λ N, lt_of_le_of_lt (hk N) k.lt_succ_self⟩ },
rintro ⟨h₁, h₂⟩,
rw frequently_at_top at h₁ h₂,
refine not_not.2 hω _,
push_neg,
intro k,
induc... | lemma | measure_theory.not_frequently_of_upcrossings_lt_top | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"exists_const",
"ih",
"lt_top_iff_ne_top",
"zero_le'"
] | If a stochastic process has bounded upcrossing from below `a` to above `b`,
then it does not frequently visit both below `a` and above `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upcrossings_eq_top_of_frequently_lt (hab : a < b)
(h₁ : ∃ᶠ n in at_top, f n ω < a) (h₂ : ∃ᶠ n in at_top, b < f n ω) :
upcrossings a b f ω = ∞ | classical.by_contradiction (λ h, not_frequently_of_upcrossings_lt_top hab h ⟨h₁, h₂⟩) | lemma | measure_theory.upcrossings_eq_top_of_frequently_lt | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [] | A stochastic process that frequently visits below `a` and above `b` have infinite
upcrossings. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_of_uncrossing_lt_top
(hf₁ : liminf (λ n, (‖f n ω‖₊ : ℝ≥0∞)) at_top < ∞)
(hf₂ : ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞) :
∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) | begin
by_cases h : is_bounded_under (≤) at_top (λ n, |f n ω|),
{ rw is_bounded_under_le_abs at h,
refine tendsto_of_no_upcrossings rat.dense_range_cast _ h.1 h.2,
{ intros a ha b hb hab,
obtain ⟨⟨a, rfl⟩, ⟨b, rfl⟩⟩ := ⟨ha, hb⟩,
exact not_frequently_of_upcrossings_lt_top hab (hf₂ a b (rat.cast_lt... | lemma | measure_theory.tendsto_of_uncrossing_lt_top | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"ennreal.exists_upcrossings_of_not_bounded_under",
"rat.dense_range_cast",
"tendsto_of_no_upcrossings"
] | A realization of a stochastic process with bounded upcrossings and bounded liminfs is
convergent.
We use the spelling `< ∞` instead of the standard `≠ ∞` in the assumptions since it is not as easy
to change `<` to `≠` under binders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.upcrossings_ae_lt_top' [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) (hab : a < b) :
∀ᵐ ω ∂μ, upcrossings a b f ω < ∞ | begin
refine ae_lt_top (hf.adapted.measurable_upcrossings hab) _,
have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b,
rw [mul_comm, ← ennreal.le_div_iff_mul_le] at this,
{ refine (lt_of_le_of_lt this (ennreal.div_lt_top _ _)).ne,
{ have hR' : ∀ n, ∫⁻ ω, ‖f n ω - a‖₊ ∂μ ≤ R + ‖a‖₊ * μ set.univ,
... | lemma | measure_theory.submartingale.upcrossings_ae_lt_top' | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"abs_of_nonneg",
"ennreal.coe_add",
"ennreal.coe_le_coe",
"ennreal.coe_ne_top",
"ennreal.coe_to_real",
"ennreal.div_lt_top",
"ennreal.le_div_iff_mul_le",
"ennreal.mul_lt_top",
"ennreal.of_real_eq_zero",
"ennreal.of_real_le_iff_le_to_real",
"ennreal.of_real_ne_top",
"le_rfl",
"measurable_cons... | An L¹-bounded submartingale has bounded upcrossings almost everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.upcrossings_ae_lt_top [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂μ, ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞ | begin
simp only [ae_all_iff, eventually_imp_distrib_left],
rintro a b hab,
exact hf.upcrossings_ae_lt_top' hbdd (rat.cast_lt.2 hab),
end | lemma | measure_theory.submartingale.upcrossings_ae_lt_top | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.exists_ae_tendsto_of_bdd [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂μ, ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) | begin
filter_upwards [hf.upcrossings_ae_lt_top hbdd, ae_bdd_liminf_at_top_of_snorm_bdd one_ne_zero
(λ n, (hf.strongly_measurable n).measurable.mono (ℱ.le n) le_rfl) hbdd] with ω h₁ h₂,
exact tendsto_of_uncrossing_lt_top h₂ h₁,
end | lemma | measure_theory.submartingale.exists_ae_tendsto_of_bdd | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"le_rfl",
"measurable.mono",
"one_ne_zero"
] | An L¹-bounded submartingale converges almost everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.exists_ae_trim_tendsto_of_bdd [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂(μ.trim (Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _) : (⨆ n, ℱ n) ≤ m0)),
∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) | begin
rw [ae_iff, trim_measurable_set_eq],
{ exact hf.exists_ae_tendsto_of_bdd hbdd },
{ exact measurable_set.compl (@measurable_set_exists_tendsto _ _ _ _ _ _ (⨆ n, ℱ n) _ _ _ _ _
(λ n, ((hf.strongly_measurable n).measurable.mono (le_Sup ⟨n, rfl⟩) le_rfl))) }
end | lemma | measure_theory.submartingale.exists_ae_trim_tendsto_of_bdd | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"Sup_le",
"le_Sup",
"le_rfl",
"measurable.mono",
"measurable_set.compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.ae_tendsto_limit_process [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (ℱ.limit_process f μ ω)) | begin
classical,
suffices : ∃ g, strongly_measurable[⨆ n, ℱ n] g ∧ ∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (g ω)),
{ rw [limit_process, dif_pos this],
exact (classical.some_spec this).2 },
set g' : Ω → ℝ := λ ω, if h : ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) then h.some else 0,
have hle : (⨆ n, ℱ n) ≤ m0... | lemma | measure_theory.submartingale.ae_tendsto_limit_process | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"Sup_le",
"ae_measurable",
"ae_measurable_of_tendsto_metrizable_ae'",
"le_Sup",
"le_rfl",
"measurable.mono"
] | **Almost everywhere martingale convergence theorem**: An L¹-bounded submartingale converges
