On the Strong Law of Large Numbers for Sequences of Pairwise Independent   Random Variables

Valery Korchevsky

arXiv:1701.02234·math.PR·January 10, 2017

On the Strong Law of Large Numbers for Sequences of Pairwise Independent Random Variables

Valery Korchevsky

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TL;DR

This paper extends the strong law of large numbers to sequences of pairwise independent, non-identically distributed random variables, providing new conditions and generalizations, including results with arbitrary norming sequences.

Contribution

It introduces new sufficient conditions for the SLLN to hold for pairwise independent variables, generalizing previous results and allowing for arbitrary norming sequences.

Findings

01

Established new SLLN conditions for pairwise independent variables

02

Generalized Etemadi's extension of Kolmogorov's SLLN

03

Some results hold with arbitrary norming sequences

Abstract

We establish new sufficient conditions for the applicability of the strong law of large numbers (SLLN) for sequences of pairwise independent non-identically distributed random variables. These results generalize Etemadi's extension of Kolmogorov's SLLN for identically distributed random variables. Some of the obtained results hold with an arbitrary norming sequence in place of the classical normalization.

Equations62

\int_{0}^{\infty} G (x) d x < \infty,

\int_{0}^{\infty} G (x) d x < \infty,

\frac{1}{n} k = 1 \sum n c_{k} (X_{k} - E X_{k}) \to 0 \mbox a . s . (n \to \infty)

\frac{1}{n} k = 1 \sum n c_{k} (X_{k} - E X_{k}) \to 0 \mbox a . s . (n \to \infty)

\int_{0}^{\infty} G (x) d x < \infty \mbox w i t h G (x) = n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \mbox f or a l l x \geq 0,

\int_{0}^{\infty} G (x) d x < \infty \mbox w i t h G (x) = n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \mbox f or a l l x \geq 0,

n = 1 \sum \infty P (∣ X_{n} ∣ > n) < \infty.

n = 1 \sum \infty P (∣ X_{n} ∣ > n) < \infty.

\frac{S _{n} - E S _{n}}{n} \to 0 \mbox a . s .

\frac{S _{n} - E S _{n}}{n} \to 0 \mbox a . s .

n \geq 1 sup E ∣ X_{n} ∣ < \infty.

n \geq 1 sup E ∣ X_{n} ∣ < \infty.

n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \leq C P (∣ X ∣ > x) \mbox f or a l l x \geq 0,

n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \leq C P (∣ X ∣ > x) \mbox f or a l l x \geq 0,

\int_{0}^{\infty} G (x) d x < \infty \mbox w i t h G (x) = n \geq 1 sup \frac{1}{a _{n}} k = 1 \sum n P (∣ X_{k} ∣ > x) \mbox f or a l l x \geq 0,

\int_{0}^{\infty} G (x) d x < \infty \mbox w i t h G (x) = n \geq 1 sup \frac{1}{a _{n}} k = 1 \sum n P (∣ X_{k} ∣ > x) \mbox f or a l l x \geq 0,

n = 1 \sum \infty P (∣ X_{n} ∣ > a_{n}) < \infty.

n = 1 \sum \infty P (∣ X_{n} ∣ > a_{n}) < \infty.

\frac{S _{n} - E S _{n}}{a _{n}} \to 0 \mbox a . s .

\frac{S _{n} - E S _{n}}{a _{n}} \to 0 \mbox a . s .

\int_{0}^{\infty} H (x) d x < \infty, \mbox an d n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \leq H (x) \mbox f or a l l x \geq 0,

\int_{0}^{\infty} H (x) d x < \infty, \mbox an d n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \leq H (x) \mbox f or a l l x \geq 0,

\frac{S _{n} - E S _{n}}{n} \to 0 \mbox a . s .

\frac{S _{n} - E S _{n}}{n} \to 0 \mbox a . s .

n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \leq C P (∣ X ∣ > x) \mbox f or a l l x \geq 0,

n \geq 1 sup \frac{1}{n} k = 1 \sum n P (∣ X_{k} ∣ > x) \leq C P (∣ X ∣ > x) \mbox f or a l l x \geq 0,

\frac{S _{n} - E S _{n}}{n} \to 0 \mbox a . s .

\frac{S _{n} - E S _{n}}{n} \to 0 \mbox a . s .

