Documentation

Mathlib.Probability.StrongLaw

The strong law of large numbers #

We prove the strong law of large numbers, in ProbabilityTheory.strong_law_ae: If X n is a sequence of independent identically distributed integrable 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.

This file also contains the Lᵖ version of the strong law of large numbers provided by ProbabilityTheory.strong_law_Lp which shows ∑ i ∈ range n, X i / n converges in Lᵖ to 𝔼[X 0] provided X n is independent identically distributed and is Lᵖ.

Implementation #

The main point is to prove the result for real-valued random variables, as the general case of Banach-space valued random variables follows from this case and approximation by simple functions. The real version is given in ProbabilityTheory.strong_law_ae_real.

We follow the proof by Etemadi [Etemadi, An elementary proof of the strong law of large numbers][etemadi_strong_law], which goes as follows.

It suffices to prove the result for nonnegative X, as one can prove the general result by splitting a general X into its positive part and negative part. Consider Xₙ a sequence of nonnegative integrable identically distributed pairwise independent random variables. Let Yₙ be the truncation of Xₙ up to n. We claim that

  ∑_k ℙ (|∑_{i=0}^{c^k - 1} Yᵢ - 𝔼[Yᵢ]| > c^k ε)
    ≤ ∑_k (c^k ε)^{-2} ∑_{i=0}^{c^k - 1} Var[Yᵢ]    (by Markov inequality)
    ≤ ∑_i (C/i^2) Var[Yᵢ]                           (as ∑_{c^k > i} 1/(c^k)^2 ≤ C/i^2)
    ≤ ∑_i (C/i^2) 𝔼[Yᵢ^2]
    ≤ 2C 𝔼[X^2]                                     (see `sum_variance_truncation_le`)

Prerequisites on truncations #

def ProbabilityTheory.truncation {α : Type u_1} (f : α) (A : ) :
α

Truncating a real-valued function to the interval (-A, A].

Equations
Instances For
    theorem ProbabilityTheory.abs_truncation_le_bound {α : Type u_1} (f : α) (A : ) (x : α) :
    @[simp]
    theorem ProbabilityTheory.abs_truncation_le_abs_self {α : Type u_1} (f : α) (A : ) (x : α) :
    theorem ProbabilityTheory.truncation_eq_self {α : Type u_1} {f : α} {A : } {x : α} (h : |f x| < A) :
    theorem ProbabilityTheory.truncation_eq_of_nonneg {α : Type u_1} {f : α} {A : } (h : ∀ (x : α), 0 f x) :
    ProbabilityTheory.truncation f A = (Set.Ioc 0 A).indicator id f
    theorem ProbabilityTheory.truncation_nonneg {α : Type u_1} {f : α} (A : ) {x : α} (h : 0 f x) :
    theorem ProbabilityTheory.moment_truncation_eq_intervalIntegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : } (hA : 0 A) {n : } (hn : n 0) :
    ∫ (x : α), ProbabilityTheory.truncation f A x ^ nμ = ∫ (y : ) in -A..A, y ^ nMeasureTheory.Measure.map f μ
    theorem ProbabilityTheory.moment_truncation_eq_intervalIntegral_of_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : } {n : } (hn : n 0) (h'f : 0 f) :
    ∫ (x : α), ProbabilityTheory.truncation f A x ^ nμ = ∫ (y : ) in 0 ..A, y ^ nMeasureTheory.Measure.map f μ
    theorem ProbabilityTheory.integral_truncation_eq_intervalIntegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : } (hA : 0 A) :
    ∫ (x : α), ProbabilityTheory.truncation f A xμ = ∫ (y : ) in -A..A, yMeasureTheory.Measure.map f μ
    theorem ProbabilityTheory.integral_truncation_eq_intervalIntegral_of_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : } (h'f : 0 f) :
    ∫ (x : α), ProbabilityTheory.truncation f A xμ = ∫ (y : ) in 0 ..A, yMeasureTheory.Measure.map f μ
    theorem ProbabilityTheory.integral_truncation_le_integral_of_nonneg {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.Integrable f μ) (h'f : 0 f) {A : } :
    ∫ (x : α), ProbabilityTheory.truncation f A xμ ∫ (x : α), f xμ
    theorem ProbabilityTheory.tendsto_integral_truncation {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α} (hf : MeasureTheory.Integrable f μ) :
    Filter.Tendsto (fun (A : ) => ∫ (x : α), ProbabilityTheory.truncation f A xμ) Filter.atTop (nhds (∫ (x : α), f xμ))

