fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
HasLaw.variance_eq {μ : Measure ℝ} {X : Ω → ℝ} (hX : HasLaw X μ P) :
Var[X; P] = Var[id; μ] := by
rw [← hX.map_eq, variance_map aemeasurable_id hX.aemeasurable, Function.id_comp] | lemma | Probability | [
"Mathlib.Probability.Density",
"Mathlib.Probability.Moments.Variance"
] | Mathlib/Probability/HasLaw.lean | HasLaw.variance_eq | null |
HasPDF.hasLaw [h : HasPDF X P μ] : HasLaw X (μ.withDensity (pdf X P μ)) P where
aemeasurable := h.aemeasurable
map_eq := map_eq_withDensity_pdf X P μ | lemma | Probability | [
"Mathlib.Probability.Density",
"Mathlib.Probability.Moments.Variance"
] | Mathlib/Probability/HasLaw.lean | HasPDF.hasLaw | null |
IdentDistrib (f : α → γ) (g : β → γ)
(μ : Measure α := by volume_tac)
(ν : Measure β := by volume_tac) : Prop where
aemeasurable_fst : AEMeasurable f μ
aemeasurable_snd : AEMeasurable g ν
map_eq : Measure.map f μ = Measure.map g ν | structure | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | IdentDistrib | Two functions defined on two (possibly different) measure spaces are identically distributed if
their image measures coincide. This only makes sense when the functions are ae measurable
(as otherwise the image measures are not defined), so we require this as well in the definition. |
protected refl (hf : AEMeasurable f μ) : IdentDistrib f f μ μ :=
{ aemeasurable_fst := hf
aemeasurable_snd := hf
map_eq := rfl } | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | refl | null |
protected symm (h : IdentDistrib f g μ ν) : IdentDistrib g f ν μ :=
{ aemeasurable_fst := h.aemeasurable_snd
aemeasurable_snd := h.aemeasurable_fst
map_eq := h.map_eq.symm } | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | symm | null |
protected trans {ρ : Measure δ} {h : δ → γ} (h₁ : IdentDistrib f g μ ν)
(h₂ : IdentDistrib g h ν ρ) : IdentDistrib f h μ ρ :=
{ aemeasurable_fst := h₁.aemeasurable_fst
aemeasurable_snd := h₂.aemeasurable_snd
map_eq := h₁.map_eq.trans h₂.map_eq } | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | trans | null |
protected comp_of_aemeasurable {u : γ → δ} (h : IdentDistrib f g μ ν)
(hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν :=
{ aemeasurable_fst := hu.comp_aemeasurable h.aemeasurable_fst
aemeasurable_snd := by rw [h.map_eq] at hu; exact hu.comp_aemeasurable h.aemeasurable_snd
map_eq... | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | comp_of_aemeasurable | null |
protected comp {u : γ → δ} (h : IdentDistrib f g μ ν) (hu : Measurable u) :
IdentDistrib (u ∘ f) (u ∘ g) μ ν :=
h.comp_of_aemeasurable hu.aemeasurable | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | comp | null |
protected of_ae_eq {g : α → γ} (hf : AEMeasurable f μ) (heq : f =ᵐ[μ] g) :
IdentDistrib f g μ μ :=
{ aemeasurable_fst := hf
aemeasurable_snd := hf.congr heq
map_eq := Measure.map_congr heq } | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | of_ae_eq | null |
_root_.MeasureTheory.AEMeasurable.identDistrib_mk
(hf : AEMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ :=
IdentDistrib.of_ae_eq hf hf.ae_eq_mk | lemma | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | _root_.MeasureTheory.AEMeasurable.identDistrib_mk | null |
_root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk
[TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]
(hf : AEStronglyMeasurable f μ) : IdentDistrib f (hf.mk f) μ μ :=
IdentDistrib.of_ae_eq hf.aemeasurable hf.ae_eq_mk | lemma | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | _root_.MeasureTheory.AEStronglyMeasurable.identDistrib_mk | null |
measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :
μ (f ⁻¹' s) = ν (g ⁻¹' s) := by
rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ←
Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]
alias measure_preimage_eq := measure_mem_eq | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | measure_mem_eq | null |
ae_snd (h : IdentDistrib f g μ ν) {p : γ → Prop} (pmeas : MeasurableSet {x | p x})
(hp : ∀ᵐ x ∂μ, p (f x)) : ∀ᵐ x ∂ν, p (g x) := by
apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [← h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | ae_snd | null |
ae_mem_snd (h : IdentDistrib f g μ ν) {t : Set γ} (tmeas : MeasurableSet t)
(ht : ∀ᵐ x ∂μ, f x ∈ t) : ∀ᵐ x ∂ν, g x ∈ t :=
h.ae_snd tmeas ht | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | ae_mem_snd | null |
aestronglyMeasurable_fst [TopologicalSpace γ] [PseudoMetrizableSpace γ]
[OpensMeasurableSpace γ] [SecondCountableTopology γ] (h : IdentDistrib f g μ ν) :
AEStronglyMeasurable f μ :=
h.aemeasurable_fst.aestronglyMeasurable | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | aestronglyMeasurable_fst | In a second countable topology, the first function in an identically distributed pair is a.e.