almost everywhere to a `⨆ n, ℱ n`-measurable function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.mem_ℒp_limit_process {p : ℝ≥0∞}
(hf : submartingale f ℱ μ) (hbdd : ∀ n, snorm (f n) p μ ≤ R) :
mem_ℒp (ℱ.limit_process f μ) p μ | mem_ℒp_limit_process_of_snorm_bdd
(λ n, ((hf.strongly_measurable n).mono (ℱ.le n)).ae_strongly_measurable) hbdd | lemma | measure_theory.submartingale.mem_ℒp_limit_process | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [] | The limiting process of an Lᵖ-bounded submartingale is Lᵖ. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.tendsto_snorm_one_limit_process
(hf : submartingale f ℱ μ) (hunif : uniform_integrable f 1 μ) :
tendsto (λ n, snorm (f n - ℱ.limit_process f μ) 1 μ) at_top (𝓝 0) | begin
obtain ⟨R, hR⟩ := hunif.2.2,
have hmeas : ∀ n, ae_strongly_measurable (f n) μ :=
λ n, ((hf.strongly_measurable n).mono (ℱ.le _)).ae_strongly_measurable,
exact tendsto_Lp_of_tendsto_in_measure _ le_rfl ennreal.one_ne_top hmeas
(mem_ℒp_limit_process_of_snorm_bdd hmeas hR) hunif.2.1
(tendsto_in_mea... | lemma | measure_theory.submartingale.tendsto_snorm_one_limit_process | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"ennreal.one_ne_top",
"le_rfl"
] | Part a of the **L¹ martingale convergence theorem**: a uniformly integrable submartingale
adapted to the filtration `ℱ` converges a.e. and in L¹ to an integrable function which is
measurable with respect to the σ-algebra `⨆ n, ℱ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.ae_tendsto_limit_process_of_uniform_integrable
(hf : submartingale f ℱ μ) (hunif : uniform_integrable f 1 μ) :
∀ᵐ ω ∂μ, tendsto (λ n, f n ω) at_top (𝓝 (ℱ.limit_process f μ ω)) | let ⟨R, hR⟩ := hunif.2.2 in hf.ae_tendsto_limit_process hR | lemma | measure_theory.submartingale.ae_tendsto_limit_process_of_uniform_integrable | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale.eq_condexp_of_tendsto_snorm {μ : measure Ω}
(hf : martingale f ℱ μ) (hg : integrable g μ)
(hgtends : tendsto (λ n, snorm (f n - g) 1 μ) at_top (𝓝 0)) (n : ℕ) :
f n =ᵐ[μ] μ[g | ℱ n] | begin
rw [← sub_ae_eq_zero, ← snorm_eq_zero_iff ((((hf.strongly_measurable n).mono (ℱ.le _)).sub
(strongly_measurable_condexp.mono (ℱ.le _))).ae_strongly_measurable) one_ne_zero],
have ht : tendsto (λ m, snorm (μ[f m - g | ℱ n]) 1 μ) at_top (𝓝 0),
{ have hint : ∀ m, integrable (f m - g) μ := λ m, (hf.integra... | lemma | measure_theory.martingale.eq_condexp_of_tendsto_snorm | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"one_ne_zero",
"tendsto_at_top_of_eventually_const",
"tendsto_const_nhds",
"tendsto_nhds_unique",
"tendsto_of_tendsto_of_tendsto_of_le_of_le"
] | If a martingale `f` adapted to `ℱ` converges in L¹ to `g`, then for all `n`, `f n` is almost
everywhere equal to `𝔼[g | ℱ n]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
martingale.ae_eq_condexp_limit_process
(hf : martingale f ℱ μ) (hbdd : uniform_integrable f 1 μ) (n : ℕ) :
f n =ᵐ[μ] μ[ℱ.limit_process f μ | ℱ n] | let ⟨R, hR⟩ := hbdd.2.2 in hf.eq_condexp_of_tendsto_snorm
((mem_ℒp_limit_process_of_snorm_bdd hbdd.1 hR).integrable le_rfl)
(hf.submartingale.tendsto_snorm_one_limit_process hbdd) n | lemma | measure_theory.martingale.ae_eq_condexp_limit_process | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"le_rfl"
] | Part b of the **L¹ martingale convergence theorem**: if `f` is a uniformly integrable martingale
adapted to the filtration `ℱ`, then for all `n`, `f n` is almost everywhere equal to the conditional
expectation of its limiting process wrt. `ℱ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable.tendsto_ae_condexp
(hg : integrable g μ) (hgmeas : strongly_measurable[⨆ n, ℱ n] g) :
∀ᵐ x ∂μ, tendsto (λ n, μ[g | ℱ n] x) at_top (𝓝 (g x)) | begin
have hle : (⨆ n, ℱ n) ≤ m0 := Sup_le (λ m ⟨n, hn⟩, hn ▸ ℱ.le _),
have hunif : uniform_integrable (λ n, μ[g | ℱ n]) 1 μ := hg.uniform_integrable_condexp_filtration,
obtain ⟨R, hR⟩ := hunif.2.2,
have hlimint : integrable (ℱ.limit_process (λ n, μ[g | ℱ n]) μ) μ :=
(mem_ℒp_limit_process_of_snorm_bdd hunif... | lemma | measure_theory.integrable.tendsto_ae_condexp | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"Sup_le",
"forall_true_left",
"le_rfl",
"measurable_set",
"measurable_set.univ",
"measurable_space.induction_on_inter",
"measurable_space.measurable_space_supr_eq",
"tsum_congr",
"with_top.zero_lt_top"
] | Part c of the **L¹ martingale convergnce theorem**: Given a integrable function `g` which
is measurable with respect to `⨆ n, ℱ n` where `ℱ` is a filtration, the martingale defined by
`𝔼[g | ℱ n]` converges almost everywhere to `g`.
This martingale also converges to `g` in L¹ and this result is provided by
`measure_t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable.tendsto_snorm_condexp
(hg : integrable g μ) (hgmeas : strongly_measurable[⨆ n, ℱ n] g) :
tendsto (λ n, snorm (μ[g | ℱ n] - g) 1 μ) at_top (𝓝 0) | tendsto_Lp_of_tendsto_in_measure _ le_rfl ennreal.one_ne_top
(λ n, (strongly_measurable_condexp.mono (ℱ.le n)).ae_strongly_measurable)
(mem_ℒp_one_iff_integrable.2 hg) (hg.uniform_integrable_condexp_filtration).2.1
(tendsto_in_measure_of_tendsto_ae
(λ n,(strongly_measurable_condexp.mono (ℱ.le n)).ae_strongl... | lemma | measure_theory.integrable.tendsto_snorm_condexp | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"ennreal.one_ne_top",
"le_rfl"
] | Part c of the **L¹ martingale convergnce theorem**: Given a integrable function `g` which
is measurable with respect to `⨆ n, ℱ n` where `ℱ` is a filtration, the martingale defined by
`𝔼[g | ℱ n]` converges in L¹ to `g`.
