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 real-valued random variables, then ∑ i in 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 in range n, X i / n converges in Lᵖ to 𝔼[X 0] provided X n is independent identically distributed and is Lᵖ.

Implementation

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

• Almost surely, Xₙ = Yₙ for all but finitely many indices. Indeed, ∑ ℙ (Xₙ ≠ Yₙ) is bounded by 1 + 𝔼[X] (see sum_prob_mem_Ioc_le and tsum_prob_mem_Ioi_lt_top).
• Let c > 1. Along the sequence n = c ^ k, then (∑_{i=0}^{n-1} Yᵢ - 𝔼[Yᵢ])/n converges almost surely to 0. This follows from a variance control, as

  ∑_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`)

• As 𝔼[Yᵢ] converges to 𝔼[X], it follows from the two previous items and Cesàro that, along the sequence n = c^k, one has (∑_{i=0}^{n-1} Xᵢ) / n → 𝔼[X] almost surely.
• To generalize it to all indices, we use the fact that ∑_{i=0}^{n-1} Xᵢ is nondecreasing and that, if c is close enough to 1, the gap between c^k and c^(k+1) is small.

Prerequisites on truncations

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

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

theorem MeasureTheory.AEStronglyMeasurable.truncation {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} : MeasureTheory.AEStronglyMeasurable (ProbabilityTheory.truncation f A) μ

theorem ProbabilityTheory.abs_truncation_le_bound {α : Type u_1} (f : α → ℝ) (A : ℝ) (x : α) : |ProbabilityTheory.truncation f A x| ≤ |A|

@[simp]

theorem ProbabilityTheory.truncation_zero {α : Type u_1} (f : α → ℝ) : ProbabilityTheory.truncation f 0 = 0

theorem ProbabilityTheory.abs_truncation_le_abs_self {α : Type u_1} (f : α → ℝ) (A : ℝ) (x : α) : |ProbabilityTheory.truncation f A x| ≤ |f x|

theorem ProbabilityTheory.truncation_eq_self {α : Type u_1} {f : α → ℝ} {A : ℝ} {x : α} (h : |f x| < A) : ProbabilityTheory.truncation f A x = f x

theorem ProbabilityTheory.truncation_eq_of_nonneg {α : Type u_1} {f : α → ℝ} {A : ℝ} (h : ∀ (x : α), 0 ≤ f x) : ProbabilityTheory.truncation f A = Set.indicator (Set.Ioc 0 A) id ∘ f

theorem ProbabilityTheory.truncation_nonneg {α : Type u_1} {f : α → ℝ} (A : ℝ) {x : α} (h : 0 ≤ f x) : 0 ≤ ProbabilityTheory.truncation f A x

theorem MeasureTheory.AEStronglyMeasurable.memℒp_truncation {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} {p : ENNReal} : MeasureTheory.Memℒp (ProbabilityTheory.truncation f A) p

theorem MeasureTheory.AEStronglyMeasurable.integrable_truncation {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} : MeasureTheory.Integrable (ProbabilityTheory.truncation f A)

theorem ProbabilityTheory.moment_truncation_eq_intervalIntegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A) {n : ℕ} (hn : n ≠ 0) : ∫ (x : α), ProbabilityTheory.truncation f A x ^ n ∂μ = ∫ (y : ℝ) in -A..A, y ^ n ∂MeasureTheory.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 → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} {n : ℕ} (hn : n ≠ 0) (h'f : 0 ≤ f) : ∫ (x : α), ProbabilityTheory.truncation f A x ^ n ∂μ = ∫ (y : ℝ) in 0 ..A, y ^ n ∂MeasureTheory.Measure.map f μ

theorem ProbabilityTheory.integral_truncation_eq_intervalIntegral {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A) : ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ = ∫ (y : ℝ) in -A..A, y ∂MeasureTheory.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 → ℝ} (hf : MeasureTheory.AEStronglyMeasurable f μ) {A : ℝ} (h'f : 0 ≤ f) : ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ = ∫ (y : ℝ) in 0 ..A, y ∂MeasureTheory.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 → ℝ} (hf : MeasureTheory.Integrable f) (h'f : 0 ≤ f) {A : ℝ} : ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ ≤ ∫ (x : α), f x ∂μ

theorem ProbabilityTheory.tendsto_integral_truncation {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf : MeasureTheory.Integrable f) : Filter.Tendsto (fun A => ∫ (x : α), ProbabilityTheory.truncation f A x ∂μ) Filter.atTop (nhds (∫ (x : α), f x ∂μ))