strongly measurable. So is the second function, but use `h.symm.aestronglyMeasurable_fst` as
`h.aestronglyMeasurable_snd` has a different meaning. |
aestronglyMeasurable_snd [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]
(h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν := by
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩
rcases (aestronglyMeasurable_iff_aemeasurable_separ... | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | aestronglyMeasurable_snd | If `f` and `g` are identically distributed and `f` is a.e. strongly measurable, so is `g`. |
aestronglyMeasurable_iff [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]
(h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g ν :=
⟨fun hf => h.aestronglyMeasurable_snd hf, fun hg => h.symm.aestronglyMeasurable_snd hg⟩ | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | aestronglyMeasurable_iff | null |
essSup_eq [ConditionallyCompleteLinearOrder γ] [TopologicalSpace γ] [OpensMeasurableSpace γ]
[OrderClosedTopology γ] (h : IdentDistrib f g μ ν) : essSup f μ = essSup g ν := by
have I : ∀ a, μ {x : α | a < f x} = ν {x : β | a < g x} := fun a =>
h.measure_mem_eq measurableSet_Ioi
simp_rw [essSup_eq_sInf, I] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | essSup_eq | null |
lintegral_eq {f : α → ℝ≥0∞} {g : β → ℝ≥0∞} (h : IdentDistrib f g μ ν) :
∫⁻ x, f x ∂μ = ∫⁻ x, g x ∂ν := by
change ∫⁻ x, id (f x) ∂μ = ∫⁻ x, id (g x) ∂ν
rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ←
lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | lintegral_eq | null |
integral_eq [NormedAddCommGroup γ] [NormedSpace ℝ γ] [BorelSpace γ]
(h : IdentDistrib f g μ ν) : ∫ x, f x ∂μ = ∫ x, g x ∂ν := by
by_cases hf : AEStronglyMeasurable f μ
· have A : AEStronglyMeasurable id (Measure.map f μ) := by
rw [aestronglyMeasurable_iff_aemeasurable_separable]
rcases (aestronglyMe... | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | integral_eq | null |
eLpNorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν)
(p : ℝ≥0∞) : eLpNorm f p μ = eLpNorm g p ν := by
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, if_true,
if_false]
apply essSup_eq
exac... | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | eLpNorm_eq | null |
memLp_snd [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : IdentDistrib f g μ ν)
(hf : MemLp f p μ) : MemLp g p ν := by
refine ⟨h.aestronglyMeasurable_snd hf.aestronglyMeasurable, ?_⟩
rw [← h.eLpNorm_eq]
exact hf.2 | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | memLp_snd | null |
memLp_iff [NormedAddCommGroup γ] [BorelSpace γ] {p : ℝ≥0∞} (h : IdentDistrib f g μ ν) :
MemLp f p μ ↔ MemLp g p ν :=
⟨fun hf => h.memLp_snd hf, fun hg => h.symm.memLp_snd hg⟩ | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | memLp_iff | null |
integrable_snd [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν)
(hf : Integrable f μ) : Integrable g ν := by
rw [← memLp_one_iff_integrable] at hf ⊢
exact h.memLp_snd hf | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | integrable_snd | null |
integrable_iff [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) :
Integrable f μ ↔ Integrable g ν :=
⟨fun hf => h.integrable_snd hf, fun hg => h.symm.integrable_snd hg⟩ | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | integrable_iff | null |
protected norm [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν) :
IdentDistrib (fun x => ‖f x‖) (fun x => ‖g x‖) μ ν :=
h.comp measurable_norm | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | norm | null |
protected nnnorm [NormedAddCommGroup γ] [OpensMeasurableSpace γ]
(h : IdentDistrib f g μ ν) :
IdentDistrib (fun x => ‖f x‖₊) (fun x => ‖g x‖₊) μ ν :=
h.comp measurable_nnnorm | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | nnnorm | null |
protected pow [Pow γ ℕ] [MeasurablePow γ ℕ] (h : IdentDistrib f g μ ν) {n : ℕ} :
IdentDistrib (fun x => f x ^ n) (fun x => g x ^ n) μ ν :=
h.comp (measurable_id.pow_const n) | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | pow | null |
protected sq [Pow γ ℕ] [MeasurablePow γ ℕ] (h : IdentDistrib f g μ ν) :
IdentDistrib (fun x => f x ^ 2) (fun x => g x ^ 2) μ ν :=
h.comp (measurable_id.pow_const 2) | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | sq | null |
protected coe_nnreal_ennreal {f : α → ℝ≥0} {g : β → ℝ≥0} (h : IdentDistrib f g μ ν) :
IdentDistrib (fun x => (f x : ℝ≥0∞)) (fun x => (g x : ℝ≥0∞)) μ ν :=
h.comp measurable_coe_nnreal_ennreal
@[to_additive] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | coe_nnreal_ennreal | null |
mul_const [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) :
IdentDistrib (fun x => f x * c) (fun x => g x * c) μ ν :=
h.comp (measurable_mul_const c)
@[to_additive] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | mul_const | null |
const_mul [Mul γ] [MeasurableMul γ] (h : IdentDistrib f g μ ν) (c : γ) :
IdentDistrib (fun x => c * f x) (fun x => c * g x) μ ν :=
h.comp (measurable_const_mul c)
@[to_additive] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | const_mul | null |
div_const [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) :
IdentDistrib (fun x => f x / c) (fun x => g x / c) μ ν :=