This martingale also converges to `g` almost everywhere and this result is provided by
`measure_t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_ae_condexp (g : Ω → ℝ) :
∀ᵐ x ∂μ, tendsto (λ n, μ[g | ℱ n] x) at_top (𝓝 (μ[g | ⨆ n, ℱ n] x)) | begin
have ht : ∀ᵐ x ∂μ, tendsto (λ n, μ[μ[g | ⨆ n, ℱ n] | ℱ n] x) at_top (𝓝 (μ[g | ⨆ n, ℱ n] x)) :=
integrable_condexp.tendsto_ae_condexp strongly_measurable_condexp,
have heq : ∀ n, ∀ᵐ x ∂μ, μ[μ[g | ⨆ n, ℱ n] | ℱ n] x = μ[g | ℱ n] x :=
λ n, condexp_condexp_of_le (le_supr _ n) (supr_le (λ n, ℱ.le n)),
r... | lemma | measure_theory.tendsto_ae_condexp | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"le_supr",
"supr_le"
] | **Lévy's upward theorem**, almost everywhere version: given a function `g` and a filtration
`ℱ`, the sequence defined by `𝔼[g | ℱ n]` converges almost everywhere to `𝔼[g | ⨆ n, ℱ n]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_snorm_condexp (g : Ω → ℝ) :
tendsto (λ n, snorm (μ[g | ℱ n] - μ[g | ⨆ n, ℱ n]) 1 μ) at_top (𝓝 0) | begin
have ht : tendsto (λ n, snorm (μ[μ[g | ⨆ n, ℱ n] | ℱ n] - μ[g | ⨆ n, ℱ n]) 1 μ) at_top (𝓝 0) :=
integrable_condexp.tendsto_snorm_condexp strongly_measurable_condexp,
have heq : ∀ n, ∀ᵐ x ∂μ, μ[μ[g | ⨆ n, ℱ n] | ℱ n] x = μ[g | ℱ n] x :=
λ n, condexp_condexp_of_le (le_supr _ n) (supr_le (λ n, ℱ.le n)),... | lemma | measure_theory.tendsto_snorm_condexp | probability.martingale | src/probability/martingale/convergence.lean | [
"probability.martingale.upcrossing",
"measure_theory.function.uniform_integrable",
"measure_theory.constructions.polish"
] | [
"le_supr",
"supr_le"
] | **Lévy's upward theorem**, L¹ version: given a function `g` and a filtration `ℱ`, the
sequence defined by `𝔼[g | ℱ n]` converges in L¹ to `𝔼[g | ⨆ n, ℱ n]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_topology.second_countable_topology_of_countable {α : Type*} [topological_space α]
[discrete_topology α] [countable α] :
second_countable_topology α | @discrete_topology.second_countable_topology_of_encodable _ _ _ (encodable.of_countable _) | instance | discrete_topology.second_countable_topology_of_countable | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"countable",
"discrete_topology",
"discrete_topology.second_countable_topology_of_encodable",
"encodable.of_countable",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_stopping_time_ae_eq_restrict_eq_const
[(filter.at_top : filter ι).is_countably_generated]
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) [sigma_finite (μ.trim hτ.measurable_space_le)]
(hin : i ≤ n) :
μ[f n | hτ.measurable_space] =ᵐ[μ.restrict {x | τ x = i}] f i | begin
refine filter.eventually_eq.trans _ (ae_restrict_of_ae (h.condexp_ae_eq hin)),
refine condexp_ae_eq_restrict_of_measurable_space_eq_on hτ.measurable_space_le (ℱ.le i)
(hτ.measurable_set_eq' i) (λ t, _),
rw [set.inter_comm _ t, is_stopping_time.measurable_set_inter_eq_iff],
end | lemma | measure_theory.martingale.condexp_stopping_time_ae_eq_restrict_eq_const | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"filter",
"filter.at_top",
"filter.eventually_eq.trans",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hτ_le : ∀ x, τ x ≤ n)
[sigma_finite (μ.trim (hτ.measurable_space_le_of_le hτ_le))] (i : ι) :
μ[f n | hτ.measurable_space] =ᵐ[μ.restrict {x | τ x = i}] f i | begin
by_cases hin : i ≤ n,
{ refine filter.eventually_eq.trans _ (ae_restrict_of_ae (h.condexp_ae_eq hin)),
refine condexp_ae_eq_restrict_of_measurable_space_eq_on (hτ.measurable_space_le_of_le hτ_le)
(ℱ.le i) (hτ.measurable_set_eq' i) (λ t, _),
rw [set.inter_comm _ t, is_stopping_time.measurable_set... | lemma | measure_theory.martingale.condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"filter.eventually_eq.trans",
"set.inter_comm",
"set.mem_empty_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_ae_eq_restrict_eq
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hτ_le : ∀ x, τ x ≤ n)
[sigma_finite (μ.trim ((hτ.measurable_space_le_of_le hτ_le)))] (i : ι) :
stopped_value f τ =ᵐ[μ.restrict {x | τ x = i}] μ[f n | hτ.measurable_space] | begin
refine filter.eventually_eq.trans _
(condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const h hτ hτ_le i).symm,
rw [filter.eventually_eq, ae_restrict_iff' (ℱ.le _ _ (hτ.measurable_set_eq i))],
refine filter.eventually_of_forall (λ x hx, _),
rw set.mem_set_of_eq at hx,
simp_rw [stopped_value, hx],... | lemma | measure_theory.martingale.stopped_value_ae_eq_restrict_eq | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"filter.eventually_eq",
"filter.eventually_eq.trans",
"filter.eventually_of_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_ae_eq_condexp_of_le_const_of_countable_range
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ)
(hτ_le : ∀ x, τ x ≤ n) (h_countable_range : (set.range τ).countable)
[sigma_finite (μ.trim (hτ.measurable_space_le_of_le hτ_le))] :
stopped_value f τ =ᵐ[μ] μ[f n | hτ.measurable_space] | begin
have : set.univ = ⋃ i ∈ (set.range τ), {x | τ x = i},
{ ext1 x,
simp only [set.mem_univ, set.mem_range, true_and, set.Union_exists, set.Union_Union_eq',
set.mem_Union, set.mem_set_of_eq, exists_apply_eq_apply'], },
nth_rewrite 0 ← @measure.restrict_univ Ω _ μ,
rw [this, ae_eq_restrict_bUnion_iff... | lemma | measure_theory.martingale.stopped_value_ae_eq_condexp_of_le_const_of_countable_range | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"countable",
"exists_apply_eq_apply'",
"set.Union_Union_eq'",
"set.Union_exists",
"set.mem_Union",
"set.mem_range",
"set.mem_univ",
"set.range"
] | The value of a martingale `f` at a stopping time `τ` bounded by `n` is the conditional
expectation of `f n` with respect to the σ-algebra generated by `τ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stopped_value_ae_eq_condexp_of_le_const [countable ι]
