statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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strongly_measurable.integral_kernel_prod_left''
{f : γ × β → E} (hf : strongly_measurable f) :
strongly_measurable (λ y, ∫ x, f (x, y) ∂(η (a, y))) | begin
change strongly_measurable ((λ y, ∫ x, (λ u : γ × (α × β), f (u.1, u.2.2)) (x, y) ∂(η y))
∘ (λ x, (a, x))),
refine strongly_measurable.comp_measurable _ measurable_prod_mk_left,
refine measure_theory.strongly_measurable.integral_kernel_prod_left' _,
exact hf.comp_measurable (measurable_fst.prod_mk mea... | lemma | measure_theory.strongly_measurable.integral_kernel_prod_left'' | probability.kernel | src/probability/kernel/measurable_integral.lean | [
"probability.kernel.basic"
] | [
"measurable_prod_mk_left",
"measure_theory.strongly_measurable.integral_kernel_prod_left'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density (κ : kernel α β) [is_s_finite_kernel κ] (f : α → β → ℝ≥0∞) :
kernel α β | @dite _ (measurable (function.uncurry f)) (classical.dec _)
(λ hf, ({ val := λ a, (κ a).with_density (f a),
property :=
begin
refine measure.measurable_of_measurable_coe _ (λ s hs, _),
simp_rw with_density_apply _ hs,
exact hf.set_lintegral_kernel_prod_right hs,
end, } : kernel α β))
(... | def | probability_theory.kernel.with_density | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"classical.dec",
"measurable"
] | Kernel with image `(κ a).with_density (f a)` if `function.uncurry f` is measurable, and
with image 0 otherwise. If `function.uncurry f` is measurable, it satisfies
`∫⁻ b, g b ∂(with_density κ f hf a) = ∫⁻ b, f a b * g b ∂(κ a)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_density_of_not_measurable (κ : kernel α β) [is_s_finite_kernel κ]
(hf : ¬ measurable (function.uncurry f)) :
with_density κ f = 0 | by { classical, exact dif_neg hf, } | lemma | probability_theory.kernel.with_density_of_not_measurable | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density_apply (κ : kernel α β) [is_s_finite_kernel κ]
(hf : measurable (function.uncurry f)) (a : α) :
with_density κ f a = (κ a).with_density (f a) | by { classical, rw [with_density, dif_pos hf], refl, } | lemma | probability_theory.kernel.with_density_apply | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density_apply' (κ : kernel α β) [is_s_finite_kernel κ]
(hf : measurable (function.uncurry f)) (a : α) {s : set β} (hs : measurable_set s) :
with_density κ f a s = ∫⁻ b in s, f a b ∂(κ a) | by rw [kernel.with_density_apply κ hf, with_density_apply _ hs] | lemma | probability_theory.kernel.with_density_apply' | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"measurable",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lintegral_with_density (κ : kernel α β) [is_s_finite_kernel κ]
(hf : measurable (function.uncurry f)) (a : α) {g : β → ℝ≥0∞} (hg : measurable g) :
∫⁻ b, g b ∂(with_density κ f a) = ∫⁻ b, f a b * g b ∂(κ a) | begin
rw [kernel.with_density_apply _ hf,
lintegral_with_density_eq_lintegral_mul _ (measurable.of_uncurry_left hf) hg],
simp_rw pi.mul_apply,
end | lemma | probability_theory.kernel.lintegral_with_density | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"measurable",
"measurable.of_uncurry_left",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_with_density {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
[complete_space E] {f : β → E} [is_s_finite_kernel κ] {a : α}
{g : α → β → ℝ≥0} (hg : measurable (function.uncurry g)) :
∫ b, f b ∂(with_density κ (λ a b, g a b) a) = ∫ b, (g a b) • f b ∂(κ a) | begin
rw [kernel.with_density_apply, integral_with_density_eq_integral_smul],
{ exact measurable.of_uncurry_left hg, },
{ exact measurable_coe_nnreal_ennreal.comp hg, },
end | lemma | probability_theory.kernel.integral_with_density | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"complete_space",
"integral_with_density_eq_integral_smul",
"measurable",
"measurable.of_uncurry_left",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density_add_left (κ η : kernel α β) [is_s_finite_kernel κ] [is_s_finite_kernel η]
(f : α → β → ℝ≥0∞) :
with_density (κ + η) f = with_density κ f + with_density η f | begin
by_cases hf : measurable (function.uncurry f),
{ ext a s hs : 2,
simp only [kernel.with_density_apply _ hf, coe_fn_add, pi.add_apply, with_density_add_measure,
measure.add_apply], },
{ simp_rw [with_density_of_not_measurable _ hf],
rw zero_add, },
end | lemma | probability_theory.kernel.with_density_add_left | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density_kernel_sum [countable ι] (κ : ι → kernel α β)
(hκ : ∀ i, is_s_finite_kernel (κ i)) (f : α → β → ℝ≥0∞) :
@with_density _ _ _ _ (kernel.sum κ) (is_s_finite_kernel_sum hκ) f
= kernel.sum (λ i, with_density (κ i) f) | begin
by_cases hf : measurable (function.uncurry f),
{ ext1 a,
simp_rw [sum_apply, kernel.with_density_apply _ hf, sum_apply,
with_density_sum (λ n, κ n a) (f a)], },
{ simp_rw [with_density_of_not_measurable _ hf],
exact sum_zero.symm, },
end | lemma | probability_theory.kernel.with_density_kernel_sum | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"countable",
"measurable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_density_tsum [countable ι] (κ : kernel α β) [is_s_finite_kernel κ]
{f : ι → α → β → ℝ≥0∞} (hf : ∀ i, measurable (function.uncurry (f i))) :
with_density κ (∑' n, f n) = kernel.sum (λ n, with_density κ (f n)) | begin
have h_sum_a : ∀ a, summable (λ n, f n a) := λ a, pi.summable.mpr (λ b, ennreal.summable),
have h_sum : summable (λ n, f n) := pi.summable.mpr h_sum_a,
ext a s hs : 2,
rw [sum_apply' _ a hs, with_density_apply' κ _ a hs],
swap,
{ have : function.uncurry (∑' n, f n) = ∑' n, function.uncurry (f n),