h.comp (MeasurableDiv.measurable_div_const c)
@[to_additive] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | div_const | null |
const_div [Div γ] [MeasurableDiv γ] (h : IdentDistrib f g μ ν) (c : γ) :
IdentDistrib (fun x => c / f x) (fun x => c / g x) μ ν :=
h.comp (MeasurableDiv.measurable_const_div c)
@[to_additive] | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | const_div | null |
inv [Inv γ] [MeasurableInv γ] (h : IdentDistrib f g μ ν) :
IdentDistrib f⁻¹ g⁻¹ μ ν := h.comp measurable_inv | lemma | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | inv | null |
evariance_eq {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) :
evariance f μ = evariance g ν := by
convert (h.sub_const (∫ x, f x ∂μ)).nnnorm.coe_nnreal_ennreal.sq.lintegral_eq
rw [h.integral_eq]
rfl | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | evariance_eq | null |
variance_eq {f : α → ℝ} {g : β → ℝ} (h : IdentDistrib f g μ ν) :
variance f μ = variance g ν := by rw [variance, h.evariance_eq]; rfl | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | variance_eq | null |
MemLp.uniformIntegrable_of_identDistrib_aux {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : MemLp (f j) p μ) (hfmeas : ∀ i, StronglyMeasurable (f i))
(hf : ∀ i, IdentDistrib (f i) (f j) μ μ) : UniformIntegrable f p μ := by
refine uniformIntegrable_of' hp hp' hfmeas fun ε hε =>... | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | MemLp.uniformIntegrable_of_identDistrib_aux | This lemma is superseded by `MemLp.uniformIntegrable_of_identDistrib` which only requires
`AEStronglyMeasurable`. |
MemLp.uniformIntegrable_of_identDistrib {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : MemLp (f j) p μ) (hf : ∀ i, IdentDistrib (f i) (f j) μ μ) :
UniformIntegrable f p μ := by
have hfmeas : ∀ i, AEStronglyMeasurable (f i) μ := fun i =>
(hf i).aestronglyMeasurable_iff.2 h... | theorem | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | MemLp.uniformIntegrable_of_identDistrib | A sequence of identically distributed Lᵖ functions is p-uniformly integrable. |
indepFun_of_identDistrib_pair
{μ : Measure γ} {μ' : Measure δ} [IsFiniteMeasure μ] [IsFiniteMeasure μ']
{X : γ → α} {X' : δ → α} {Y : γ → β} {Y' : δ → β} (h_indep : IndepFun X Y μ)
(h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ') :
IndepFun X' Y' μ' := by
rw [indepFun_iff_map_... | lemma | Probability | [
"Mathlib.Probability.Moments.Variance",
"Mathlib.MeasureTheory.Function.UniformIntegrable"
] | Mathlib/Probability/IdentDistrib.lean | indepFun_of_identDistrib_pair | If `X` and `Y` are independent and `(X, Y)` and `(X', Y')` are identically distributed,
then `X'` and `Y'` are independent. |
to construct the measure over a product indexed by `ℕ`, which is `infinitePiNat`. This
is an implementation detail and should not be used directly. Then we construct the product measure
over an arbitrary type by extending `piContent μ` thanks to Carathéodory's theorem. The key lemma
to do so is `piContent_tendsto_zero`... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | to | null |
isProjectiveMeasureFamily_pi :
IsProjectiveMeasureFamily (fun I : Finset ι ↦ (Measure.pi (fun i : I ↦ μ i))) := by
refine fun I J hJI ↦ Measure.pi_eq (fun s ms ↦ ?_)
classical
simp_rw [Measure.map_apply (measurable_restrict₂ hJI) (.univ_pi ms), restrict₂_preimage hJI,
Measure.pi_pi, prod_eq_prod_extend]
... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | isProjectiveMeasureFamily_pi | Consider a family of probability measures. You can take their products for any finite
subfamily. This gives a projective family of measures. |
noncomputable piContent : AddContent (measurableCylinders X) :=
projectiveFamilyContent (isProjectiveMeasureFamily_pi μ) | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | piContent | Consider a family of probability measures. You can take their products for any finite
subfamily. This gives an additive content on the measurable cylinders. |
piContent_cylinder {I : Finset ι} {S : Set (Π i : I, X i)} (hS : MeasurableSet S) :
piContent μ (cylinder I S) = Measure.pi (fun i : I ↦ μ i) S :=
projectiveFamilyContent_cylinder _ hS | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | piContent_cylinder | null |
piContent_eq_measure_pi [Fintype ι] {s : Set (Π i, X i)} (hs : MeasurableSet s) :
piContent μ s = Measure.pi μ s := by
let e : @Finset.univ ι _ ≃ ι :=
{ toFun i := i
invFun i := ⟨i, mem_univ i⟩ }
have : s = cylinder univ (MeasurableEquiv.piCongrLeft X e ⁻¹' s) := rfl
nth_rw 1 [this]
dsimp [e]
rw... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | piContent_eq_measure_pi | null |
noncomputable infinitePiNat : Measure (Π n, X n) :=
(traj (fun n ↦ const _ (μ (n + 1))) 0) ∘ₘ (Measure.pi (fun i : Iic 0 ↦ μ i)) | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePiNat | Infinite product measure indexed by `ℕ`. This is an auxiliary construction, you should use
the generic product measure `Measure.infinitePi`. |
pi_prod_map_IocProdIoc {a b c : ℕ} (hab : a ≤ b) (hbc : b ≤ c) :
((Measure.pi (fun i : Ioc a b ↦ μ i)).prod (Measure.pi (fun i : Ioc b c ↦ μ i))).map