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hτ_le : ∀ x, τ x ≤ n)
[sigma_finite (μ.trim (hτ.measurable_space_le_of_le hτ_le))] :
stopped_value f τ =ᵐ[μ] μ[f n | hτ.measurable_space] | h.stopped_value_ae_eq_condexp_of_le_const_of_countable_range hτ hτ_le (set.to_countable _) | lemma | measure_theory.martingale.stopped_value_ae_eq_condexp_of_le_const | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"countable",
"set.to_countable"
] | The value of a martingale `f` at a stopping time `τ` bounded by `n` is the conditional
expectation of `f n` with respect to the σ-algebra generated by `τ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stopped_value_ae_eq_condexp_of_le_of_countable_range
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hσ : is_stopping_time ℱ σ)
(hσ_le_τ : σ ≤ τ) (hτ_le : ∀ x, τ x ≤ n)
(hτ_countable_range : (set.range τ).countable) (hσ_countable_range : (set.range σ).countable)
[sigma_finite (μ.trim (hσ.measurable_space_le... | begin
haveI : sigma_finite (μ.trim (hτ.measurable_space_le_of_le hτ_le)),
{ exact sigma_finite_trim_mono _ (is_stopping_time.measurable_space_mono hσ hτ hσ_le_τ), },
have : μ[stopped_value f τ|hσ.measurable_space]
=ᵐ[μ] μ[μ[f n|hτ.measurable_space] | hσ.measurable_space],
from condexp_congr_ae (h.stoppe... | lemma | measure_theory.martingale.stopped_value_ae_eq_condexp_of_le_of_countable_range | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"countable",
"filter.eventually_eq.trans",
"set.range"
] | If `τ` and `σ` are two stopping times with `σ ≤ τ` and `τ` is bounded, then the value of a
martingale `f` at `σ` is the conditional expectation of its value at `τ` with respect to the
σ-algebra generated by `σ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stopped_value_ae_eq_condexp_of_le [countable ι]
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hσ : is_stopping_time ℱ σ)
(hσ_le_τ : σ ≤ τ) (hτ_le : ∀ x, τ x ≤ n) [sigma_finite (μ.trim hσ.measurable_space_le)] :
stopped_value f σ =ᵐ[μ] μ[stopped_value f τ | hσ.measurable_space] | h.stopped_value_ae_eq_condexp_of_le_of_countable_range hτ hσ hσ_le_τ hτ_le
(set.to_countable _) (set.to_countable _) | lemma | measure_theory.martingale.stopped_value_ae_eq_condexp_of_le | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"countable",
"set.to_countable"
] | If `τ` and `σ` are two stopping times with `σ ≤ τ` and `τ` is bounded, then the value of a
martingale `f` at `σ` is the conditional expectation of its value at `τ` with respect to the
σ-algebra generated by `σ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
condexp_stopped_value_stopping_time_ae_eq_restrict_le
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hσ : is_stopping_time ℱ σ)
[sigma_finite (μ.trim hσ.measurable_space_le)] (hτ_le : ∀ x, τ x ≤ n) :
μ[stopped_value f τ | hσ.measurable_space] =ᵐ[μ.restrict {x : Ω | τ x ≤ σ x}] stopped_value f τ | begin
rw ae_eq_restrict_iff_indicator_ae_eq
(hτ.measurable_space_le _ (hτ.measurable_set_le_stopping_time hσ)),
swap, apply_instance,
refine (condexp_indicator (integrable_stopped_value ι hτ h.integrable hτ_le)
(hτ.measurable_set_stopping_time_le hσ)).symm.trans _,
have h_int : integrable ({ω : Ω | τ ω ... | lemma | measure_theory.martingale.condexp_stopped_value_stopping_time_ae_eq_restrict_le | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"ae_eq_restrict_iff_indicator_ae_eq",
"measurable.strongly_measurable",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_min_ae_eq_condexp [sigma_finite_filtration μ ℱ]
(h : martingale f ℱ μ) (hτ : is_stopping_time ℱ τ) (hσ : is_stopping_time ℱ σ) {n : ι}
(hτ_le : ∀ x, τ x ≤ n) [h_sf_min : sigma_finite (μ.trim (hτ.min hσ).measurable_space_le)] :
stopped_value f (λ x, min (σ x) (τ x)) =ᵐ[μ] μ[stopped_value f τ | hσ.mea... | begin
refine (h.stopped_value_ae_eq_condexp_of_le hτ (hσ.min hτ) (λ x, min_le_right _ _) hτ_le).trans _,
refine ae_of_ae_restrict_of_ae_restrict_compl {x | σ x ≤ τ x} _ _,
{ exact condexp_min_stopping_time_ae_eq_restrict_le hσ hτ, },
{ suffices : μ[stopped_value f τ|(hσ.min hτ).measurable_space]
=ᵐ[μ.rest... | lemma | measure_theory.martingale.stopped_value_min_ae_eq_condexp | probability.martingale | src/probability/martingale/optional_sampling.lean | [
"order.succ_pred.linear_locally_finite",
"probability.martingale.basic"
] | [
"filter.eventually_eq",
"filter.eventually_eq.trans",
"inf_comm",
"measurable.strongly_measurable",
"measurable_space",
"set.mem_compl_iff"
] | **Optional Sampling theorem**. If `τ` is a bounded stopping time and `σ` is another stopping
time, then the value of a martingale `f` at the stopping time `min τ σ` is almost everywhere equal
to the conditional expectation of `f` stopped at `τ` with respect to the σ-algebra generated
by `σ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.expected_stopped_value_mono [sigma_finite_filtration μ 𝒢]
(hf : submartingale f 𝒢 μ) (hτ : is_stopping_time 𝒢 τ) (hπ : is_stopping_time 𝒢 π) (hle : τ ≤ π)
{N : ℕ} (hbdd : ∀ ω, π ω ≤ N) :
μ[stopped_value f τ] ≤ μ[stopped_value f π] | begin
rw [← sub_nonneg, ← integral_sub', stopped_value_sub_eq_sum' hle hbdd],
{ simp only [finset.sum_apply],
have : ∀ i, measurable_set[𝒢 i] {ω : Ω | τ ω ≤ i ∧ i < π ω},
{ intro i,
refine (hτ i).inter _,
convert (hπ i).compl,
ext x,
simpa },
rw integral_finset_sum,
{ refine... | lemma | measure_theory.submartingale.expected_stopped_value_mono | probability.martingale | src/probability/martingale/optional_stopping.lean | [
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"measurable_set"
] | Given a submartingale `f` and bounded stopping times `τ` and `π` such that `τ ≤ π`, the
expectation of `stopped_value f τ` is less than or equal to the expectation of `stopped_value f π`.