V a r (S_{n}) \leq C k = 1 \sum n V a r (X_{k}) \mbox f or a l l n \geq 1,

V a r (S_{n}) \leq C k = 1 \sum n V a r (X_{k}) \mbox f or a l l n \geq 1,

n = 1 \sum \infty \frac{V a r ( X _{n} )}{a _{n}^{2}} < \infty,

n = 1 \sum \infty \frac{V a r ( X _{n} )}{a _{n}^{2}} < \infty,

n \geq 1 sup \frac{1}{a _{n}} k = 1 \sum n E X_{n} < \infty.

n \geq 1 sup \frac{1}{a _{n}} k = 1 \sum n E X_{n} < \infty.

\frac{S _{n} - E S _{n}}{a _{n}} \to 0 \mbox a . s .

\frac{S _{n} - E S _{n}}{a _{n}} \to 0 \mbox a . s .

\frac{E S _{n} - E T _{n}}{a _{n}} \to 0 (n \to \infty),

\frac{E S _{n} - E T _{n}}{a _{n}} \to 0 (n \to \infty),

\frac{T _{n} - E T _{n}}{a _{n}} \to 0 \mbox a . s .,

\frac{T _{n} - E T _{n}}{a _{n}} \to 0 \mbox a . s .,

\frac{S _{n} - T _{n}}{a _{n}} \to 0 \mbox a . s .

\frac{S _{n} - T _{n}}{a _{n}} \to 0 \mbox a . s .

E (Z I_{{Z > a}}) = a P (Z > a) + \int_{a}^{\infty} P (Z > x) d x

E (Z I_{{Z > a}}) = a P (Z > a) + \int_{a}^{\infty} P (Z > x) d x

E (Z I_{{Z \leq a}}) \leq \int_{0}^{a} P (Z > x) d x .

E (Z I_{{Z \leq a}}) \leq \int_{0}^{a} P (Z > x) d x .

E S_{n} - E T_{n} = k = 1 \sum n E (X_{k} I_{{X_{k} > a_{k}}}) = k = 1 \sum n a_{k} P (X_{k} > a_{k}) + k = 1 \sum n \int_{a_{k}}^{\infty} P (X_{k} > x) d x = k = 1 \sum n a_{k} P (X_{k} > a_{k}) + k = 1 \sum N \int_{a_{k}}^{\infty} P (X_{k} > x) d x + k = N + 1 \sum n \int_{a_{k}}^{\infty} P (X_{k} > x) d x \leq k = 1 \sum n a_{k} P (X_{k} > a_{k}) + k = 1 \sum N \int_{0}^{\infty} P (X_{k} > x) d x + k = 1 \sum n \int_{a_{N}}^{\infty} P (X_{k} > x) d x \leq k = 1 \sum n a_{k} P (X_{k} > a_{k}) + a_{N} \int_{0}^{\infty} G (x) d x + a_{n} \int_{a_{N}}^{\infty} G (x) d x .

E S_{n} - E T_{n} = k = 1 \sum n E (X_{k} I_{{X_{k} > a_{k}}}) = k = 1 \sum n a_{k} P (X_{k} > a_{k}) + k = 1 \sum n \int_{a_{k}}^{\infty} P (X_{k} > x) d x = k = 1 \sum n a_{k} P (X_{k} > a_{k}) + k = 1 \sum N \int_{a_{k}}^{\infty} P (X_{k} > x) d x + k = N + 1 \sum n \int_{a_{k}}^{\infty} P (X_{k} > x) d x \leq k = 1 \sum n a_{k} P (X_{k} > a_{k}) + k = 1 \sum N \int_{0}^{\infty} P (X_{k} > x) d x + k = 1 \sum n \int_{a_{N}}^{\infty} P (X_{k} > x) d x \leq k = 1 \sum n a_{k} P (X_{k} > a_{k}) + a_{N} \int_{0}^{\infty} G (x) d x + a_{n} \int_{a_{N}}^{\infty} G (x) d x .

\frac{1}{a _{n}} k = 1 \sum n a_{k} P (X_{k} > a_{k}) \to 0 (n \to \infty) .