(IocProdIoc a b c) = Measure.pi (fun i : Ioc a c ↦ μ i) := by
refine (Measure.pi_eq fun s ms ↦ ?_).symm
simp_rw [Measure.map_apply measurable_IocProdIoc (.u... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | pi_prod_map_IocProdIoc | Let `μ : (i : Ioc a c) → Measure (X i)` be a family of measures. Up to an equivalence,
`(⨂ i : Ioc a b, μ i) ⊗ (⨂ i : Ioc b c, μ i) = ⨂ i : Ioc a c, μ i`, where `⊗` denotes the
product of measures. |
pi_prod_map_IicProdIoc {a b : ℕ} :
((Measure.pi (fun i : Iic a ↦ μ i)).prod (Measure.pi (fun i : Ioc a b ↦ μ i))).map
(IicProdIoc a b) = Measure.pi (fun i : Iic b ↦ μ i) := by
obtain hab | hba := le_total a b
· refine (Measure.pi_eq fun s ms ↦ ?_).symm
simp_rw [Measure.map_apply measurable_IicProdIoc ... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | pi_prod_map_IicProdIoc | Let `μ : (i : Iic b) → Measure (X i)` be a family of measures. Up to an equivalence,
`(⨂ i : Iic a, μ i) ⊗ (⨂ i : Ioc a b, μ i) = ⨂ i : Iic b, μ i`, where `⊗` denotes the
product of measures. |
map_piSingleton (μ : (n : ℕ) → Measure (X n)) [∀ n, SigmaFinite (μ n)] (n : ℕ) :
(μ (n + 1)).map (piSingleton n) = Measure.pi (fun i : Ioc n (n + 1) ↦ μ i) := by
refine (Measure.pi_eq fun s hs ↦ ?_).symm
have : Subsingleton (Ioc n (n + 1)) := by rw [Nat.Ioc_succ_singleton]; infer_instance
rw [Fintype.prod_sub... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | map_piSingleton | Let `μ (i + 1) : Measure (X (i + 1))` be a measure. Up to an equivalence,
`μ i = ⨂ j : Ioc i (i + 1), μ i`, where `⊗` denotes the product of measures. |
partialTraj_const_restrict₂ {a b : ℕ} :
(partialTraj (fun n ↦ const _ (μ (n + 1))) a b).map (restrict₂ Ioc_subset_Iic_self) =
const _ (Measure.pi (fun i : Ioc a b ↦ μ i)) := by
obtain hab | hba := lt_or_ge a b
· refine Nat.le_induction ?_ (fun n hn hind ↦ ?_) b (Nat.succ_le.2 hab) <;> ext1 x₀
· rw [part... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | partialTraj_const_restrict₂ | `partialTraj κ a b` is a kernel which up to an equivalence is equal to
`Kernel.id ×ₖ (κ a ⊗ₖ ... ⊗ₖ κ (b - 1))`. This lemma therefore states that if the kernels `κ`
are constant then their composition-product is the product measure. |
partialTraj_const {a b : ℕ} :
partialTraj (fun n ↦ const _ (μ (n + 1))) a b =
(Kernel.id ×ₖ (const _ (Measure.pi (fun i : Ioc a b ↦ μ i)))).map (IicProdIoc a b) := by
rw [partialTraj_eq_prod, partialTraj_const_restrict₂] | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | partialTraj_const | `partialTraj κ a b` is a kernel which up to an equivalence is equal to
`Kernel.id ×ₖ (κ a ⊗ₖ ... ⊗ₖ κ (b - 1))`. This lemma therefore states that if the kernel `κ i`
is constant equal to `μ i` for all `i`, then up to an equivalence
`partialTraj κ a b = Kernel.id ×ₖ Kernel.const (⨂ μ i)`. |
isProjectiveLimit_infinitePiNat :
IsProjectiveLimit (infinitePiNat μ) (fun I : Finset ℕ ↦ (Measure.pi (fun i : I ↦ μ i))) := by
intro I
rw [isProjectiveMeasureFamily_pi μ _ _ I.subset_Iic_sup_id,
← restrict₂_comp_restrict I.subset_Iic_sup_id, ← map_map, ← frestrictLe, infinitePiNat,
map_comp, traj_map_f... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | isProjectiveLimit_infinitePiNat | null |
infinitePiNat_map_restrict (I : Finset ℕ) :
(infinitePiNat μ).map I.restrict = Measure.pi fun i : I ↦ μ i :=
isProjectiveLimit_infinitePiNat μ I | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePiNat_map_restrict | Restricting the product measure to a product indexed by a finset yields the usual
product measure. |
piContent_eq_infinitePiNat {A : Set (Π n, X n)} (hA : A ∈ measurableCylinders X) :
piContent μ A = infinitePiNat μ A := by
obtain ⟨s, S, mS, rfl⟩ : ∃ s S, MeasurableSet S ∧ A = cylinder s S := by
simpa [mem_measurableCylinders] using hA
rw [piContent_cylinder _ mS, cylinder, ← map_apply (measurable_restrict... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | piContent_eq_infinitePiNat | null |
Measure.infinitePiNat_map_piCongrLeft (e : ℕ ≃ ι) {s : Set (Π i, X i)}
(hs : s ∈ measurableCylinders X) :
(infinitePiNat (fun n ↦ μ (e n))).map (piCongrLeft X e) s = piContent μ s := by
obtain ⟨I, S, hS, rfl⟩ := (mem_measurableCylinders s).1 hs
rw [map_apply _ hS.cylinder, cylinder, ← Set.preimage_comp, coe... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | Measure.infinitePiNat_map_piCongrLeft | If we push the product measure forward by a reindexing equivalence, we get a product measure
on the reindexed product in the sense that it coincides with `piContent μ` over
measurable cylinders. See `infinitePi_map_piCongrLeft` for a general version. |
piContent_tendsto_zero {A : ℕ → Set (Π i, X i)} (A_mem : ∀ n, A n ∈ measurableCylinders X)
(A_anti : Antitone A) (A_inter : ⋂ n, A n = ∅) :
Tendsto (fun n ↦ piContent μ (A n)) atTop (𝓝 0) := by
have : ∀ i, Nonempty (X i) := fun i ↦ ProbabilityMeasure.nonempty ⟨μ i, hμ i⟩
have A_cyl n : ∃ s S, MeasurableSet... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | piContent_tendsto_zero | This is the key theorem to build the product of an arbitrary family of probability measures:
the `piContent` of a decreasing sequence of cylinders with empty intersection converges to `0`.