This is the forward direction of the optional stopping theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale_of_expected_stopped_value_mono [is_finite_measure μ]
(hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ τ π : Ω → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ ω, π ω ≤ N) →
μ[stopped_value f τ] ≤ μ[stopped_value f π]) :
submartingale f 𝒢 μ | begin
refine submartingale_of_set_integral_le hadp hint (λ i j hij s hs, _),
classical,
specialize hf (s.piecewise (λ _, i) (λ _, j)) _
(is_stopping_time_piecewise_const hij hs)
(is_stopping_time_const 𝒢 j) (λ x, (ite_le_sup _ _ _).trans (max_eq_right hij).le)
⟨j, λ x, le_rfl⟩,
rwa [stopped_value_c... | lemma | measure_theory.submartingale_of_expected_stopped_value_mono | probability.martingale | src/probability/martingale/optional_stopping.lean | [
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ite_le_sup"
] | The converse direction of the optional stopping theorem, i.e. an adapted integrable process `f`
is a submartingale if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the
stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale_iff_expected_stopped_value_mono [is_finite_measure μ]
(hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ) :
submartingale f 𝒢 μ ↔
∀ τ π : Ω → ℕ, is_stopping_time 𝒢 τ → is_stopping_time 𝒢 π → τ ≤ π → (∃ N, ∀ x, π x ≤ N) →
μ[stopped_value f τ] ≤ μ[stopped_value f π] | ⟨λ hf _ _ hτ hπ hle ⟨N, hN⟩, hf.expected_stopped_value_mono hτ hπ hle hN,
submartingale_of_expected_stopped_value_mono hadp hint⟩ | lemma | measure_theory.submartingale_iff_expected_stopped_value_mono | probability.martingale | src/probability/martingale/optional_stopping.lean | [
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | **The optional stopping theorem** (fair game theorem): an adapted integrable process `f`
is a submartingale if and only if for all bounded stopping times `τ` and `π` such that `τ ≤ π`, the
stopped value of `f` at `τ` has expectation smaller than its stopped value at `π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.stopped_process [is_finite_measure μ]
(h : submartingale f 𝒢 μ) (hτ : is_stopping_time 𝒢 τ) :
submartingale (stopped_process f τ) 𝒢 μ | begin
rw submartingale_iff_expected_stopped_value_mono,
{ intros σ π hσ hπ hσ_le_π hπ_bdd,
simp_rw stopped_value_stopped_process,
obtain ⟨n, hπ_le_n⟩ := hπ_bdd,
exact h.expected_stopped_value_mono (hσ.min hτ) (hπ.min hτ)
(λ ω, min_le_min (hσ_le_π ω) le_rfl) (λ ω, (min_le_left _ _).trans (hπ_le_n ω... | lemma | measure_theory.submartingale.stopped_process | probability.martingale | src/probability/martingale/optional_stopping.lean | [
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"le_rfl",
"min_le_min"
] | The stopped process of a submartingale with respect to a stopping time is a submartingale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_le_stopped_value_hitting [is_finite_measure μ]
(hsub : submartingale f 𝒢 μ) {ε : ℝ≥0} (n : ℕ) :
ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)} ≤
ennreal.of_real (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)},
stopped_value f (hitting f {y : ℝ |... | begin
have hn : set.Icc 0 n = {k | k ≤ n},
{ ext x, simp },
have : ∀ ω, ((ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)) →
(ε : ℝ) ≤ stopped_value f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω,
{ intros x hx,
simp_rw [le_sup'_iff, mem_range, nat.lt_succ_iff] at hx,
refine stopped_value_hitti... | lemma | measure_theory.smul_le_stopped_value_hitting | probability.martingale | src/probability/martingale/optional_stopping.lean | [
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ennreal.le_of_real_iff_to_real_le",
"ennreal.mul_ne_top",
"ennreal.of_real",
"ennreal.to_real_nonneg",
"ennreal.to_real_smul",
"exists_prop",
"finset.measurable_range_sup''",
"measurable.le",
"measurable_const",
"measurable_set_Ici",
"measurable_set_le",
"nat.lt_succ_iff",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maximal_ineq [is_finite_measure μ]
(hsub : submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0} (n : ℕ) :
ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)} ≤
ennreal.of_real (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)},
f n ω ∂μ) | begin
suffices : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)} +
ennreal.of_real (∫ ω in {ω | ((range (n + 1)).sup' nonempty_range_succ (λ k, f k ω)) < ε},
f n ω ∂μ) ≤ ennreal.of_real (μ[f n]),
{ have hadd : ennreal.of_real (∫ ω, f n ω ∂μ) =
ennreal.of_real (∫ ω in
... | lemma | measure_theory.maximal_ineq | probability.martingale | src/probability/martingale/optional_stopping.lean | [
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"and_imp",
"disjoint_iff_inf_le",
"ennreal.add_le_add_iff_right",
"ennreal.of_real",
"ennreal.of_real_add",
"ennreal.of_real_le_of_real",
"ennreal.of_real_ne_top",
"exists_prop",
"finset.measurable_range_sup''",
"forall_exists_index",
"ite_eq_right_iff",
"le_rfl",
"measurable.le",
"measura... | **Doob's maximal inequality**: Given a non-negative submartingale `f`, for all `ε : ℝ≥0`,
we have `ε • μ {ε ≤ f* n} ≤ ∫ ω in {ε ≤ f* n}, f n` where `f* n ω = max_{k ≤ n}, f k ω`.
In some literature, the Doob's maximal inequality refers to what we call Doob's Lp inequality
(which is a corollary of this lemma and will b... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_crossing_time_aux [preorder ι] [has_Inf ι] (a : ℝ) (f : ι → Ω → ℝ) (c N : ι) : Ω → ι | hitting f (set.Iic a) c N | def | measure_theory.lower_crossing_time_aux | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"has_Inf",
"set.Iic"
] | `lower_crossing_time_aux a f c N` is the first time `f` reached below `a` after time `c` before
time `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_crossing_time [preorder ι] [order_bot ι] [has_Inf ι]
(a b : ℝ) (f : ι → Ω → ℝ) (N : ι) : ℕ → Ω → ι | | 0 := ⊥
| (n + 1) := λ ω, hitting f (set.Ici b)
(lower_crossing_time_aux a f (upper_crossing_time n ω) N ω) N ω | def | measure_theory.upper_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"has_Inf",
"order_bot",
"set.Ici"
] | `upper_crossing_time a b f N n` is the first time before time `N`, `f` reaches
above `b` after `f` reached below `a` for the `n - 1`-th time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lower_crossing_time [preorder ι] [order_bot ι] [has_Inf ι]
(a b : ℝ) (f : ι → Ω → ℝ) (N : ι) (n : ℕ) : Ω → ι | λ ω, hitting f (set.Iic a) (upper_crossing_time a b f N n ω) N ω | def | measure_theory.lower_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"has_Inf",
"order_bot",
"set.Iic"
] | `lower_crossing_time a b f N n` is the first time before time `N`, `f` reaches