... | lemma | probability_theory.kernel.with_density_tsum | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"ae_measurable",
"countable",
"ennreal.summable",
"measurable",
"measurable.ennreal_tsum'",
"measurable.of_uncurry_left",
"summable",
"tsum_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_kernel_with_density_of_bounded (κ : kernel α β) [is_finite_kernel κ]
{B : ℝ≥0∞} (hB_top : B ≠ ∞) (hf_B : ∀ a b, f a b ≤ B) :
is_finite_kernel (with_density κ f) | begin
by_cases hf : measurable (function.uncurry f),
{ exact
⟨⟨B * is_finite_kernel.bound κ, ennreal.mul_lt_top hB_top (is_finite_kernel.bound_ne_top κ),
λ a,
begin
rw with_density_apply' κ hf a measurable_set.univ,
calc ∫⁻ b in set.univ, f a b ∂(κ a)
≤ ∫⁻ b i... | lemma | probability_theory.kernel.is_finite_kernel_with_density_of_bounded | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"ennreal.mul_lt_top",
"measurable",
"measurable_set.univ",
"measure_theory.lintegral_const",
"mul_le_mul_left'"
] | If a kernel `κ` is finite and a function `f : α → β → ℝ≥0∞` is bounded, then `with_density κ f`
is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_s_finite_kernel_with_density_of_is_finite_kernel (κ : kernel α β) [is_finite_kernel κ]
(hf_ne_top : ∀ a b, f a b ≠ ∞) :
is_s_finite_kernel (with_density κ f) | begin
-- We already have that for `f` bounded from above and a `κ` a finite kernel,
-- `with_density κ f` is finite. We write any function as a countable sum of bounded
-- functions, and decompose an s-finite kernel as a sum of finite kernels. We then use that
-- `with_density` commutes with sums for both argum... | lemma | probability_theory.kernel.is_s_finite_kernel_with_density_of_is_finite_kernel | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [
"add_tsub_cancel_iff_le",
"algebra_map.coe_zero",
"ennreal.coe_ne_top",
"ennreal.le_of_real_iff_to_real_le",
"ennreal.of_real_coe_nat",
"ennreal.tsum_eq_liminf_sum_nat",
"filter.eventually_at_top",
"filter.eventually_eq",
"filter.tendsto.congr'",
"filter.tendsto.liminf_eq",
"finset.mem_range",
... | Auxiliary lemma for `is_s_finite_kernel_with_density`.
If a kernel `κ` is finite, then `with_density κ f` is s-finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_s_finite_kernel.with_density (κ : kernel α β) [is_s_finite_kernel κ]
(hf_ne_top : ∀ a b, f a b ≠ ∞) :
is_s_finite_kernel (with_density κ f) | begin
have h_eq_sum : with_density κ f = kernel.sum (λ i, with_density (seq κ i) f),
{ rw ← with_density_kernel_sum _ _,
congr,
exact (kernel_sum_seq κ).symm, },
rw h_eq_sum,
exact is_s_finite_kernel_sum
(λ n, is_s_finite_kernel_with_density_of_is_finite_kernel (seq κ n) hf_ne_top),
end | theorem | probability_theory.kernel.is_s_finite_kernel.with_density | probability.kernel | src/probability/kernel/with_density.lean | [
"probability.kernel.measurable_integral",
"measure_theory.integral.set_integral"
] | [] | For a s-finite kernel `κ` and a function `f : α → β → ℝ≥0∞` which is everywhere finite,
`with_density κ f` is s-finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
martingale (f : ι → Ω → E) (ℱ : filtration ι m0) (μ : measure Ω . volume_tac) : Prop | adapted ℱ f ∧ ∀ i j, i ≤ j → μ[f j | ℱ i] =ᵐ[μ] f i | def | measure_theory.martingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | A family of functions `f : ι → Ω → E` is a martingale with respect to a filtration `ℱ` if `f`
is adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ i] =ᵐ[μ] f i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supermartingale [has_le E] (f : ι → Ω → E) (ℱ : filtration ι m0) (μ : measure Ω . volume_tac) :
Prop | adapted ℱ f ∧ (∀ i j, i ≤ j → μ[f j | ℱ i] ≤ᵐ[μ] f i) ∧ ∀ i, integrable (f i) μ | def | measure_theory.supermartingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | A family of integrable functions `f : ι → Ω → E` is a supermartingale with respect to a
filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`μ[f j | ℱ.le i] ≤ᵐ[μ] f i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale [has_le E] (f : ι → Ω → E) (ℱ : filtration ι m0) (μ : measure Ω . volume_tac) :
Prop | adapted ℱ f ∧ (∀ i j, i ≤ j → f i ≤ᵐ[μ] μ[f j | ℱ i]) ∧ ∀ i, integrable (f i) μ | def | measure_theory.submartingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | A family of integrable functions `f : ι → Ω → E` is a submartingale with respect to a
filtration `ℱ` if `f` is adapted with respect to `ℱ` and for all `i ≤ j`,
`f i ≤ᵐ[μ] μ[f j | ℱ.le i]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
martingale_const (ℱ : filtration ι m0) (μ : measure Ω) [is_finite_measure μ] (x : E) :
martingale (λ _ _, x) ℱ μ | ⟨adapted_const ℱ _, λ i j hij, by rw condexp_const (ℱ.le _)⟩ | lemma | measure_theory.martingale_const | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_const_fun [order_bot ι]
(ℱ : filtration ι m0) (μ : measure Ω) [is_finite_measure μ]
{f : Ω → E} (hf : strongly_measurable[ℱ ⊥] f) (hfint : integrable f μ) :
martingale (λ _, f) ℱ μ | begin
refine ⟨λ i, hf.mono $ ℱ.mono bot_le, λ i j hij, _⟩,
rw condexp_of_strongly_measurable (ℱ.le _) (hf.mono $ ℱ.mono bot_le) hfint,
apply_instance,
end | lemma | measure_theory.martingale_const_fun | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"bot_le",
"order_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_zero (ℱ : filtration ι m0) (μ : measure Ω) :
martingale (0 : ι → Ω → E) ℱ μ | ⟨adapted_zero E ℱ, λ i j hij, by { rw [pi.zero_apply, condexp_zero], simp, }⟩ | lemma | measure_theory.martingale_zero | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted (hf : martingale f ℱ μ) : adapted ℱ f | hf.1 | lemma | measure_theory.martingale.adapted | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable (hf : martingale f ℱ μ) (i : ι) : strongly_measurable[ℱ i] (f i) | hf.adapted i | lemma | measure_theory.martingale.strongly_measurable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_ae_eq (hf : martingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