    If a function is integrable, then the integral of its truncated versions converges to the integral of the whole function.

    theorem ProbabilityTheory.sum_prob_mem_Ioc_le {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X : Ω} (hint : MeasureTheory.Integrable X MeasureTheory.volume) (hnonneg : 0 X) {K : } {N : } (hKN : K N) :
    jFinset.range K, MeasureTheory.volume {ω : Ω | X ω Set.Ioc j N} ENNReal.ofReal ((∫ (a : Ω), X a) + 1)
    theorem ProbabilityTheory.tsum_prob_mem_Ioi_lt_top {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X : Ω} (hint : MeasureTheory.Integrable X MeasureTheory.volume) (hnonneg : 0 X) :
    ∑' (j : ), MeasureTheory.volume {ω : Ω | X ω Set.Ioi j} <
    theorem ProbabilityTheory.sum_variance_truncation_le {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X : Ω} (hint : MeasureTheory.Integrable X MeasureTheory.volume) (hnonneg : 0 X) (K : ) :
    jFinset.range K, (j ^ 2)⁻¹ * ∫ (a : Ω), (ProbabilityTheory.truncation X j ^ 2) a 2 * ∫ (a : Ω), X a

    Proof of the strong law of large numbers (almost sure version, assuming only pairwise independence) for nonnegative random variables, following Etemadi's proof.

    theorem ProbabilityTheory.strong_law_aux1 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (hnonneg : ∀ (i : ) (ω : Ω), 0 X i ω) {c : } (c_one : 1 < c) {ε : } (εpos : 0 < ε) :
    ∀ᵐ (ω : Ω), ∀ᶠ (n : ) in Filter.atTop, |iFinset.range c ^ n⌋₊, ProbabilityTheory.truncation (X i) (i) ω - ∫ (a : Ω), (iFinset.range c ^ n⌋₊, ProbabilityTheory.truncation (X i) i) a| < ε * c ^ n⌋₊

    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.

    theorem ProbabilityTheory.strong_law_aux2 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (hnonneg : ∀ (i : ) (ω : Ω), 0 X i ω) {c : } (c_one : 1 < c) :
    ∀ᵐ (ω : Ω), (fun (n : ) => iFinset.range c ^ n⌋₊, ProbabilityTheory.truncation (X i) (i) ω - ∫ (a : Ω), (iFinset.range c ^ n⌋₊, ProbabilityTheory.truncation (X i) i) a) =o[Filter.atTop] fun (n : ) => c ^ n⌋₊
    theorem ProbabilityTheory.strong_law_aux3 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) :
    (fun (n : ) => (∫ (a : Ω), (iFinset.range n, ProbabilityTheory.truncation (X i) i) a) - n * ∫ (a : Ω), X 0 a) =o[Filter.atTop] Nat.cast

    The expectation of the truncated version of Xᵢ behaves asymptotically like the whole expectation. This follows from convergence and Cesàro averaging.

    theorem ProbabilityTheory.strong_law_aux4 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (hnonneg : ∀ (i : ) (ω : Ω), 0 X i ω) {c : } (c_one : 1 < c) :
    ∀ᵐ (ω : Ω), (fun (n : ) => iFinset.range c ^ n⌋₊, ProbabilityTheory.truncation (X i) (i) ω - c ^ n⌋₊ * ∫ (a : Ω), X 0 a) =o[Filter.atTop] fun (n : ) => c ^ n⌋₊
    theorem ProbabilityTheory.strong_law_aux5 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (hnonneg : ∀ (i : ) (ω : Ω), 0 X i ω) :
    ∀ᵐ (ω : Ω), (fun (n : ) => iFinset.range n, ProbabilityTheory.truncation (X i) (i) ω - iFinset.range n, X i ω) =o[Filter.atTop] fun (n : ) => n