below `a` after `f` reached above `b` for the `n`-th time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upper_crossing_time_zero : upper_crossing_time a b f N 0 = ⊥ | rfl | lemma | measure_theory.upper_crossing_time_zero | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_zero : lower_crossing_time a b f N 0 = hitting f (set.Iic a) ⊥ N | rfl | lemma | measure_theory.lower_crossing_time_zero | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_succ :
upper_crossing_time a b f N (n + 1) ω =
hitting f (set.Ici b) (lower_crossing_time_aux a f (upper_crossing_time a b f N n ω) N ω) N ω | by rw upper_crossing_time | lemma | measure_theory.upper_crossing_time_succ | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_succ_eq (ω : Ω) :
upper_crossing_time a b f N (n + 1) ω =
hitting f (set.Ici b) (lower_crossing_time a b f N n ω) N ω | begin
simp only [upper_crossing_time_succ],
refl,
end | lemma | measure_theory.upper_crossing_time_succ_eq | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_le : upper_crossing_time a b f N n ω ≤ N | begin
cases n,
{ simp only [upper_crossing_time_zero, pi.bot_apply, bot_le] },
{ simp only [upper_crossing_time_succ, hitting_le] },
end | lemma | measure_theory.upper_crossing_time_le | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"bot_le",
"pi.bot_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_zero' : upper_crossing_time a b f ⊥ n ω = ⊥ | eq_bot_iff.2 upper_crossing_time_le | lemma | measure_theory.upper_crossing_time_zero' | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_le : lower_crossing_time a b f N n ω ≤ N | by simp only [lower_crossing_time, hitting_le ω] | lemma | measure_theory.lower_crossing_time_le | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_le_lower_crossing_time :
upper_crossing_time a b f N n ω ≤ lower_crossing_time a b f N n ω | by simp only [lower_crossing_time, le_hitting upper_crossing_time_le ω] | lemma | measure_theory.upper_crossing_time_le_lower_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_le_upper_crossing_time_succ :
lower_crossing_time a b f N n ω ≤ upper_crossing_time a b f N (n + 1) ω | begin
rw upper_crossing_time_succ,
exact le_hitting lower_crossing_time_le ω,
end | lemma | measure_theory.lower_crossing_time_le_upper_crossing_time_succ | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_mono (hnm : n ≤ m) :
lower_crossing_time a b f N n ω ≤ lower_crossing_time a b f N m ω | begin
suffices : monotone (λ n, lower_crossing_time a b f N n ω),
{ exact this hnm },
exact monotone_nat_of_le_succ
(λ n, le_trans lower_crossing_time_le_upper_crossing_time_succ
upper_crossing_time_le_lower_crossing_time)
end | lemma | measure_theory.lower_crossing_time_mono | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"monotone",
"monotone_nat_of_le_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_mono (hnm : n ≤ m) :
upper_crossing_time a b f N n ω ≤ upper_crossing_time a b f N m ω | begin
suffices : monotone (λ n, upper_crossing_time a b f N n ω),
{ exact this hnm },
exact monotone_nat_of_le_succ
(λ n, le_trans upper_crossing_time_le_lower_crossing_time
lower_crossing_time_le_upper_crossing_time_succ),
end | lemma | measure_theory.upper_crossing_time_mono | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"monotone",
"monotone_nat_of_le_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_lower_crossing_time (h : lower_crossing_time a b f N n ω ≠ N) :
stopped_value f (lower_crossing_time a b f N n) ω ≤ a | begin
obtain ⟨j, hj₁, hj₂⟩ :=
(hitting_le_iff_of_lt _ (lt_of_le_of_ne lower_crossing_time_le h)).1 le_rfl,
exact stopped_value_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lower_crossing_time_le⟩, hj₂⟩,
end | lemma | measure_theory.stopped_value_lower_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_upper_crossing_time (h : upper_crossing_time a b f N (n + 1) ω ≠ N) :
b ≤ stopped_value f (upper_crossing_time a b f N (n + 1)) ω | begin
obtain ⟨j, hj₁, hj₂⟩ :=
(hitting_le_iff_of_lt _ (lt_of_le_of_ne upper_crossing_time_le h)).1 le_rfl,
exact stopped_value_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩,
end | lemma | measure_theory.stopped_value_upper_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_lt_lower_crossing_time
(hab : a < b) (hn : lower_crossing_time a b f N (n + 1) ω ≠ N) :
upper_crossing_time a b f N (n + 1) ω < lower_crossing_time a b f N (n + 1) ω | begin
refine lt_of_le_of_ne upper_crossing_time_le_lower_crossing_time
(λ h, not_le.2 hab $ le_trans _ (stopped_value_lower_crossing_time hn)),
simp only [stopped_value],
rw ← h,
exact stopped_value_upper_crossing_time (h.symm ▸ hn),
end | lemma | measure_theory.upper_crossing_time_lt_lower_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_lt_upper_crossing_time
(hab : a < b) (hn : upper_crossing_time a b f N (n + 1) ω ≠ N) :
lower_crossing_time a b f N n ω < upper_crossing_time a b f N (n + 1) ω | begin
refine lt_of_le_of_ne lower_crossing_time_le_upper_crossing_time_succ
(λ h, not_le.2 hab $ le_trans (stopped_value_upper_crossing_time hn) _),
simp only [stopped_value],
rw ← h,
exact stopped_value_lower_crossing_time (h.symm ▸ hn),
end | lemma | measure_theory.lower_crossing_time_lt_upper_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_lt_succ (hab : a < b) (hn : upper_crossing_time a b f N (n + 1) ω ≠ N) :
upper_crossing_time a b f N n ω < upper_crossing_time a b f N (n + 1) ω | lt_of_le_of_lt upper_crossing_time_le_lower_crossing_time
(lower_crossing_time_lt_upper_crossing_time hab hn) | lemma | measure_theory.upper_crossing_time_lt_succ | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_stabilize (hnm : n ≤ m) (hn : lower_crossing_time a b f N n ω = N) :
lower_crossing_time a b f N m ω = N | le_antisymm lower_crossing_time_le (le_trans (le_of_eq hn.symm) (lower_crossing_time_mono hnm)) | lemma | measure_theory.lower_crossing_time_stabilize | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_stabilize (hnm : n ≤ m) (hn : upper_crossing_time a b f N n ω = N) :
upper_crossing_time a b f N m ω = N | le_antisymm upper_crossing_time_le (le_trans (le_of_eq hn.symm) (upper_crossing_time_mono hnm)) | lemma | measure_theory.upper_crossing_time_stabilize | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_crossing_time_stabilize' (hnm : n ≤ m) (hn : N ≤ lower_crossing_time a b f N n ω) :
lower_crossing_time a b f N m ω = N | lower_crossing_time_stabilize hnm (le_antisymm lower_crossing_time_le hn) | lemma | measure_theory.lower_crossing_time_stabilize' | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_stabilize' (hnm : n ≤ m) (hn : N ≤ upper_crossing_time a b f N n ω) :