μ[f j | ℱ i] =ᵐ[μ] f i | hf.2 i j hij | lemma | measure_theory.martingale.condexp_ae_eq | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable (hf : martingale f ℱ μ) (i : ι) : integrable (f i) μ | integrable_condexp.congr (hf.condexp_ae_eq (le_refl i)) | lemma | measure_theory.martingale.integrable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_eq [sigma_finite_filtration μ ℱ] (hf : martingale f ℱ μ) {i j : ι} (hij : i ≤ j)
{s : set Ω} (hs : measurable_set[ℱ i] s) :
∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ | begin
rw ← @set_integral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs,
refine set_integral_congr_ae (ℱ.le i s hs) _,
filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm,
end | lemma | measure_theory.martingale.set_integral_eq | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add (hf : martingale f ℱ μ) (hg : martingale g ℱ μ) : martingale (f + g) ℱ μ | begin
refine ⟨hf.adapted.add hg.adapted, λ i j hij, _⟩,
exact (condexp_add (hf.integrable j) (hg.integrable j)).trans
((hf.2 i j hij).add (hg.2 i j hij)),
end | lemma | measure_theory.martingale.add | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (hf : martingale f ℱ μ) : martingale (-f) ℱ μ | ⟨hf.adapted.neg, λ i j hij, (condexp_neg (f j)).trans ((hf.2 i j hij).neg)⟩ | lemma | measure_theory.martingale.neg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub (hf : martingale f ℱ μ) (hg : martingale g ℱ μ) : martingale (f - g) ℱ μ | by { rw sub_eq_add_neg, exact hf.add hg.neg, } | lemma | measure_theory.martingale.sub | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul (c : ℝ) (hf : martingale f ℱ μ) : martingale (c • f) ℱ μ | begin
refine ⟨hf.adapted.smul c, λ i j hij, _⟩,
refine (condexp_smul c (f j)).trans ((hf.2 i j hij).mono (λ x hx, _)),
rw [pi.smul_apply, hx, pi.smul_apply, pi.smul_apply],
end | lemma | measure_theory.martingale.smul | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supermartingale [preorder E] (hf : martingale f ℱ μ) : supermartingale f ℱ μ | ⟨hf.1, λ i j hij, (hf.2 i j hij).le, λ i, hf.integrable i⟩ | lemma | measure_theory.martingale.supermartingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale [preorder E] (hf : martingale f ℱ μ) : submartingale f ℱ μ | ⟨hf.1, λ i j hij, (hf.2 i j hij).symm.le, λ i, hf.integrable i⟩ | lemma | measure_theory.martingale.submartingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_iff [partial_order E] : martingale f ℱ μ ↔
supermartingale f ℱ μ ∧ submartingale f ℱ μ | ⟨λ hf, ⟨hf.supermartingale, hf.submartingale⟩,
λ ⟨hf₁, hf₂⟩, ⟨hf₁.1, λ i j hij, (hf₁.2.1 i j hij).antisymm (hf₂.2.1 i j hij)⟩⟩ | lemma | measure_theory.martingale_iff | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_condexp (f : Ω → E) (ℱ : filtration ι m0) (μ : measure Ω)
[sigma_finite_filtration μ ℱ] :
martingale (λ i, μ[f | ℱ i]) ℱ μ | ⟨λ i, strongly_measurable_condexp, λ i j hij, condexp_condexp_of_le (ℱ.mono hij) (ℱ.le j)⟩ | lemma | measure_theory.martingale_condexp | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted [has_le E] (hf : supermartingale f ℱ μ) : adapted ℱ f | hf.1 | lemma | measure_theory.supermartingale.adapted | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable [has_le E] (hf : supermartingale f ℱ μ) (i : ι) :
strongly_measurable[ℱ i] (f i) | hf.adapted i | lemma | measure_theory.supermartingale.strongly_measurable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable [has_le E] (hf : supermartingale f ℱ μ) (i : ι) : integrable (f i) μ | hf.2.2 i | lemma | measure_theory.supermartingale.integrable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condexp_ae_le [has_le E] (hf : supermartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
μ[f j | ℱ i] ≤ᵐ[μ] f i | hf.2.1 i j hij | lemma | measure_theory.supermartingale.condexp_ae_le | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → Ω → ℝ} (hf : supermartingale f ℱ μ)
{i j : ι} (hij : i ≤ j) {s : set Ω} (hs : measurable_set[ℱ i] s) :
∫ ω in s, f j ω ∂μ ≤ ∫ ω in s, f i ω ∂μ | begin
rw ← set_integral_condexp (ℱ.le i) (hf.integrable j) hs,
refine set_integral_mono_ae integrable_condexp.integrable_on (hf.integrable i).integrable_on _,
filter_upwards [hf.2.1 i j hij] with _ heq using heq,
end | lemma | measure_theory.supermartingale.set_integral_le | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add [preorder E] [covariant_class E E (+) (≤)]
(hf : supermartingale f ℱ μ) (hg : supermartingale g ℱ μ) : supermartingale (f + g) ℱ μ | begin
refine ⟨hf.1.add hg.1, λ i j hij, _, λ i, (hf.2.2 i).add (hg.2.2 i)⟩,
refine (condexp_add (hf.integrable j) (hg.integrable j)).le.trans _,
filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij],
intros,
refine add_le_add _ _; assumption,
end | lemma | measure_theory.supermartingale.add | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_martingale [preorder E] [covariant_class E E (+) (≤)]
(hf : supermartingale f ℱ μ) (hg : martingale g ℱ μ) : supermartingale (f + g) ℱ μ | hf.add hg.supermartingale | lemma | measure_theory.supermartingale.add_martingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg [preorder E] [covariant_class E E (+) (≤)]
(hf : supermartingale f ℱ μ) : submartingale (-f) ℱ μ | begin
refine ⟨hf.1.neg, λ i j hij, _, λ i, (hf.2.2 i).neg⟩,
refine eventually_le.trans _ (condexp_neg (f j)).symm.le,
filter_upwards [hf.2.1 i j hij] with _ _,
simpa,