\frac{1}{a _{n}} k = 1 \sum n a_{k} P (X_{k} > a_{k}) \to 0 (n \to \infty) .

n \to \infty lim sup \frac{1}{a _{n}} (E S_{n} - E T_{n}) \leq \int_{a_{N}}^{\infty} G (x) d x

n \to \infty lim sup \frac{1}{a _{n}} (E S_{n} - E T_{n}) \leq \int_{a_{N}}^{\infty} G (x) d x

n = 1 \sum \infty \frac{V a r ( Y _{n} )}{a _{n}^{2}} \leq n = 1 \sum \infty \frac{E ( X _{n}^{2} I _{{X_{n} \leq a_{n}}} )}{a _{n}^{2}} \leq n = 1 \sum \infty \frac{1}{a _{n}^{2}} \int_{0}^{a_{n}^{2}} P (X_{n} > x^{1/2}) d x \leq 2 n = 1 \sum \infty \frac{1}{a _{n}^{2}} \int_{0}^{a_{n}} y P (X_{n} > y) d y \leq 4 n = 1 \sum \infty \frac{1}{([ a _{n} ] + 1 ) ^{2}} \int_{0}^{[a_{n}] + 1} y P (X_{n} > y) d y \leq 8 n = 1 \sum \infty j = [a_{n}] + 1 \sum \infty \frac{1}{j ^{3}} k = 1 \sum [a_{n}] + 1 \int_{k - 1}^{k} y P (X_{n} > y) d y \leq 8 n = 1 \sum \infty j = [a_{n}] + 1 \sum \infty \frac{1}{j ^{3}} k = 1 \sum j \int_{k - 1}^{k} y P (X_{n} > y) d y \leq 8 j = [a_{1}] + 1 \sum \infty n : a_{n} \leq j \sum \frac{1}{j ^{3}} k = 1 \sum j \int_{k - 1}^{k} y P (X_{n} > y) d y \leq 8 j = 1 \sum \infty k = 1 \sum j \frac{1}{j ^{2}} \int_{k - 1}^{k} y \frac{\sum _{n : a_{n} \leq j} P ( X _{n} > y )}{j} d y \leq 8 j = 1 \sum \infty k = 1 \sum j \frac{1}{j ^{2}} \int_{k - 1}^{k} y G (y) d y = 8 k = 1 \sum \infty j = k \sum \infty \frac{1}{j ^{2}} \int_{k - 1}^{k} y G (y) d y \leq 16 k = 1 \sum \infty \frac{1}{k} \int_{k - 1}^{k} y G (y) d y \leq 16 k = 1 \sum \infty \int_{k - 1}^{k} G (y) d y = 16 \int_{0}^{\infty} G (y) d y < \infty.

n = 1 \sum \infty \frac{V a r ( Y _{n} )}{a _{n}^{2}} \leq n = 1 \sum \infty \frac{E ( X _{n}^{2} I _{{X_{n} \leq a_{n}}} )}{a _{n}^{2}} \leq n = 1 \sum \infty \frac{1}{a _{n}^{2}} \int_{0}^{a_{n}^{2}} P (X_{n} > x^{1/2}) d x \leq 2 n = 1 \sum \infty \frac{1}{a _{n}^{2}} \int_{0}^{a_{n}} y P (X_{n} > y) d y \leq 4 n = 1 \sum \infty \frac{1}{([ a _{n} ] + 1 ) ^{2}} \int_{0}^{[a_{n}] + 1} y P (X_{n} > y) d y \leq 8 n = 1 \sum \infty j = [a_{n}] + 1 \sum \infty \frac{1}{j ^{3}} k = 1 \sum [a_{n}] + 1 \int_{k - 1}^{k} y P (X_{n} > y) d y \leq 8 n = 1 \sum \infty j = [a_{n}] + 1 \sum \infty \frac{1}{j ^{3}} k = 1 \sum j \int_{k - 1}^{k} y P (X_{n} > y) d y \leq 8 j = [a_{1}] + 1 \sum \infty n : a_{n} \leq j \sum \frac{1}{j ^{3}} k = 1 \sum j \int_{k - 1}^{k} y P (X_{n} > y) d y \leq 8 j = 1 \sum \infty k = 1 \sum j \frac{1}{j ^{2}} \int_{k - 1}^{k} y \frac{\sum _{n : a_{n} \leq j} P ( X _{n} > y )}{j} d y \leq 8 j = 1 \sum \infty k = 1 \sum j \frac{1}{j ^{2}} \int_{k - 1}^{k} y G (y) d y = 8 k = 1 \sum \infty j = k \sum \infty \frac{1}{j ^{2}} \int_{k - 1}^{k} y G (y) d y \leq 16 k = 1 \sum \infty \frac{1}{k} \int_{k - 1}^{k} y G (y) d y \leq 16 k = 1 \sum \infty \int_{k - 1}^{k} G (y) d y = 16 \int_{0}^{\infty} G (y) d y < \infty.

\frac{1}{a _{n}} k = 1 \sum n E Y_{k} \leq \frac{1}{a _{n}} k = 1 \sum n E X_{k} = \frac{1}{a _{n}} k = 1 \sum n \int_{0}^{\infty} P (X_{k} > x) d x \leq \int_{0}^{\infty} G (x) d x < \infty.