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

theorem ProbabilityTheory.IdentDistrib.truncation {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_2} [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α → ℝ} {g : β → ℝ} (h : ProbabilityTheory.IdentDistrib f g) {A : ℝ} : ProbabilityTheory.IdentDistrib (ProbabilityTheory.truncation f A) (ProbabilityTheory.truncation g A)

theorem ProbabilityTheory.sum_prob_mem_Ioc_le {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X : Ω → ℝ} (hint : MeasureTheory.Integrable X) (hnonneg : 0 ≤ X) {K : ℕ} {N : ℕ} (hKN : K ≤ N) : (Finset.sum (Finset.range K) fun j => ↑↑MeasureTheory.volume {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 → ℝ} (hint : MeasureTheory.Integrable X) (hnonneg : 0 ≤ X) : ∑' (j : ℕ), ↑↑MeasureTheory.volume {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 → ℝ} (hint : MeasureTheory.Integrable X) (hnonneg : 0 ≤ X) (K : ℕ) : (Finset.sum (Finset.range K) fun j => (↑j ^ 2)⁻¹ * ∫ (a : Ω), (ProbabilityTheory.truncation X ↑j ^ 2) a) ≤ 2 * ∫ (a : Ω), X a

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

theorem ProbabilityTheory.strong_law_aux2 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) {c : ℝ} (c_one : 1 < c) : ∀ᵐ (ω : Ω), (fun n => (Finset.sum (Finset.range ⌊c ^ n⌋₊) fun i => ProbabilityTheory.truncation (X i) (↑i) ω) - ∫ (a : Ω), Finset.sum (Finset.range ⌊c ^ n⌋₊) (fun i => 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 : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) : (fun n => (∫ (a : Ω), Finset.sum (Finset.range n) (fun i => 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 : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) {c : ℝ} (c_one : 1 < c) : ∀ᵐ (ω : Ω), (fun n => (Finset.sum (Finset.range ⌊c ^ n⌋₊) fun i => 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 : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) : ∀ᵐ (ω : Ω), (fun n => (Finset.sum (Finset.range n) fun i => ProbabilityTheory.truncation (X i) (↑i) ω) - Finset.sum (Finset.range n) fun i => 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 : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) {c : ℝ} (c_one : 1 < c) : ∀ᵐ (ω : Ω), Filter.Tendsto (fun n => (Finset.sum (Finset.range ⌊c ^ n⌋₊) fun i => 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 : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) (hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω) : ∀ᵐ (ω : Ω), Filter.Tendsto (fun n => (Finset.sum (Finset.range n) fun i => 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 {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X : ℕ → Ω → ℝ) (hint : MeasureTheory.Integrable (X 0)) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) : ∀ᵐ (ω : Ω), Filter.Tendsto (fun n => (Finset.sum (Finset.range n) fun i => 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 in 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.

theorem ProbabilityTheory.strong_law_Lp {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {p : ENNReal} (hp : 1 ≤ p) (hp' : p ≠ ⊤) (X : ℕ → Ω → ℝ) (hℒp : MeasureTheory.Memℒp (X 0) p) (hindep : Pairwise fun i j => ProbabilityTheory.IndepFun (X i) (X j)) (hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0)) : Filter.Tendsto (fun n => MeasureTheory.snorm (fun ω => (Finset.sum (Finset.range n) fun i => X i ω) / ↑n - ∫ (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 real-valued random variables in Lᵖ, then ∑ i in range n, X i / n converges in Lᵖ to 𝔼[X 0].