end | lemma | measure_theory.supermartingale.neg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted [has_le E] (hf : submartingale f ℱ μ) : adapted ℱ f | hf.1 | lemma | measure_theory.submartingale.adapted | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable [has_le E] (hf : submartingale f ℱ μ) (i : ι) :
strongly_measurable[ℱ i] (f i) | hf.adapted i | lemma | measure_theory.submartingale.strongly_measurable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable [has_le E] (hf : submartingale f ℱ μ) (i : ι) : integrable (f i) μ | hf.2.2 i | lemma | measure_theory.submartingale.integrable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_le_condexp [has_le E] (hf : submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
f i ≤ᵐ[μ] μ[f j | ℱ i] | hf.2.1 i j hij | lemma | measure_theory.submartingale.ae_le_condexp | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add [preorder E] [covariant_class E E (+) (≤)]
(hf : submartingale f ℱ μ) (hg : submartingale g ℱ μ) : submartingale (f + g) ℱ μ | begin
refine ⟨hf.1.add hg.1, λ i j hij, _, λ i, (hf.2.2 i).add (hg.2.2 i)⟩,
refine eventually_le.trans _ (condexp_add (hf.integrable j) (hg.integrable j)).symm.le,
filter_upwards [hf.2.1 i j hij, hg.2.1 i j hij],
intros,
refine add_le_add _ _; assumption,
end | lemma | measure_theory.submartingale.add | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_martingale [preorder E] [covariant_class E E (+) (≤)]
(hf : submartingale f ℱ μ) (hg : martingale g ℱ μ) : submartingale (f + g) ℱ μ | hf.add hg.submartingale | lemma | measure_theory.submartingale.add_martingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg [preorder E] [covariant_class E E (+) (≤)]
(hf : submartingale f ℱ μ) : supermartingale (-f) ℱ μ | begin
refine ⟨hf.1.neg, λ i j hij, (condexp_neg (f j)).le.trans _, λ i, (hf.2.2 i).neg⟩,
filter_upwards [hf.2.1 i j hij] with _ _,
simpa,
end | lemma | measure_theory.submartingale.neg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_integral_le [sigma_finite_filtration μ ℱ] {f : ι → Ω → ℝ} (hf : submartingale f ℱ μ)
{i j : ι} (hij : i ≤ j) {s : set Ω} (hs : measurable_set[ℱ i] s) :
∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f j ω ∂μ | begin
rw [← neg_le_neg_iff, ← integral_neg, ← integral_neg],
exact supermartingale.set_integral_le hf.neg hij hs,
end | lemma | measure_theory.submartingale.set_integral_le | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"measurable_set"
] | The converse of this lemma is `measure_theory.submartingale_of_set_integral_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sub_supermartingale [preorder E] [covariant_class E E (+) (≤)]
(hf : submartingale f ℱ μ) (hg : supermartingale g ℱ μ) : submartingale (f - g) ℱ μ | by { rw sub_eq_add_neg, exact hf.add hg.neg } | lemma | measure_theory.submartingale.sub_supermartingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_martingale [preorder E] [covariant_class E E (+) (≤)]
(hf : submartingale f ℱ μ) (hg : martingale g ℱ μ) : submartingale (f - g) ℱ μ | hf.sub_supermartingale hg.supermartingale | lemma | measure_theory.submartingale.sub_martingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup {f g : ι → Ω → ℝ} (hf : submartingale f ℱ μ) (hg : submartingale g ℱ μ) :
submartingale (f ⊔ g) ℱ μ | begin
refine ⟨λ i, @strongly_measurable.sup _ _ _ _ (ℱ i) _ _ _ (hf.adapted i) (hg.adapted i),
λ i j hij, _, λ i, integrable.sup (hf.integrable _) (hg.integrable _)⟩,
refine eventually_le.sup_le _ _,
{ exact eventually_le.trans (hf.2.1 i j hij)
(condexp_mono (hf.integrable _) (integrable.sup (hf.integra... | lemma | measure_theory.submartingale.sup | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos {f : ι → Ω → ℝ} (hf : submartingale f ℱ μ) :
submartingale (f⁺) ℱ μ | hf.sup (martingale_zero _ _ _).submartingale | lemma | measure_theory.submartingale.pos | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale_of_set_integral_le [is_finite_measure μ]
{f : ι → Ω → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i j : ι, i ≤ j → ∀ s : set Ω, measurable_set[ℱ i] s →
∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f j ω ∂μ) :
submartingale f ℱ μ | begin
refine ⟨hadp, λ i j hij, _, hint⟩,
suffices : f i ≤ᵐ[μ.trim (ℱ.le i)] μ[f j| ℱ i],
{ exact ae_le_of_ae_le_trim this },
suffices : 0 ≤ᵐ[μ.trim (ℱ.le i)] μ[f j| ℱ i] - f i,
{ filter_upwards [this] with x hx,
rwa ← sub_nonneg },
refine ae_nonneg_of_forall_set_integral_nonneg
((integrable_condexp.... | lemma | measure_theory.submartingale_of_set_integral_le | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale_of_condexp_sub_nonneg [is_finite_measure μ]
{f : ι → Ω → ℝ} (hadp : adapted ℱ f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ℱ i]) :
submartingale f ℱ μ | begin
refine ⟨hadp, λ i j hij, _, hint⟩,
rw [← condexp_of_strongly_measurable (ℱ.le _) (hadp _) (hint _), ← eventually_sub_nonneg],
exact eventually_le.trans (hf i j hij) (condexp_sub (hint _) (hint _)).le,
apply_instance
end | lemma | measure_theory.submartingale_of_condexp_sub_nonneg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.condexp_sub_nonneg
{f : ι → Ω → ℝ} (hf : submartingale f ℱ μ) {i j : ι} (hij : i ≤ j) :
0 ≤ᵐ[μ] μ[f j - f i | ℱ i] | begin
by_cases h : sigma_finite (μ.trim (ℱ.le i)),
swap, { rw condexp_of_not_sigma_finite (ℱ.le i) h },
refine eventually_le.trans _ (condexp_sub (hf.integrable _) (hf.integrable _)).symm.le,
rw [eventually_sub_nonneg,
condexp_of_strongly_measurable (ℱ.le _) (hf.adapted _) (hf.integrable _)],
{ exact hf.2... | lemma | measure_theory.submartingale.condexp_sub_nonneg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale_iff_condexp_sub_nonneg [is_finite_measure μ] {f : ι → Ω → ℝ} :