\frac{1}{a _{n}} k = 1 \sum n E Y_{k} \leq \frac{1}{a _{n}} k = 1 \sum n E X_{k} = \frac{1}{a _{n}} k = 1 \sum n \int_{0}^{\infty} P (X_{k} > x) d x \leq \int_{0}^{\infty} G (x) d x < \infty.

n = 1 \sum \infty P (X_{n} \neq = Y_{n}) = n = 1 \sum \infty P (X_{n} > a_{n}) < \infty.

n = 1 \sum \infty P (X_{n} \neq = Y_{n}) = n = 1 \sum \infty P (X_{n} > a_{n}) < \infty.

\frac{S _{n} - E T _{n}}{a _{n}} \to 0 \mbox a . s .

\frac{S _{n} - E T _{n}}{a _{n}} \to 0 \mbox a . s .

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Full text

On the Strong Law of Large Numbers

for Sequences of Pairwise Independent

Random Variables

Valery Korchevsky Saint-Petersburg State University of Aerospace Instrumentation, Saint-Petersburg. E-mail: [email protected]

Abstract

We establish new sufficient conditions for the applicability of the strong law of large numbers (SLLN) for sequences of pairwise independent non-identically distributed random variables. These results generalize Etemadi’s extension of Kolmogorov’s SLLN for identically distributed random variables. Some of the obtained results hold with an arbitrary norming sequence in place of the classical normalization.

Keywords: strong law of large numbers, pairwise independent random variables.

1. Introduction

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of random variables defined on the same probability space and put $S_{n}=\sum_{k=1}^{n}X_{k}$ .

The classical Kolmogorov’s theorem states that if $\{X_{n}\}_{n=1}^{\infty}$ is a sequence of independent identically distributed random variables and $E|X_{1}|<\infty$ then $S_{n}/n\to EX_{1}$ almost surely. Etemadi [4] generalized the Kolmogorov theorem replacing the mutual independence assumption by the pairwise independence assumption.

Theorem A ([4]).

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent identically distributed random variables. If $E|X_{1}|<\infty$ then $S_{n}/n\to EX_{1}$ almost surely.

One can find further extensions of the Kolmogorov theorem to wide classes of dependent random variables in the papers [6] and [7].

In the present work we generalize Theorem A to non-identically distributed random variables. This problem was considered in the several papers. Chandra and Goswami [3] established the following result.

Theorem B ([3]).

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent random variables and put $G(x)=\sup_{n\geq 1}P(|X_{n}|>x)$ for $x\geq 0$ . If

[TABLE]

Then

[TABLE]

for each bounded sequence $\{c_{n}\}$ .

The following result was obtained by Bose and Chandra [1].

Theorem C ([1]).

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent random variables. Suppose that

[TABLE]

Then

[TABLE]

Kruglov [5] proved the next generalization of Theorem A.

Theorem D ([5]).

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent random variables. Assume that

[TABLE]

If there exists a random variable $X$ such that $E|X|<\infty$ and

[TABLE]

where $C$ is a positive constant, then relation (2) holds.

The aim of present work is to generalize Theorems C and D. We present a generalization of Theorem C using an arbitrary norming sequence in place of the classical normalization. Furthermore we show that condition (3) in Theorems D can be dropped.

In order to prove the theorems in the present work, we use methods developed by Bose and Chandra [1] (see also Chandra [2]).

2. Main results

Theorem 1.

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent random variables. Assume that $\{a_{n}\}_{n=1}^{\infty}$ is non-decreasing unbounded sequence of positive numbers. Suppose that

[TABLE]

Then

[TABLE]

Theorem 1 generalizes Theorem C, which corresponds to the case $a_{n}=n$ for all ${n\geq 1}$ .

Theorem 2.

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent random variables. If there exists function $H(x)$ such that $H(x)$ is non-increasing in the interval $x\geq 0$ ,

[TABLE]

then

[TABLE]

As a consequence of Theorem 2 we immediately obtain the following result.

Corollary 1.

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of pairwise independent random variables. If there exists a random variable $X$ such that $E|X|<\infty$ and

[TABLE]

where $C$ is a positive constant, then

[TABLE]

Corollary 1 shows that we can omit condition (3) in Theorem D.

3. Proofs

To prove Theorems 1 we need the following proposition that is a consequence of Theorem 1 in [3].

Lemma 1.