upper_crossing_time a b f N m ω = N | upper_crossing_time_stabilize hnm (le_antisymm upper_crossing_time_le hn) | lemma | measure_theory.upper_crossing_time_stabilize' | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_upper_crossing_time_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
∃ n, upper_crossing_time a b f N n ω = N | begin
by_contra h, push_neg at h,
have : strict_mono (λ n, upper_crossing_time a b f N n ω) :=
strict_mono_nat_of_lt_succ (λ n, upper_crossing_time_lt_succ hab (h _)),
obtain ⟨_, ⟨k, rfl⟩, hk⟩ :
∃ m (hm : m ∈ set.range (λ n, upper_crossing_time a b f N n ω)), N < m :=
⟨upper_crossing_time a b f N (N +... | lemma | measure_theory.exists_upper_crossing_time_eq | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"by_contra",
"set.range",
"strict_mono",
"strict_mono.id_le",
"strict_mono_nat_of_lt_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_lt_bdd_above (hab : a < b) :
bdd_above {n | upper_crossing_time a b f N n ω < N} | begin
obtain ⟨k, hk⟩ := exists_upper_crossing_time_eq f N ω hab,
refine ⟨k, λ n (hn : upper_crossing_time a b f N n ω < N), _⟩,
by_contra hn',
exact hn.ne (upper_crossing_time_stabilize (not_le.1 hn').le hk)
end | lemma | measure_theory.upper_crossing_time_lt_bdd_above | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"bdd_above",
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_lt_nonempty (hN : 0 < N) :
{n | upper_crossing_time a b f N n ω < N}.nonempty | ⟨0, hN⟩ | lemma | measure_theory.upper_crossing_time_lt_nonempty | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_bound_eq (f : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) (hab : a < b) :
upper_crossing_time a b f N N ω = N | begin
by_cases hN' : N < nat.find (exists_upper_crossing_time_eq f N ω hab),
{ refine le_antisymm upper_crossing_time_le _,
have hmono : strict_mono_on (λ n, upper_crossing_time a b f N n ω)
(set.Iic (nat.find (exists_upper_crossing_time_eq f N ω hab)).pred),
{ refine strict_mono_on_Iic_of_lt_succ (λ ... | lemma | measure_theory.upper_crossing_time_bound_eq | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"nat.le_pred_of_lt",
"nat.lt_pred_iff",
"set.Iic",
"strict_mono_on",
"strict_mono_on.Iic_id_le",
"strict_mono_on_Iic_of_lt_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_eq_of_bound_le (hab : a < b) (hn : N ≤ n) :
upper_crossing_time a b f N n ω = N | le_antisymm upper_crossing_time_le
((le_trans (upper_crossing_time_bound_eq f N ω hab).symm.le (upper_crossing_time_mono hn))) | lemma | measure_theory.upper_crossing_time_eq_of_bound_le | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.is_stopping_time_crossing (hf : adapted ℱ f) :
is_stopping_time ℱ (upper_crossing_time a b f N n) ∧
is_stopping_time ℱ (lower_crossing_time a b f N n) | begin
induction n with k ih,
{ refine ⟨is_stopping_time_const _ 0, _⟩,
simp [hitting_is_stopping_time hf measurable_set_Iic] },
{ obtain ⟨ih₁, ih₂⟩ := ih,
have : is_stopping_time ℱ (upper_crossing_time a b f N (k + 1)),
{ intro n,
simp_rw upper_crossing_time_succ_eq,
exact is_stopping_time... | lemma | measure_theory.adapted.is_stopping_time_crossing | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ih",
"measurable_set_Ici",
"measurable_set_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.is_stopping_time_upper_crossing_time (hf : adapted ℱ f) :
is_stopping_time ℱ (upper_crossing_time a b f N n) | hf.is_stopping_time_crossing.1 | lemma | measure_theory.adapted.is_stopping_time_upper_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.is_stopping_time_lower_crossing_time (hf : adapted ℱ f) :
is_stopping_time ℱ (lower_crossing_time a b f N n) | hf.is_stopping_time_crossing.2 | lemma | measure_theory.adapted.is_stopping_time_lower_crossing_time | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossing_strat (a b : ℝ) (f : ℕ → Ω → ℝ) (N n : ℕ) (ω : Ω) : ℝ | ∑ k in finset.range N,
(set.Ico (lower_crossing_time a b f N k ω) (upper_crossing_time a b f N (k + 1) ω)).indicator 1 n | def | measure_theory.upcrossing_strat | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.range",
"set.Ico"
] | `upcrossing_strat a b f N n` is 1 if `n` is between a consecutive pair of lower and upper
crossings and is 0 otherwise. `upcrossing_strat` is shifted by one index so that it is adapted
rather than predictable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upcrossing_strat_nonneg : 0 ≤ upcrossing_strat a b f N n ω | finset.sum_nonneg (λ i hi, set.indicator_nonneg (λ ω hω, zero_le_one) _) | lemma | measure_theory.upcrossing_strat_nonneg | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossing_strat_le_one : upcrossing_strat a b f N n ω ≤ 1 | begin
rw [upcrossing_strat, ← set.indicator_finset_bUnion_apply],
{ exact set.indicator_le_self' (λ _ _, zero_le_one) _ },
{ intros i hi j hj hij,
rw set.Ico_disjoint_Ico,
obtain (hij' | hij') := lt_or_gt_of_ne hij,
{ rw [min_eq_left ((upper_crossing_time_mono (nat.succ_le_succ hij'.le)) :
u... | lemma | measure_theory.upcrossing_strat_le_one | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"gt_iff_lt",
"set.Ico_disjoint_Ico",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.upcrossing_strat_adapted (hf : adapted ℱ f) :
adapted ℱ (upcrossing_strat a b f N) | begin
intro n,
change strongly_measurable[ℱ n] (λ ω, ∑ k in finset.range N,
({n | lower_crossing_time a b f N k ω ≤ n} ∩
{n | n < upper_crossing_time a b f N (k + 1) ω}).indicator 1 n),
refine finset.strongly_measurable_sum _ (λ i hi,
strongly_measurable_const.indicator ((hf.is_stopping_time_lower_cr... | lemma | measure_theory.adapted.upcrossing_strat_adapted | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.sum_upcrossing_strat_mul [is_finite_measure μ] (hf : submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) :
submartingale
(λ n : ℕ, ∑ k in finset.range n, upcrossing_strat a b f N k * (f (k + 1) - f k)) ℱ μ | hf.sum_mul_sub hf.adapted.upcrossing_strat_adapted
(λ _ _, upcrossing_strat_le_one) (λ _ _, upcrossing_strat_nonneg) | lemma | measure_theory.submartingale.sum_upcrossing_strat_mul | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.sum_sub_upcrossing_strat_mul [is_finite_measure μ] (hf : submartingale f ℱ μ)
(a b : ℝ) (N : ℕ) :
submartingale
(λ n : ℕ, ∑ k in finset.range n, (1 - upcrossing_strat a b f N k) * (f (k + 1) - f k)) ℱ μ | begin
refine hf.sum_mul_sub (λ n, (adapted_const ℱ 1 n).sub (hf.adapted.upcrossing_strat_adapted n))
(_ : ∀ n ω, (1 - upcrossing_strat a b f N n) ω ≤ 1) _,
{ exact λ n ω, sub_le_self _ upcrossing_strat_nonneg },
{ intros n ω,
simp [upcrossing_strat_le_one] }