submartingale f ℱ μ ↔ adapted ℱ f ∧ (∀ i, integrable (f i) μ) ∧ ∀ i j, i ≤ j →
0 ≤ᵐ[μ] μ[f j - f i | ℱ i] | ⟨λ h, ⟨h.adapted, h.integrable, λ i j, h.condexp_sub_nonneg⟩,
λ ⟨hadp, hint, h⟩, submartingale_of_condexp_sub_nonneg hadp hint h⟩ | lemma | measure_theory.submartingale_iff_condexp_sub_nonneg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_submartingale [preorder E] [covariant_class E E (+) (≤)]
(hf : supermartingale f ℱ μ) (hg : submartingale g ℱ μ) : supermartingale (f - g) ℱ μ | by { rw sub_eq_add_neg, exact hf.add hg.neg } | lemma | measure_theory.supermartingale.sub_submartingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_martingale [preorder E] [covariant_class E E (+) (≤)]
(hf : supermartingale f ℱ μ) (hg : martingale g ℱ μ) : supermartingale (f - g) ℱ μ | hf.sub_submartingale hg.submartingale | lemma | measure_theory.supermartingale.sub_martingale | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"covariant_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonneg {f : ι → Ω → F}
{c : ℝ} (hc : 0 ≤ c) (hf : supermartingale f ℱ μ) :
supermartingale (c • f) ℱ μ | begin
refine ⟨hf.1.smul c, λ i j hij, _, λ i, (hf.2.2 i).smul c⟩,
refine (condexp_smul c (f j)).le.trans _,
filter_upwards [hf.2.1 i j hij] with _ hle,
simp_rw [pi.smul_apply],
exact smul_le_smul_of_nonneg hle hc,
end | lemma | measure_theory.supermartingale.smul_nonneg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"pi.smul_apply",
"smul_le_smul_of_nonneg",
"smul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonpos {f : ι → Ω → F}
{c : ℝ} (hc : c ≤ 0) (hf : supermartingale f ℱ μ) :
submartingale (c • f) ℱ μ | begin
rw [← neg_neg c, (by { ext i x, simp } : - -c • f = -(-c • f))],
exact (hf.smul_nonneg $ neg_nonneg.2 hc).neg,
end | lemma | measure_theory.supermartingale.smul_nonpos | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonneg {f : ι → Ω → F}
{c : ℝ} (hc : 0 ≤ c) (hf : submartingale f ℱ μ) :
submartingale (c • f) ℱ μ | begin
rw [← neg_neg c, (by { ext i x, simp } : - -c • f = -(c • -f))],
exact supermartingale.neg (hf.neg.smul_nonneg hc),
end | lemma | measure_theory.submartingale.smul_nonneg | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"smul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_nonpos {f : ι → Ω → F}
{c : ℝ} (hc : c ≤ 0) (hf : submartingale f ℱ μ) :
supermartingale (c • f) ℱ μ | begin
rw [← neg_neg c, (by { ext i x, simp } : - -c • f = -(-c • f))],
exact (hf.smul_nonneg $ neg_nonneg.2 hc).neg,
end | lemma | measure_theory.submartingale.smul_nonpos | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale_of_set_integral_le_succ [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, ∀ s : set Ω, measurable_set[𝒢 i] s → ∫ ω in s, f i ω ∂μ ≤ ∫ ω in s, f (i + 1) ω ∂μ) :
submartingale f 𝒢 μ | begin
refine submartingale_of_set_integral_le hadp hint (λ i j hij s hs, _),
induction hij with k hk₁ hk₂,
{ exact le_rfl },
{ exact le_trans hk₂ (hf k s (𝒢.mono hk₁ _ hs)) }
end | lemma | measure_theory.submartingale_of_set_integral_le_succ | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"le_rfl",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supermartingale_of_set_integral_succ_le [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, ∀ s : set Ω, measurable_set[𝒢 i] s → ∫ ω in s, f (i + 1) ω ∂μ ≤ ∫ ω in s, f i ω ∂μ) :
supermartingale f 𝒢 μ | begin
rw ← neg_neg f,
refine (submartingale_of_set_integral_le_succ hadp.neg (λ i, (hint i).neg) _).neg,
simpa only [integral_neg, pi.neg_apply, neg_le_neg_iff],
end | lemma | measure_theory.supermartingale_of_set_integral_succ_le | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_of_set_integral_eq_succ [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, ∀ s : set Ω, measurable_set[𝒢 i] s → ∫ ω in s, f i ω ∂μ = ∫ ω in s, f (i + 1) ω ∂μ) :
martingale f 𝒢 μ | martingale_iff.2
⟨supermartingale_of_set_integral_succ_le hadp hint $ λ i s hs, (hf i s hs).ge,
submartingale_of_set_integral_le_succ hadp hint $ λ i s hs, (hf i s hs).le⟩ | lemma | measure_theory.martingale_of_set_integral_eq_succ | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale_nat [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, f i ≤ᵐ[μ] μ[f (i + 1) | 𝒢 i]) :
submartingale f 𝒢 μ | begin
refine submartingale_of_set_integral_le_succ hadp hint (λ i s hs, _),
have : ∫ ω in s, f (i + 1) ω ∂μ = ∫ ω in s, μ[f (i + 1)|𝒢 i] ω ∂μ :=
(set_integral_condexp (𝒢.le i) (hint _) hs).symm,
rw this,
exact set_integral_mono_ae (hint i).integrable_on integrable_condexp.integrable_on (hf i),
end | lemma | measure_theory.submartingale_nat | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supermartingale_nat [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, μ[f (i + 1) | 𝒢 i] ≤ᵐ[μ] f i) :
supermartingale f 𝒢 μ | begin
rw ← neg_neg f,
refine (submartingale_nat hadp.neg (λ i, (hint i).neg) $ λ i,
eventually_le.trans _ (condexp_neg _).symm.le).neg,
filter_upwards [hf i] with x hx using neg_le_neg hx,
end | lemma | measure_theory.supermartingale_nat | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_nat [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, f i =ᵐ[μ] μ[f (i + 1) | 𝒢 i]) :
martingale f 𝒢 μ | martingale_iff.2 ⟨supermartingale_nat hadp hint $ λ i, (hf i).symm.le,
submartingale_nat hadp hint $ λ i, (hf i).le⟩ | lemma | measure_theory.martingale_nat | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale_of_condexp_sub_nonneg_nat [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, 0 ≤ᵐ[μ] μ[f (i + 1) - f i | 𝒢 i]) :
submartingale f 𝒢 μ | begin
refine submartingale_nat hadp hint (λ i, _),
rw [← condexp_of_strongly_measurable (𝒢.le _) (hadp _) (hint _), ← eventually_sub_nonneg],
