# Limit theorems for delayed sums of random sequence

## Abstract

For a sequence of arbitrarily dependent random variables (X n )nNand Borel sets (B n )nN, on real line the strong limit theorems, represented by inequalities, i.e. the strong deviation theorems of the delayed average $S n . k n ω$ are investigated by using the notion of asymptotic delayed log-likelihood ratio. The results obtained popularizes the methods proposed by Liu.

Mathematics Subject Classification 2000: Primary, 60F15.

## 1. Introduction

Let (a n )nNbe a sequence of real numbers and let (k n )nNbe a sequence of positive integers. The numbers

$ρ n , k n = ∑ j = 1 k n a n + j - 1 / k n$

are called the (forward) delayed first arithmetic means (See [1]). In [2], using the limiting behavior of delayed average, Chow found necessary and sufficient conditions for the Borel summability of i.i.d. random variables and also obtained very simple proofs of a number of well-known results such as the Hsu-Robbins-Spitzer-Katz theorem. In [3], Lai studied the analogues of the law of the iterated logarithim for delayed sums of independent random variables. Recently, Chen [4] has presented an accurate description the limiting behavior of delayed sums under a non-identically distribution setup, and has deduced Chover-type laws of the iterated logarithm for them.

Our aim in this article is to establish strong deviation theorems (limit theorem expressed by inequalities, see [5]) of delayed average for the dependent absolutely continuous random variables. By using the notion of asymptotic delayed log-likelihood ratio, we extend the analytic technique proposed by Liu [5] to the case of delayed sums. The crucial part of the proof is to construct a delayed likelihood ratio depending on a parameter, and then applies the Borel-Cantelli lemma.

Throughout, let (X n )nNbe a sequence of absolutely continuous random variables on a fixed probability space $Ω , F , P$ with the joint density function g1, n(x1,..., x n ), n N, and f j (x), j = 1, 2,... be the the marginal density function of random variable X j . (k n )nNbe a subsequence of positive integers, such that, for every ε > 0, $∑ n = 1 ∞ exp - k n ε <∞$.

Definition 1. The delayed likelihood ratio is defined by

$L n ω = Π j = n n + k n - 1 f j X j g n , n + k n - 1 X n , . . . , X n + k n - 1 , if denominator > 0 0 , otherwise$
(1.1)

Let

$L ω = - lim inf n 1 k n log L n ω$
(1.2)

$L ω$ is called asymptotic delayed log-likelihood ratio, where $g n , n + k n - 1 x n , … , x n + k n - 1$ denotes the joint density function of random vector $X n , … , X n + k n - 1$, ω is a sample point (with log 0 = -∞).

It will be shown in Lemma 1 that $L ω ≥0$a.e. in any case.

Remark 1. It will be seen below that $L ω$ has the analogous properties of the likelihood ratio in [5], Although $L ω$ is not a proper metric among probability measures, we nevertheless consider it as a measure of "discrimination" between the dependence (their joint distribution) and independence (the product of their marginals). Obviously, $L n ω =1$, a.e. n N if (X n )nNis independent. In view of the above discussion of the asymptotic logarithmic delayed likelihood ratio, it is natural for us to think of $L ω$ as a measure how far (the random deviation) of (X n )nNis from being independent and how dependent they are. The closer $L ω$ approaches to 0, the smaller the deviation is.

Lemma 1. Let$L n ω$be define as above, then

$lim sup n 1 k n log L n ω ≤ 0 , a . e .$
(1.3)

Proof. Let $B= x n , … , x n + k n - 1 : g n , n + k n - 1 x n , … , x n + k n - 1 > 0 .$ Since

$E L n ω = ∫ ⋯ ∫ x n , … , x n + k n - 1 ∈ B ∏ j = n n + k n - 1 f j x j g n , n + k n - 1 x n , … , x n + k n - 1 . g n , n + k n - 1 x n , … , x n + k n - 1 d x n … d x n + k n - 1 = ∫ ⋯ ∫ x n , … , x n + k n - 1 ∈ B ∏ j = n n + k n - 1 f j x j d x n … d x n + k n - 1 ≤ ∫ ⋯ ∫ x n , … , x n + k n - 1 ∈ R k n ∏ j = n n + k n - 1 f j x j d x n … d x n + k n - 1 = 1 .$