This implies the `σ`-additivity of `piContent` (see `addContent_iUnion_eq_sum_of_tendsto_zero`),
which allows to extend it to the ... |
isSigmaSubadditive_piContent : (piContent μ).IsSigmaSubadditive := by
refine isSigmaSubadditive_of_addContent_iUnion_eq_tsum
isSetRing_measurableCylinders (fun f hf hf_Union hf' ↦ ?_)
exact addContent_iUnion_eq_sum_of_tendsto_zero isSetRing_measurableCylinders
(piContent μ) (fun s hs ↦ projectiveFamilyConte... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | isSigmaSubadditive_piContent | The `projectiveFamilyContent` associated to a family of probability measures is
σ-subadditive. |
noncomputable infinitePi : Measure (Π i, X i) :=
(piContent μ).measure isSetSemiring_measurableCylinders
generateFrom_measurableCylinders.ge (isSigmaSubadditive_piContent μ) | def | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi | The product measure of an arbitrary family of probability measures. It is defined as the unique
extension of the function which gives to cylinders the measure given by the associated product
measure. |
isProjectiveLimit_infinitePi :
IsProjectiveLimit (infinitePi μ) (fun I : Finset ι ↦ (Measure.pi (fun i : I ↦ μ i))) := by
intro I
ext s hs
rw [map_apply (measurable_restrict I) hs, infinitePi, AddContent.measure_eq, ← cylinder,
piContent_cylinder μ hs]
· exact generateFrom_measurableCylinders.symm
· e... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | isProjectiveLimit_infinitePi | The product measure is the projective limit of the partial product measures. This ensures
uniqueness and expresses the value of the product measure applied to cylinders. |
infinitePi_map_restrict {I : Finset ι} :
(Measure.infinitePi μ).map I.restrict = Measure.pi fun i : I ↦ μ i :=
isProjectiveLimit_infinitePi μ I | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi_map_restrict | Restricting the product measure to a product indexed by a finset yields the usual
product measure. |
eq_infinitePi {ν : Measure (Π i, X i)}
(hν : ∀ s : Finset ι, ∀ t : (i : ι) → Set (X i),
(∀ i ∈ s, MeasurableSet (t i)) → ν (Set.pi s t) = ∏ i ∈ s, μ i (t i)) :
ν = infinitePi μ := by
refine (isProjectiveLimit_infinitePi μ).unique ?_ |>.symm
refine fun s ↦ (pi_eq fun t ht ↦ ?_).symm
classical
rw [M... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | eq_infinitePi | To prove that a measure is equal to the product measure it is enough to check that it
it gives the same measure to measurable boxes. |
infinitePi_pi {s : Finset ι} {t : (i : ι) → Set (X i)}
(mt : ∀ i ∈ s, MeasurableSet (t i)) :
infinitePi μ (Set.pi s t) = ∏ i ∈ s, (μ i) (t i) := by
have : Set.pi s t = cylinder s ((@Set.univ s).pi (fun i : s ↦ t i)) := by
ext x
simp
rw [this, cylinder, ← map_apply, infinitePi_map_restrict, pi_pi]
... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi_pi | null |
_root_.measurePreserving_eval_infinitePi (i : ι) :
MeasurePreserving (Function.eval i) (infinitePi μ) (μ i) where
measurable := by fun_prop
map_eq := by
ext s hs
have : @Function.eval ι X i =
(@Function.eval ({i} : Finset ι) (fun j ↦ X j) ⟨i, by simp⟩) ∘
(Finset.restrict {i}) := by ext; ... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | _root_.measurePreserving_eval_infinitePi | null |
infinitePi_map_pi {Y : ι → Type*} [∀ i, MeasurableSpace (Y i)] {f : (i : ι) → X i → Y i}
(hf : ∀ i, Measurable (f i)) :
haveI (i : ι) : IsProbabilityMeasure ((μ i).map (f i)) :=
isProbabilityMeasure_map (hf i).aemeasurable
(infinitePi μ).map (fun x i ↦ f i (x i)) = infinitePi (fun i ↦ (μ i).map (f i))... | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi_map_pi | null |
infinitePi_map_piCongrLeft {α : Type*} (e : α ≃ ι) :
(infinitePi (fun i ↦ μ (e i))).map (piCongrLeft X e) = infinitePi μ := by
refine eq_infinitePi μ fun s t ht ↦ ?_
conv_lhs => enter [2, 1]; rw [← e.image_preimage s, ← coe_preimage _ e.injective.injOn]
rw [map_apply, coe_piCongrLeft, Equiv.piCongrLeft_preima... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi_map_piCongrLeft | If we push the product measure forward by a reindexing equivalence, we get a product measure
on the reindexed product. |
infinitePi_eq_pi [Fintype ι] : infinitePi μ = Measure.pi μ := by
refine (pi_eq fun s hs ↦ ?_).symm
rw [← coe_univ, infinitePi_pi]
simpa | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi_eq_pi | null |
infinitePi_cylinder {s : Finset ι} {S : Set (Π i : s, X i)} (mS : MeasurableSet S) :
infinitePi μ (cylinder s S) = Measure.pi (fun i : s ↦ μ i) S := by
rw [cylinder, ← Measure.map_apply (measurable_restrict _) mS, infinitePi_map_restrict] | lemma | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | infinitePi_cylinder | null |