exact eventually_le.trans (hf i) (condexp_sub (hint _) (hint _)).le,
apply_instance
end | lemma | measure_theory.submartingale_of_condexp_sub_nonneg_nat | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supermartingale_of_condexp_sub_nonneg_nat [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, 0 ≤ᵐ[μ] μ[f i - f (i + 1) | 𝒢 i]) :
supermartingale f 𝒢 μ | begin
rw ← neg_neg f,
refine (submartingale_of_condexp_sub_nonneg_nat hadp.neg (λ i, (hint i).neg) _).neg,
simpa only [pi.zero_apply, pi.neg_apply, neg_sub_neg]
end | lemma | measure_theory.supermartingale_of_condexp_sub_nonneg_nat | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_of_condexp_sub_eq_zero_nat [is_finite_measure μ]
{f : ℕ → Ω → ℝ} (hadp : adapted 𝒢 f) (hint : ∀ i, integrable (f i) μ)
(hf : ∀ i, μ[f (i + 1) - f i | 𝒢 i] =ᵐ[μ] 0) :
martingale f 𝒢 μ | begin
refine martingale_iff.2 ⟨supermartingale_of_condexp_sub_nonneg_nat hadp hint $ λ i, _,
submartingale_of_condexp_sub_nonneg_nat hadp hint $ λ i, (hf i).symm.le⟩,
rw ← neg_sub,
refine (eventually_eq.trans _ (condexp_neg _).symm).le,
filter_upwards [hf i] with x hx,
simpa only [pi.zero_apply, pi.neg_ap... | lemma | measure_theory.martingale_of_condexp_sub_eq_zero_nat | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.zero_le_of_predictable [preorder E] [sigma_finite_filtration μ 𝒢]
{f : ℕ → Ω → E} (hfmgle : submartingale f 𝒢 μ) (hfadp : adapted 𝒢 (λ n, f (n + 1))) (n : ℕ) :
f 0 ≤ᵐ[μ] f n | begin
induction n with k ih,
{ refl },
{ exact ih.trans ((hfmgle.2.1 k (k + 1) k.le_succ).trans_eq $ germ.coe_eq.mp $ congr_arg coe $
condexp_of_strongly_measurable (𝒢.le _) (hfadp _) $ hfmgle.integrable _) }
end | lemma | measure_theory.submartingale.zero_le_of_predictable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"ih"
] | A predictable submartingale is a.e. greater equal than its initial state. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
supermartingale.le_zero_of_predictable [preorder E] [sigma_finite_filtration μ 𝒢]
{f : ℕ → Ω → E} (hfmgle : supermartingale f 𝒢 μ) (hfadp : adapted 𝒢 (λ n, f (n + 1))) (n : ℕ) :
f n ≤ᵐ[μ] f 0 | begin
induction n with k ih,
{ refl },
{ exact ((germ.coe_eq.mp $ congr_arg coe $ condexp_of_strongly_measurable (𝒢.le _) (hfadp _) $
hfmgle.integrable _).symm.trans_le (hfmgle.2.1 k (k + 1) k.le_succ)).trans ih }
end | lemma | measure_theory.supermartingale.le_zero_of_predictable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"ih"
] | A predictable supermartingale is a.e. less equal than its initial state. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
martingale.eq_zero_of_predictable [sigma_finite_filtration μ 𝒢]
{f : ℕ → Ω → E} (hfmgle : martingale f 𝒢 μ) (hfadp : adapted 𝒢 (λ n, f (n + 1))) (n : ℕ) :
f n =ᵐ[μ] f 0 | begin
induction n with k ih,
{ refl },
{ exact ((germ.coe_eq.mp (congr_arg coe $ condexp_of_strongly_measurable (𝒢.le _) (hfadp _)
(hfmgle.integrable _))).symm.trans (hfmgle.2 k (k + 1) k.le_succ)).trans ih }
end | lemma | measure_theory.martingale.eq_zero_of_predictable | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"ih"
] | A predictable martingale is a.e. equal to its initial state. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integrable_stopped_value [has_le E] {f : ℕ → Ω → E} (hf : submartingale f 𝒢 μ) {τ : Ω → ℕ}
(hτ : is_stopping_time 𝒢 τ) {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) :
integrable (stopped_value f τ) μ | integrable_stopped_value ℕ hτ hf.integrable hbdd | lemma | measure_theory.submartingale.integrable_stopped_value | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.sum_mul_sub [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ}
(hf : submartingale f 𝒢 μ) (hξ : adapted 𝒢 ξ)
(hbdd : ∀ n ω, ξ n ω ≤ R) (hnonneg : ∀ n ω, 0 ≤ ξ n ω) :
submartingale (λ n, ∑ k in finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ | begin
have hξbdd : ∀ i, ∃ C, ∀ ω, |ξ i ω| ≤ C :=
λ i, ⟨R, λ ω, (abs_of_nonneg (hnonneg i ω)).trans_le (hbdd i ω)⟩,
have hint : ∀ m, integrable (∑ k in finset.range m, ξ k * (f (k + 1) - f k)) μ :=
λ m, integrable_finset_sum' _
(λ i hi, integrable.bdd_mul ((hf.integrable _).sub (hf.integrable _))
... | lemma | measure_theory.submartingale.sum_mul_sub | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"abs_of_nonneg",
"finset.mem_range",
"finset.range",
"nat.Ico_succ_singleton",
"pi.mul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.sum_mul_sub' [is_finite_measure μ] {R : ℝ} {ξ f : ℕ → Ω → ℝ}
(hf : submartingale f 𝒢 μ) (hξ : adapted 𝒢 (λ n, ξ (n + 1)))
(hbdd : ∀ n ω, ξ n ω ≤ R) (hnonneg : ∀ n ω, 0 ≤ ξ n ω) :
submartingale (λ n, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)) 𝒢 μ | hf.sum_mul_sub hξ (λ n, hbdd _) (λ n, hnonneg _) | lemma | measure_theory.submartingale.sum_mul_sub' | probability.martingale | src/probability/martingale/basic.lean | [
"probability.notation",
"probability.process.stopping"
] | [
"finset.range"
] | Given a discrete submartingale `f` and a predictable process `ξ` (i.e. `ξ (n + 1)` is adapted)
the process defined by `λ n, ∑ k in finset.range n, ξ (k + 1) * (f (k + 1) - f k)` is also a
submartingale. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
least_ge (f : ℕ → Ω → ℝ) (r : ℝ) (n : ℕ) | hitting f (set.Ici r) 0 n | def | measure_theory.least_ge | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"set.Ici"
] | `least_ge f r n` is the stopping time corresponding to the first time `f ≥ r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adapted.is_stopping_time_least_ge (r : ℝ) (n : ℕ) (hf : adapted ℱ f) :
is_stopping_time ℱ (least_ge f r n) | hitting_is_stopping_time hf measurable_set_Ici | lemma | measure_theory.adapted.is_stopping_time_least_ge | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"measurable_set_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