end | lemma | measure_theory.submartingale.sum_sub_upcrossing_strat_mul | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.sum_mul_upcrossing_strat_le [is_finite_measure μ] (hf : submartingale f ℱ μ) :
μ[∑ k in finset.range n, upcrossing_strat a b f N k * (f (k + 1) - f k)] ≤
μ[f n] - μ[f 0] | begin
have h₁ : (0 : ℝ) ≤
μ[∑ k in finset.range n, (1 - upcrossing_strat a b f N k) * (f (k + 1) - f k)],
{ have := (hf.sum_sub_upcrossing_strat_mul a b N).set_integral_le (zero_le n) measurable_set.univ,
rw [integral_univ, integral_univ] at this,
refine le_trans _ this,
simp only [finset.range_zero... | lemma | measure_theory.submartingale.sum_mul_upcrossing_strat_le | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.range",
"finset.range_zero",
"measurable_set.univ",
"one_mul",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before [preorder ι] [order_bot ι] [has_Inf ι]
(a b : ℝ) (f : ι → Ω → ℝ) (N : ι) (ω : Ω) : ℕ | Sup {n | upper_crossing_time a b f N n ω < N} | def | measure_theory.upcrossings_before | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"has_Inf",
"order_bot"
] | The number of upcrossings (strictly) before time `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upcrossings_before_bot [preorder ι] [order_bot ι] [has_Inf ι]
{a b : ℝ} {f : ι → Ω → ℝ} {ω : Ω} :
upcrossings_before a b f ⊥ ω = ⊥ | by simp [upcrossings_before] | lemma | measure_theory.upcrossings_before_bot | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"has_Inf",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before_zero :
upcrossings_before a b f 0 ω = 0 | by simp [upcrossings_before] | lemma | measure_theory.upcrossings_before_zero | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before_zero' :
upcrossings_before a b f 0 = 0 | by { ext ω, exact upcrossings_before_zero } | lemma | measure_theory.upcrossings_before_zero' | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_lt_of_le_upcrossings_before
(hN : 0 < N) (hab : a < b) (hn : n ≤ upcrossings_before a b f N ω) :
upper_crossing_time a b f N n ω < N | begin
have : upper_crossing_time a b f N (upcrossings_before a b f N ω) ω < N :=
(upper_crossing_time_lt_nonempty hN).cSup_mem
((order_bot.bdd_below _).finite_of_bdd_above (upper_crossing_time_lt_bdd_above hab)),
exact lt_of_le_of_lt (upper_crossing_time_mono hn) this,
end | lemma | measure_theory.upper_crossing_time_lt_of_le_upcrossings_before | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"order_bot.bdd_below"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_eq_of_upcrossings_before_lt
(hab : a < b) (hn : upcrossings_before a b f N ω < n) :
upper_crossing_time a b f N n ω = N | begin
refine le_antisymm upper_crossing_time_le (not_lt.1 _),
convert not_mem_of_cSup_lt hn (upper_crossing_time_lt_bdd_above hab),
end | lemma | measure_theory.upper_crossing_time_eq_of_upcrossings_before_lt | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"not_mem_of_cSup_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
upcrossings_before a b f N ω ≤ N | begin
by_cases hN : N = 0,
{ subst hN,
rw upcrossings_before_zero },
{ refine cSup_le ⟨0, zero_lt_iff.2 hN⟩ (λ n (hn : _ < _), _),
by_contra hnN,
exact hn.ne (upper_crossing_time_eq_of_bound_le hab (not_le.1 hnN).le) },
end | lemma | measure_theory.upcrossings_before_le | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"by_contra",
"cSup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
crossing_eq_crossing_of_lower_crossing_time_lt {M : ℕ} (hNM : N ≤ M)
(h : lower_crossing_time a b f N n ω < N) :
upper_crossing_time a b f M n ω = upper_crossing_time a b f N n ω ∧
lower_crossing_time a b f M n ω = lower_crossing_time a b f N n ω | begin
have h' : upper_crossing_time a b f N n ω < N :=
lt_of_le_of_lt upper_crossing_time_le_lower_crossing_time h,
induction n with k ih,
{ simp only [nat.nat_zero_eq_zero, upper_crossing_time_zero, bot_eq_zero', eq_self_iff_true,
lower_crossing_time_zero, true_and, eq_comm],
refine hitting_eq_hitt... | lemma | measure_theory.crossing_eq_crossing_of_lower_crossing_time_lt | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ih",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
crossing_eq_crossing_of_upper_crossing_time_lt {M : ℕ} (hNM : N ≤ M)
(h : upper_crossing_time a b f N (n + 1) ω < N) :
upper_crossing_time a b f M (n + 1) ω = upper_crossing_time a b f N (n + 1) ω ∧
lower_crossing_time a b f M n ω = lower_crossing_time a b f N n ω | begin
have := (crossing_eq_crossing_of_lower_crossing_time_lt hNM
(lt_of_le_of_lt lower_crossing_time_le_upper_crossing_time_succ h)).2,
refine ⟨_, this⟩,
rw [upper_crossing_time_succ_eq, upper_crossing_time_succ_eq, eq_comm, this],
refine hitting_eq_hitting_of_exists hNM _,
simp only [upper_crossing_time... | lemma | measure_theory.crossing_eq_crossing_of_upper_crossing_time_lt | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_crossing_time_eq_upper_crossing_time_of_lt {M : ℕ} (hNM : N ≤ M)
(h : upper_crossing_time a b f N n ω < N) :
upper_crossing_time a b f M n ω = upper_crossing_time a b f N n ω | begin
cases n,
{ simp },
{ exact (crossing_eq_crossing_of_upper_crossing_time_lt hNM h).1 }
end | lemma | measure_theory.upper_crossing_time_eq_upper_crossing_time_of_lt | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before_mono (hab : a < b) :
monotone (λ N ω, upcrossings_before a b f N ω) | begin
intros N M hNM ω,
simp only [upcrossings_before],
by_cases hemp : {n : ℕ | upper_crossing_time a b f N n ω < N}.nonempty,
{ refine cSup_le_cSup (upper_crossing_time_lt_bdd_above hab) hemp (λ n hn, _),
rw [set.mem_set_of_eq, upper_crossing_time_eq_upper_crossing_time_of_lt hNM hn],
exact lt_of_lt_o... | lemma | measure_theory.upcrossings_before_mono | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"cSup_empty",
"cSup_le_cSup",
"monotone",
"set.not_nonempty_iff_eq_empty",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before_lt_of_exists_upcrossing (hab : a < b) {N₁ N₂ : ℕ}
(hN₁: N ≤ N₁) (hN₁': f N₁ ω < a) (hN₂: N₁ ≤ N₂) (hN₂': b < f N₂ ω) :
upcrossings_before a b f N ω < upcrossings_before a b f (N₂ + 1) ω | begin
refine lt_of_lt_of_le (nat.lt_succ_self _) (le_cSup (upper_crossing_time_lt_bdd_above hab) _),
rw [set.mem_set_of_eq, upper_crossing_time_succ_eq, hitting_lt_iff _ le_rfl],
swap,
{ apply_instance },
{ refine ⟨N₂, ⟨_, nat.lt_succ_self _⟩, hN₂'.le⟩,
rw [lower_crossing_time, hitting_le_iff_of_lt _ (nat... | lemma | measure_theory.upcrossings_before_lt_of_exists_upcrossing | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"le_cSup",
"le_rfl",
"le_zero_iff",
"nat.Sup_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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