integral_restrict_infinitePi {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{s : Finset ι} {f : (Π i : s, X i) → E}
(hf : AEStronglyMeasurable f (Measure.pi (fun i : s ↦ μ i))) :
∫ y, f (s.restrict y) ∂infinitePi μ = ∫ y, f y ∂Measure.pi (fun i : s ↦ μ i) := by
rw [← integral_map, infinitePi_map_res... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | integral_restrict_infinitePi | null |
lintegral_restrict_infinitePi {s : Finset ι}
{f : (Π i : s, X i) → ℝ≥0∞} (hf : Measurable f) :
∫⁻ y, f (s.restrict y) ∂infinitePi μ = ∫⁻ y, f y ∂Measure.pi (fun i : s ↦ μ i) := by
rw [← lintegral_map hf (measurable_restrict _), isProjectiveLimit_infinitePi μ]
open Filtration | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | lintegral_restrict_infinitePi | null |
integral_infinitePi_of_piFinset [DecidableEq ι] {E : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] {s : Finset ι} {f : (Π i, X i) → E}
(mf : StronglyMeasurable[piFinset s] f) (x : Π i, X i) :
∫ y, f y ∂infinitePi μ =
∫ y, f (Function.updateFinset x s y) ∂Measure.pi (fun i : s ↦ μ i) := by
let g : (Π... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | integral_infinitePi_of_piFinset | null |
lintegral_infinitePi_of_piFinset [DecidableEq ι] {s : Finset ι}
{f : (Π i, X i) → ℝ≥0∞} (mf : Measurable[piFinset s] f)
(x : Π i, X i) : ∫⁻ y, f y ∂infinitePi μ = (∫⋯∫⁻_s, f ∂μ) x := by
let g : (Π i : s, X i) → ℝ≥0∞ := fun y ↦ f (Function.updateFinset x _ y)
have this y : g (s.restrict y) = f y :=
mf.de... | theorem | Probability | [
"Mathlib.Probability.Kernel.Composition.MeasureComp",
"Mathlib.Probability.Kernel.IonescuTulcea.Traj"
] | Mathlib/Probability/ProductMeasure.lean | lintegral_infinitePi_of_piFinset | null |
truncation (f : α → ℝ) (A : ℝ) :=
indicator (Set.Ioc (-A) A) id ∘ f
variable {m : MeasurableSpace α} {μ : Measure α} {f : α → ℝ} | def | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | truncation | Truncating a real-valued function to the interval `(-A, A]`. |
_root_.MeasureTheory.AEStronglyMeasurable.truncation (hf : AEStronglyMeasurable f μ)
{A : ℝ} : AEStronglyMeasurable (truncation f A) μ := by
apply AEStronglyMeasurable.comp_aemeasurable _ hf.aemeasurable
exact (stronglyMeasurable_id.indicator measurableSet_Ioc).aestronglyMeasurable | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | _root_.MeasureTheory.AEStronglyMeasurable.truncation | null |
abs_truncation_le_bound (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |A| := by
simp only [truncation, Set.indicator, id, Function.comp_apply]
split_ifs with h
· exact abs_le_abs h.2 (neg_le.2 h.1.le)
· simp [abs_nonneg]
@[simp] | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | abs_truncation_le_bound | null |
truncation_zero (f : α → ℝ) : truncation f 0 = 0 := by simp [truncation]; rfl | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | truncation_zero | null |
abs_truncation_le_abs_self (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |f x| := by
simp only [truncation, indicator, id, Function.comp_apply]
split_ifs
· exact le_rfl
· simp [abs_nonneg] | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | abs_truncation_le_abs_self | null |
truncation_eq_self {f : α → ℝ} {A : ℝ} {x : α} (h : |f x| < A) :
truncation f A x = f x := by
simp only [truncation, indicator, id, Function.comp_apply, ite_eq_left_iff]
intro H
apply H.elim
simp [(abs_lt.1 h).1, (abs_lt.1 h).2.le] | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | truncation_eq_self | null |
truncation_eq_of_nonneg {f : α → ℝ} {A : ℝ} (h : ∀ x, 0 ≤ f x) :
truncation f A = indicator (Set.Ioc 0 A) id ∘ f := by
ext x
rcases (h x).lt_or_eq with (hx | hx)
· simp only [truncation, indicator, hx, Set.mem_Ioc, id, Function.comp_apply]
by_cases h'x : f x ≤ A
· have : -A < f x := by linarith [h x]
... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | truncation_eq_of_nonneg | null |
truncation_nonneg {f : α → ℝ} (A : ℝ) {x : α} (h : 0 ≤ f x) : 0 ≤ truncation f A x :=
Set.indicator_apply_nonneg fun _ => h | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | truncation_nonneg | null |
_root_.MeasureTheory.AEStronglyMeasurable.memLp_truncation [IsFiniteMeasure μ]
(hf : AEStronglyMeasurable f μ) {A : ℝ} {p : ℝ≥0∞} : MemLp (truncation f A) p μ :=
MemLp.of_bound hf.truncation |A| (Eventually.of_forall fun _ => abs_truncation_le_bound _ _ _) | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | _root_.MeasureTheory.AEStronglyMeasurable.memLp_truncation | null |
_root_.MeasureTheory.AEStronglyMeasurable.integrable_truncation [IsFiniteMeasure μ]
(hf : AEStronglyMeasurable f μ) {A : ℝ} : Integrable (truncation f A) μ := by
rw [← memLp_one_iff_integrable]; exact hf.memLp_truncation | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | _root_.MeasureTheory.AEStronglyMeasurable.integrable_truncation | null |