least_ge_le {i : ℕ} {r : ℝ} (ω : Ω) : least_ge f r i ω ≤ i | hitting_le ω | lemma | measure_theory.least_ge_le | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
least_ge_mono {n m : ℕ} (hnm : n ≤ m) (r : ℝ) (ω : Ω) :
least_ge f r n ω ≤ least_ge f r m ω | hitting_mono hnm | lemma | measure_theory.least_ge_mono | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
least_ge_eq_min (π : Ω → ℕ) (r : ℝ) (ω : Ω)
{n : ℕ} (hπn : ∀ ω, π ω ≤ n) :
least_ge f r (π ω) ω = min (π ω) (least_ge f r n ω) | begin
classical,
refine le_antisymm (le_min (least_ge_le _) (least_ge_mono (hπn ω) r ω)) _,
by_cases hle : π ω ≤ least_ge f r n ω,
{ rw [min_eq_left hle, least_ge],
by_cases h : ∃ j ∈ set.Icc 0 (π ω), f j ω ∈ set.Ici r,
{ refine hle.trans (eq.le _),
rw [least_ge, ← hitting_eq_hitting_of_exists (hπ... | lemma | measure_theory.least_ge_eq_min | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"set.Icc",
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stopped_value_stopped_value_least_ge (f : ℕ → Ω → ℝ) (π : Ω → ℕ) (r : ℝ)
{n : ℕ} (hπn : ∀ ω, π ω ≤ n) :
stopped_value (λ i, stopped_value f (least_ge f r i)) π
= stopped_value (stopped_process f (least_ge f r n)) π | by { ext1 ω, simp_rw [stopped_process, stopped_value], rw least_ge_eq_min _ _ _ hπn, } | lemma | measure_theory.stopped_value_stopped_value_least_ge | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.stopped_value_least_ge [is_finite_measure μ]
(hf : submartingale f ℱ μ) (r : ℝ) :
submartingale (λ i, stopped_value f (least_ge f r i)) ℱ μ | begin
rw submartingale_iff_expected_stopped_value_mono,
{ intros σ π hσ hπ hσ_le_π hπ_bdd,
obtain ⟨n, hπ_le_n⟩ := hπ_bdd,
simp_rw stopped_value_stopped_value_least_ge f σ r (λ i, (hσ_le_π i).trans (hπ_le_n i)),
simp_rw stopped_value_stopped_value_least_ge f π r hπ_le_n,
refine hf.expected_stopped_va... | lemma | measure_theory.submartingale.stopped_value_least_ge | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"le_rfl",
"min_le_min"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_stopped_value_least_ge_le (hr : 0 ≤ r) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
∀ᵐ ω ∂μ, stopped_value f (least_ge f r i) ω ≤ r + R | begin
filter_upwards [hbdd] with ω hbddω,
change f (least_ge f r i ω) ω ≤ r + R,
by_cases heq : least_ge f r i ω = 0,
{ rw [heq, hf0, pi.zero_apply],
exact add_nonneg hr R.coe_nonneg },
{ obtain ⟨k, hk⟩ := nat.exists_eq_succ_of_ne_zero heq,
rw [hk, add_comm, ← sub_le_iff_le_add],
have := not_mem_o... | lemma | measure_theory.norm_stopped_value_least_ge_le | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"le_abs_self",
"not_or_distrib",
"set.mem_Ici",
"set.mem_Iic",
"set.mem_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.stopped_value_least_ge_snorm_le [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hr : 0 ≤ r) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
snorm (stopped_value f (least_ge f r i)) 1 μ ≤ 2 * μ set.univ * ennreal.of_real (r + R) | begin
refine snorm_one_le_of_le' ((hf.stopped_value_least_ge r).integrable _) _
(norm_stopped_value_least_ge_le hr hf0 hbdd i),
rw ← integral_univ,
refine le_trans _ ((hf.stopped_value_least_ge r).set_integral_le (zero_le _)
measurable_set.univ),
simp_rw [stopped_value, least_ge, hitting_of_le le_rfl, h... | lemma | measure_theory.submartingale.stopped_value_least_ge_snorm_le | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"ennreal.of_real",
"le_rfl",
"measurable_set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.stopped_value_least_ge_snorm_le' [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hr : 0 ≤ r) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
snorm (stopped_value f (least_ge f r i)) 1 μ ≤
ennreal.to_nnreal (2 * μ set.univ * ennreal.of_real (r + R)) | begin
refine (hf.stopped_value_least_ge_snorm_le hr hf0 hbdd i).trans _,
simp [ennreal.coe_to_nnreal (measure_ne_top μ _), ennreal.coe_to_nnreal],
end | lemma | measure_theory.submartingale.stopped_value_least_ge_snorm_le' | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"ennreal.coe_to_nnreal",
"ennreal.of_real",
"ennreal.to_nnreal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.exists_tendsto_of_abs_bdd_above_aux [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, bdd_above (set.range $ λ n, f n ω) → ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) | begin
have ht : ∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, tendsto (λ n, stopped_value f (least_ge f i n) ω) at_top (𝓝 c),
{ rw ae_all_iff,
exact λ i, submartingale.exists_ae_tendsto_of_bdd (hf.stopped_value_least_ge i)
(hf.stopped_value_least_ge_snorm_le' i.cast_nonneg hf0 hbdd) },
filter_upwards [ht] with ω hω hωb,
rw... | lemma | measure_theory.submartingale.exists_tendsto_of_abs_bdd_above_aux | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"bdd_above",
"exists_nat_gt",
"set.mem_Icc",
"set.mem_Ici",
"set.mem_union",
"set.range"
] | This lemma is superceded by `submartingale.bdd_above_iff_exists_tendsto`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submartingale.bdd_above_iff_exists_tendsto_aux [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, bdd_above (set.range $ λ n, f n ω) ↔ ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) | by filter_upwards [hf.exists_tendsto_of_abs_bdd_above_aux hf0 hbdd] with ω hω using
⟨hω, λ ⟨c, hc⟩, hc.bdd_above_range⟩ | lemma | measure_theory.submartingale.bdd_above_iff_exists_tendsto_aux | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"bdd_above",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.bdd_above_iff_exists_tendsto [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, bdd_above (set.range $ λ n, f n ω) ↔ ∃ c, tendsto (λ n, f n ω) at_top (𝓝 c) | begin