    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.

    theorem ProbabilityTheory.strong_law_aux6 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (hnonneg : ∀ (i : ) (ω : Ω), 0 X i ω) {c : } (c_one : 1 < c) :
    ∀ᵐ (ω : Ω), Filter.Tendsto (fun (n : ) => (iFinset.range c ^ n⌋₊, X i ω) / c ^ n⌋₊) Filter.atTop (nhds (∫ (a : Ω), X 0 a))
    theorem ProbabilityTheory.strong_law_aux7 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (hnonneg : ∀ (i : ) (ω : Ω), 0 X i ω) :
    ∀ᵐ (ω : Ω), Filter.Tendsto (fun (n : ) => (iFinset.range n, X i ω) / n) Filter.atTop (nhds (∫ (a : Ω), X 0 a))

    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_tendsto_div_floor_pow).

    theorem ProbabilityTheory.strong_law_ae_real {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : Ω) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) :
    ∀ᵐ (ω : Ω), Filter.Tendsto (fun (n : ) => (iFinset.range n, X i ω) / n) Filter.atTop (nhds (∫ (a : Ω), X 0 a))

    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. Superseded by strong_law_ae, which works for random variables taking values in any Banach space.

    theorem ProbabilityTheory.strong_law_ae_simpleFunc_comp {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [MeasurableSpace E] (X : ΩE) (h' : Measurable (X 0)) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) (φ : MeasureTheory.SimpleFunc E E) :
    ∀ᵐ (ω : Ω), Filter.Tendsto (fun (n : ) => (n)⁻¹ iFinset.range n, φ (X i ω)) Filter.atTop (nhds (∫ (a : Ω), (φ X 0) a))

    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.

    theorem ProbabilityTheory.strong_law_ae_of_measurable {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] (X : ΩE) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (h' : MeasureTheory.StronglyMeasurable (X 0)) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) :
    ∀ᵐ (ω : Ω), Filter.Tendsto (fun (n : ) => (n)⁻¹ iFinset.range n, X i ω) Filter.atTop (nhds (∫ (a : Ω), X 0 a))

    Preliminary lemma for the strong law of large numbers for vector-valued random variables, assuming measurability in addition to integrability. This is weakened to ae measurability in the full version ProbabilityTheory.strong_law_ae.

    theorem ProbabilityTheory.strong_law_ae {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] (X : ΩE) (hint : MeasureTheory.Integrable (X 0) MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) :
    ∀ᵐ (ω : Ω), Filter.Tendsto (fun (n : ) => (n)⁻¹ iFinset.range n, X i ω) Filter.atTop (nhds (∫ (a : Ω), X 0 a))

    Strong law of large numbers, almost sure version: if X n is a sequence of independent identically distributed integrable random variables taking values in a Banach space, then n⁻¹ • ∑ i ∈ range n, X i converges almost surely to 𝔼[X 0]. We give here the strong version, due to Etemadi, that only requires pairwise independence.

    theorem ProbabilityTheory.strong_law_Lp {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] {p : ENNReal} (hp : 1 p) (hp' : p ) (X : ΩE) (hℒp : MeasureTheory.Memℒp (X 0) p MeasureTheory.volume) (hindep : Pairwise fun (i j : ) => ProbabilityTheory.IndepFun (X i) (X j) MeasureTheory.volume) (hident : ∀ (i : ), ProbabilityTheory.IdentDistrib (X i) (X 0) MeasureTheory.volume MeasureTheory.volume) :
    Filter.Tendsto (fun (n : ) => MeasureTheory.snorm (fun (ω : Ω) => (n)⁻¹ iFinset.range n, X i ω - ∫ (a : Ω), X 0 a) p MeasureTheory.volume) Filter.atTop (nhds 0)

    Strong law of large numbers, Lᵖ version: if X n is a sequence of independent identically distributed random variables in Lᵖ, then n⁻¹ • ∑ i ∈ range n, X i converges in Lᵖ to 𝔼[X 0].