moment_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A)
{n : ℕ} (hn : n ≠ 0) : ∫ x, truncation f A x ^ n ∂μ = ∫ y in -A..A, y ^ n ∂Measure.map f μ := by
have M : MeasurableSet (Set.Ioc (-A) A) := measurableSet_Ioc
change ∫ x, (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (f x... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | moment_truncation_eq_intervalIntegral | null |
moment_truncation_eq_intervalIntegral_of_nonneg (hf : AEStronglyMeasurable f μ) {A : ℝ}
{n : ℕ} (hn : n ≠ 0) (h'f : 0 ≤ f) :
∫ x, truncation f A x ^ n ∂μ = ∫ y in 0..A, y ^ n ∂Measure.map f μ := by
have M : MeasurableSet (Set.Ioc 0 A) := measurableSet_Ioc
have M' : MeasurableSet (Set.Ioc A 0) := measurableS... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | moment_truncation_eq_intervalIntegral_of_nonneg | null |
integral_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f μ) {A : ℝ}
(hA : 0 ≤ A) : ∫ x, truncation f A x ∂μ = ∫ y in -A..A, y ∂Measure.map f μ := by
simpa using moment_truncation_eq_intervalIntegral hf hA one_ne_zero | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | integral_truncation_eq_intervalIntegral | null |
integral_truncation_eq_intervalIntegral_of_nonneg (hf : AEStronglyMeasurable f μ) {A : ℝ}
(h'f : 0 ≤ f) : ∫ x, truncation f A x ∂μ = ∫ y in 0..A, y ∂Measure.map f μ := by
simpa using moment_truncation_eq_intervalIntegral_of_nonneg hf one_ne_zero h'f | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | integral_truncation_eq_intervalIntegral_of_nonneg | null |
integral_truncation_le_integral_of_nonneg (hf : Integrable f μ) (h'f : 0 ≤ f) {A : ℝ} :
∫ x, truncation f A x ∂μ ≤ ∫ x, f x ∂μ := by
apply integral_mono_of_nonneg
(Eventually.of_forall fun x => ?_) hf (Eventually.of_forall fun x => ?_)
· exact truncation_nonneg _ (h'f x)
· calc
truncation f A x ≤ |t... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | integral_truncation_le_integral_of_nonneg | null |
tendsto_integral_truncation {f : α → ℝ} (hf : Integrable f μ) :
Tendsto (fun A => ∫ x, truncation f A x ∂μ) atTop (𝓝 (∫ x, f x ∂μ)) := by
refine tendsto_integral_filter_of_dominated_convergence (fun x => abs (f x)) ?_ ?_ ?_ ?_
· exact Eventually.of_forall fun A ↦ hf.aestronglyMeasurable.truncation
· filter_u... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | tendsto_integral_truncation | If a function is integrable, then the integral of its truncated versions converges to the
integral of the whole function. |
IdentDistrib.truncation {β : Type*} [MeasurableSpace β] {ν : Measure β} {f : α → ℝ}
{g : β → ℝ} (h : IdentDistrib f g μ ν) {A : ℝ} :
IdentDistrib (truncation f A) (truncation g A) μ ν :=
h.comp (measurable_id.indicator measurableSet_Ioc) | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | IdentDistrib.truncation | null |
sum_prob_mem_Ioc_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) {K : ℕ} {N : ℕ}
(hKN : K ≤ N) :
∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc (j : ℝ) N} ≤ ENNReal.ofReal (𝔼[X] + 1) := by
let ρ : Measure ℝ := Measure.map X ℙ
haveI : IsProbabilityMeasure ρ := Measure.isProbabilityMeasure_map hint.aemeasurable
... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | sum_prob_mem_Ioc_le | null |
tsum_prob_mem_Ioi_lt_top {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) :
(∑' j : ℕ, ℙ {ω | X ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
suffices ∀ K : ℕ, ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi (j : ℝ)} ≤ ENNReal.ofReal (𝔼[X] + 1) from
(le_of_tendsto_of_tendsto (ENNReal.tendsto_nat_tsum _) tendsto_const_nhds
(E... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | tsum_prob_mem_Ioi_lt_top | null |
sum_variance_truncation_le {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) :
∑ j ∈ range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[truncation X j ^ 2] ≤ 2 * 𝔼[X] := by
set Y := fun n : ℕ => truncation X n
let ρ : Measure ℝ := Measure.map X ℙ
have Y2 : ∀ n, 𝔼[Y n ^ 2] = ∫ x in 0..n, x ^ 2 ∂ρ := by
intro n
... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | sum_variance_truncation_le | null |
strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) : ∀ᵐ ω, ∀ᶠ n : ℕ in atTop,
|∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω - 𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]| <
ε * ⌊c ^ n⌋₊ := by
/- Let `S n = ∑ i ∈ range n, Y i` where `Y i = truncation (X i) i`. We should show that
`|S k - 𝔼[S k]... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux1 | The truncation of `Xᵢ` up to `i` satisfies the strong law of large numbers (with respect to
the truncated expectation) along the sequence `c^n`, for any `c > 1`, up to a given `ε > 0`.