set g : ℕ → Ω → ℝ := λ n ω, f n ω - f 0 ω with hgdef,
have hg : submartingale g ℱ μ :=
hf.sub_martingale (martingale_const_fun _ _ (hf.adapted 0) (hf.integrable 0)),
have hg0 : g 0 = 0,
{ ext ω,
simp only [hgdef, sub_self, pi.zero_apply] },
have hgbdd : ∀ᵐ ω ∂μ, ∀ (i : ℕ), |g (i + 1) ω - g i ω| ... | lemma | measure_theory.submartingale.bdd_above_iff_exists_tendsto | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"bdd_above",
"eq_iff_iff",
"le_abs_self",
"neg_le_abs_self",
"set.range"
] | One sided martingale bound: If `f` is a submartingale which has uniformly bounded differences,
then for almost every `ω`, `f n ω` is bounded above (in `n`) if and only if it converges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
martingale.bdd_above_range_iff_bdd_below_range [is_finite_measure μ]
(hf : martingale f ℱ μ) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, bdd_above (set.range (λ n, f n ω)) ↔ bdd_below (set.range (λ n, f n ω)) | begin
have hbdd' : ∀ᵐ ω ∂μ, ∀ i, |(-f) (i + 1) ω - (-f) i ω| ≤ R,
{ filter_upwards [hbdd] with ω hω i,
erw [← abs_neg, neg_sub, sub_neg_eq_add, neg_add_eq_sub],
exact hω i },
have hup := hf.submartingale.bdd_above_iff_exists_tendsto hbdd,
have hdown := hf.neg.submartingale.bdd_above_iff_exists_tendsto h... | lemma | measure_theory.martingale.bdd_above_range_iff_bdd_below_range | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"abs_neg",
"bdd_above",
"bdd_below",
"mem_lower_bounds",
"mem_upper_bounds",
"set.mem_range",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale.ae_not_tendsto_at_top_at_top [is_finite_measure μ]
(hf : martingale f ℱ μ) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ¬ tendsto (λ n, f n ω) at_top at_top | by filter_upwards [hf.bdd_above_range_iff_bdd_below_range hbdd] with ω hω htop using
unbounded_of_tendsto_at_top htop (hω.2 $ bdd_below_range_of_tendsto_at_top_at_top htop) | lemma | measure_theory.martingale.ae_not_tendsto_at_top_at_top | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale.ae_not_tendsto_at_top_at_bot [is_finite_measure μ]
(hf : martingale f ℱ μ) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, ¬ tendsto (λ n, f n ω) at_top at_bot | by filter_upwards [hf.bdd_above_range_iff_bdd_below_range hbdd] with ω hω htop using
unbounded_of_tendsto_at_bot htop (hω.1 $ bdd_above_range_of_tendsto_at_top_at_bot htop) | lemma | measure_theory.martingale.ae_not_tendsto_at_top_at_bot | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
process (s : ℕ → set Ω) (n : ℕ) : Ω → ℝ | ∑ k in finset.range n, (s (k + 1)).indicator 1 | def | measure_theory.borel_cantelli.process | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range"
] | Auxiliary definition required to prove Lévy's generalization of the Borel-Cantelli lemmas for
which we will take the martingale part. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
process_zero : process s 0 = 0 | by rw [process, finset.range_zero, finset.sum_empty] | lemma | measure_theory.borel_cantelli.process_zero | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted_process (hs : ∀ n, measurable_set[ℱ n] (s n)) :
adapted ℱ (process s) | λ n, finset.strongly_measurable_sum' _ $ λ k hk, strongly_measurable_one.indicator $
ℱ.mono (finset.mem_range.1 hk) _ $ hs _ | lemma | measure_theory.borel_cantelli.adapted_process | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
martingale_part_process_ae_eq (ℱ : filtration ℕ m0) (μ : measure Ω) (s : ℕ → set Ω) (n : ℕ) :
martingale_part (process s) ℱ μ n =
∑ k in finset.range n, ((s (k + 1)).indicator 1 - μ[(s (k + 1)).indicator 1 | ℱ k]) | begin
simp only [martingale_part_eq_sum, process_zero, zero_add],
refine finset.sum_congr rfl (λ k hk, _),
simp only [process, finset.sum_range_succ_sub_sum],
end | lemma | measure_theory.borel_cantelli.martingale_part_process_ae_eq | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
predictable_part_process_ae_eq (ℱ : filtration ℕ m0) (μ : measure Ω) (s : ℕ → set Ω) (n : ℕ) :
predictable_part (process s) ℱ μ n =
∑ k in finset.range n, μ[(s (k + 1)).indicator (1 : Ω → ℝ) | ℱ k] | begin
have := martingale_part_process_ae_eq ℱ μ s n,
simp_rw [martingale_part, process, finset.sum_sub_distrib] at this,
exact sub_right_injective this,
end | lemma | measure_theory.borel_cantelli.predictable_part_process_ae_eq | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
process_difference_le (s : ℕ → set Ω) (ω : Ω) (n : ℕ) : | |process s (n + 1) ω - process s n ω| ≤ (1 : ℝ≥0) :=
begin
rw [nonneg.coe_one, process, process, finset.sum_apply, finset.sum_apply,
finset.sum_range_succ_sub_sum, ← real.norm_eq_abs, norm_indicator_eq_indicator_norm],
refine set.indicator_le' (λ _ _, _) (λ _ _, zero_le_one) _,
rw [pi.one_apply, norm_one]
end | lemma | measure_theory.borel_cantelli.process_difference_le | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"nonneg.coe_one",
"norm_indicator_eq_indicator_norm",
"pi.one_apply",
"real.norm_eq_abs",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_process (μ : measure Ω) [is_finite_measure μ]
(hs : ∀ n, measurable_set[ℱ n] (s n)) (n : ℕ) :
integrable (process s n) μ | integrable_finset_sum' _ $ λ k hk,
integrable_on.integrable_indicator (integrable_const 1) $ ℱ.le _ _ $ hs _ | lemma | measure_theory.borel_cantelli.integrable_process | probability.martingale | src/probability/martingale/borel_cantelli.lean | [
"probability.martingale.convergence",
"probability.martingale.optional_stopping",
"probability.martingale.centering"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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