Let $\{X_{n}\}_{n=1}^{\infty}$ be a sequence of non-negative random variables with finite variances. Assume that $\{a_{n}\}_{n=1}^{\infty}$ is non-decreasing unbounded sequence of positive numbers. Suppose that

[TABLE]

where $C$ is a positive constant,

[TABLE]

Then

[TABLE]

Proof of Theorem 1.

Note that for pairwise independent random variables $X_{n}$ , the positive parts $X_{n}^{+}:=max\{0,X_{n}\}$ are pairwise independent. Likewise, the $X_{n}^{-}:=max\{0,-X_{n}\}$ are pairwise independent. Thus it is enough to prove the theorem separately for the positive and negative parts. So we can assume that $X_{n}\geq 0$ for all $n\geq 1$ .

Let $Y_{n}=X_{n}\mathbb{I}_{\{X_{n}\leq a_{n}\}}$ , $T_{n}=\sum_{k=1}^{n}Y_{k}$ for every $n\geq 1$ . To prove the theorem , it is sufficient to show that

[TABLE]

Note that for any non-negative random variable $Z$ and $a>0$

[TABLE]

and

[TABLE]

Fix an integer $N\geq 1$ . Then, using (4) and (12), for $n>N$ we obtain

[TABLE]

Condition (5) and Kronecker’s lemma (see, for example, [8]) imply that

[TABLE]

Thus for each $N\geq 1$ we have

[TABLE]

so we get assertion (9) by letting $N\to\infty$ .

To establish (10) we shall prove that conditions of Lemma 1 are satisfied for sequence $\{Y_{n}\}_{n=1}^{\infty}$ . It follows from pairwise independence of random variables $Y_{n}$ that assertion (7) is satisfied for sequence $\{Y_{n}\}_{n=1}^{\infty}$ .

Using (4) and (13), we obtain

[TABLE]

where $[a_{n}]$ is the integer part of $a_{n}$ . Hence condition (8) is satisfied for sequence $\{Y_{n}\}_{n=1}^{\infty}$ .

For each $n\geq 1$ we have

[TABLE]

Thus the sequence of random variables $\{Y_{n}\}_{n=1}^{\infty}$ satisfies conditions of Lemma 1, so relation (10) holds.

To complete the proof it remains to verify assertion (11). Using (5), we obtain

[TABLE]

From Borel–Cantelli lemma and relation (10) it follows that

[TABLE]

Thus (11) follows from (10) and (14). ∎

Proof of Theorem 2.

As $H(x)$ is non-increasing in the interval $x\geq 0$ , condition $\int_{0}^{\infty}H(x)\,dx<\infty$ implies that $\sum_{k=0}^{\infty}2^{k}H(2^{k})<\infty$ . Thus using (6) we have

[TABLE]

Now, taking into account that (6) implies (1), we can conclude that the desired result follows from Theorem C. ∎

Bibliography8

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Bose A., Chandra T.K. A note on the strong law of large numbers. Calcutta Statistical Association Bulletin 44 , 115–122 (1994)
2[2] Chandra T.K. Laws of large numbers. Narosa Publishing House. New Delhi (2012)
3[3] Chandra T.K., Goswami A. Cesáro uniform integrability and a strong laws of large numbers. Sankhyā, Ser. A 54 , 215–231 (1992)
4[4] Etemadi N. An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheorie verw. Geb. 55 , 119–122 (1981)
5[5] Kruglov V.M. Strong law of large numbers. Stability Problems for Stochastic Models (Zolotarev V.M., Kruglov V.M., Korolev V.Yu., eds.) TVP/VSP. Moscow–Utrecht, 139–150 (1994)
6[6] Matula P. A note on the almost sure convergence of sums negatively dependent random variables. Statist. Probab. Lett. 15 , 209–213 (1992)
7[7] Matula P. On some families of AQSI random variables and related strong law of large numbers. Appl. Math. E-Notes, 5 , 31–35 (2005)
8[8] Petrov V.V. Limit theorems of probability theory. Clarendon Press. Oxford (1995)

TL;DR

Contribution

Findings

Abstract

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On the Strong Law of Large Numbers

Abstract

Theorem A** ([4]).**

Theorem B** ([3]).**

Theorem C** ([1]).**

Theorem D** ([5]).**

Theorem 1**.**

Theorem 2**.**

Corollary 1**.**

Lemma 1**.**

Proof of Theorem 1.

Proof of Theorem 2.

Theorem A ([4]).

Theorem B ([3]).

Theorem C ([1]).

Theorem D ([5]).

Theorem 1.

Theorem 2.

Corollary 1.

Lemma 1.