This follows from a variance control. |
strong_law_aux2 {c : ℝ} (c_one : 1 < c) :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω -
𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]) =o[atTop] fun n : ℕ => (⌊c ^ n⌋₊ : ℝ) := by
obtain ⟨v, -, v_pos, v_lim⟩ :
∃ v : ℕ → ℝ, StrictAnti v ∧ (∀ n : ℕ, 0 < v n) ∧ Tendsto v atTop (𝓝 0) :=
... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux2 | The truncation of `Xᵢ` up to `i` satisfies the strong law of large numbers
(with respect to the truncated expectation) along the sequence
`c^n`, for any `c > 1`. This follows from `strong_law_aux1` by varying `ε`. |
strong_law_aux3 :
(fun n => 𝔼[∑ i ∈ range n, truncation (X i) i] - n * 𝔼[X 0]) =o[atTop] ((↑) : ℕ → ℝ) := by
have A : Tendsto (fun i => 𝔼[truncation (X i) i]) atTop (𝓝 𝔼[X 0]) := by
convert (tendsto_integral_truncation hint).comp tendsto_natCast_atTop_atTop using 1
ext i
exact (hident i).truncati... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux3 | The expectation of the truncated version of `Xᵢ` behaves asymptotically like the whole
expectation. This follows from convergence and Cesàro averaging. |
strong_law_aux4 {c : ℝ} (c_one : 1 < c) :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω - ⌊c ^ n⌋₊ * 𝔼[X 0]) =o[atTop]
fun n : ℕ => (⌊c ^ n⌋₊ : ℝ) := by
filter_upwards [strong_law_aux2 X hint hindep hident hnonneg c_one] with ω hω
have A : Tendsto (fun n : ℕ => ⌊c ^ n⌋₊) atTop atTop :=
... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux4 | The truncation of `Xᵢ` up to `i` satisfies the strong law of large numbers
(with respect to the original expectation) along the sequence
`c^n`, for any `c > 1`. This follows from the version from the truncated expectation, and the
fact that the truncated and the original expectations have the same asymptotic behavior. |
strong_law_aux5 :
∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range n, truncation (X i) i ω - ∑ i ∈ range n, X i ω) =o[atTop]
fun n : ℕ => (n : ℝ) := by
have A : (∑' j : ℕ, ℙ {ω | X j ω ∈ Set.Ioi (j : ℝ)}) < ∞ := by
convert tsum_prob_mem_Ioi_lt_top hint (hnonneg 0) using 2
ext1 j
exact (hident j).measure_mem_eq mea... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux5 | The truncated and non-truncated versions of `Xᵢ` have the same asymptotic behavior, as they
almost surely coincide at all but finitely many steps. This follows from a probability computation
and Borel-Cantelli. |
strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i ∈ range ⌊c ^ n⌋₊, X i ω) / ⌊c ^ n⌋₊) atTop (𝓝 𝔼[X 0]) := by
have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
intro n
refine zero_lt_one.trans_le ?_
simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow₀ c_one.le]
filter_up... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux6 | `Xᵢ` satisfies the strong law of large numbers along the sequence
`c^n`, for any `c > 1`. This follows from the version for the truncated `Xᵢ`, and the fact that
`Xᵢ` and its truncated version have the same asymptotic behavior. |
strong_law_aux7 :
∀ᵐ ω, Tendsto (fun n : ℕ => (∑ i ∈ range n, X i ω) / n) atTop (𝓝 𝔼[X 0]) := by
obtain ⟨c, -, cone, clim⟩ :
∃ c : ℕ → ℝ, StrictAnti c ∧ (∀ n : ℕ, 1 < c n) ∧ Tendsto c atTop (𝓝 1) :=
exists_seq_strictAnti_tendsto (1 : ℝ)
have : ∀ k, ∀ᵐ ω,
Tendsto (fun n : ℕ => (∑ i ∈ range ⌊c ... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_aux7 | `Xᵢ` satisfies the strong law of large numbers along all integers. This follows from the
corresponding fact along the sequences `c^n`, and the fact that any integer can be sandwiched
between `c^n` and `c^(n+1)` with comparably small error if `c` is close enough to `1`
(which is formalized in `tendsto_div_of_monotone_of... |
strong_law_ae_real {Ω : Type*} {m : MeasurableSpace Ω} {μ : Measure Ω}
(X : ℕ → Ω → ℝ) (hint : Integrable (X 0) μ)
(hindep : Pairwise ((IndepFun · · μ) on X))
(hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) :
∀ᵐ ω ∂μ, Tendsto (fun n : ℕ => (∑ i ∈ range n, X i ω) / n) atTop (𝓝 μ[X 0]) := by
let mΩ : Meas... | theorem | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_ae_real | **Strong law of large numbers**, almost sure version: if `X n` is a sequence of independent
identically distributed integrable real-valued random variables, then `∑ i ∈ range n, X i / n`
converges almost surely to `𝔼[X 0]`. We give here the strong version, due to Etemadi, that only
requires pairwise independence. Supe... |
strong_law_ae_simpleFunc_comp (X : ℕ → Ω → E) (h' : Measurable (X 0))
(hindep : Pairwise ((IndepFun · · μ) on X))
(hident : ∀ i, IdentDistrib (X i) (X 0) μ μ) (φ : SimpleFunc E E) :
∀ᵐ ω ∂μ,
Tendsto (fun n : ℕ ↦ (n : ℝ) ⁻¹ • (∑ i ∈ range n, φ (X i ω))) atTop (𝓝 μ[φ ∘ (X 0)]) := by
classical
refin... | lemma | Probability | [
"Mathlib.Probability.IdentDistrib",
"Mathlib.Probability.Independence.Integrable",
"Mathlib.MeasureTheory.Integral.DominatedConvergence",
"Mathlib.Analysis.SpecificLimits.FloorPow",
"Mathlib.Analysis.PSeries",
"Mathlib.Analysis.Asymptotics.SpecificAsymptotics"
] | Mathlib/Probability/StrongLaw.lean | strong_law_ae_simpleFunc_comp | Preliminary lemma for the strong law of large numbers for vector-valued random variables:
the composition of the random variables with a simple function satisfies the strong law of large
numbers. |
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