From Markov inequality, for every ε > 0, we have

$P 1 k n log L n ω ≥ ε = P L n ω ≥ exp k n ε ≤ 1 ⋅ exp - k n ε .$

Hence

$∑ n = 1 ∞ P 1 k n log L n ω ≥ ε ≤ ∑ n = 1 ∞ exp - k n ε < ∞ .$

By Borel-Cantelli lemma, we have

$P lim sup n 1 k n log L n ω ≥ 2 ε = 0 ,$

for any ε > 0, (1.3) follows immediately. □

## 2. Main results and proofs

Theorem 1. Let (X n )nN, $L n ω$, $L ω$be defined as above, (B n )nNbe a sequence of Borel sets of the real line. Let $S n , k n ω = 1 k n ∑ j = n n + k n - 1 1 B j X j$, and assume

$c= lim sup n 1 k n ∑ j = n n + k n - 1 P X j ∈ B j ,$
(2.1)

then

$lim sup n S n , k n ω ≤ L ω + c 2 ,a.e.$
(2.2)

where$1 B n ( ⋅ )$be the indicator function of B n .

Proof. Assume s > 0 to be a constant, and let

$h j x j = s 1 B j x j f j x j 1 + s - 1 ∫ B j f j x j d x j , j = 1 , 2 , …$
(2.3)

It is not difficult to see that $∫ h j x j d x j =1$, j = 1, 2,... Let

$Λ n s , ω = Π j = n n + k n - 1 h j X j g n , n + k n - 1 X n , . . . , X n + k n - 1 , if denominator > 0 0 , otherwise$
(2.4)

From Lemma 1, there exists $A s ∈F$, P(A(s)) = 1, such that

$lim sup n 1 k n log Λ n s , ω ≤0,ω∈A s$
(2.5)

Since $∫ B j f j x j d x j =P X j ∈ B j$, by (2.3) we have

$∏ j = n n + k n - 1 h j x j = ∏ j = n n + k n - 1 s 1 B j x j f j x j 1 + s - 1 ∫ B j f j x j d x j = s ∑ j = n n + k n - 1 1 B j x j ∏ j = n n + k n - 1 f j x j 1 + s - 1 P X j ∈ B j$
(2.6)

It follows from (1.1), (2.4) and (2.6) that

$log Λ n s , ω = ∑ j = n n + k n - 1 1 B j X j log s - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j + log L n ω$
(2.7)

(2.5) and (2.7) yield

$lim sup n 1 k n log s ∑ j = n n + k n - 1 1 B j X j - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j + log L n ω ≤ 0 , ω ∈ A s$
(2.8)

Let s > 1, dividing the two sides of (2.8) by log s, we have

$lim sup n 1 k n ∑ j = n n + k n - 1 1 B j X j - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s + log L n ω log s ≤ 0 , ω ∈ A s$
(2.9)

By (1.2), (2.9) and the property lim sup n (a n - b n ) ≤ d lim sup n (a n - c n ) ≤ lim sup n (b n - c n ) + d, one gets

$lim sup n 1 k n ∑ j = n n + k n - 1 1 B j X j - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s ≤ L ω log s , ω ∈ A s$
(2.10)

By (2.10) and the property of the superior above and the inequality 0 < log(1+x) ≤ x(x > 0), we obtain

$lim sup n S n , k n ω ≤ lim sup n 1 k n ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s + L ω log s ≤ lim sup n 1 k n ∑ j = n n + k n - 1 s - 1 P X j ∈ B j log s + L ω log s ≤ c s - 1 log s + L ω log s , ω ∈ A s$
(2.11)

(2.11) and the inequality $1- 1 s < log s s > 1$ imply

$lim sup n S n , k n ω ≤c⋅s+ s L ω s - 1 ,ω∈A s$
(2.12)

Let D be a set of countable real numbers dense in the interval (1, +∞), and let A* = ∩sDA(s), g(s, x) = cs + sx/(s - 1), then we have by (2.12)

$lim sup n S n , k n ω ≤ g s , L ω , ω ∈ A * , s ∈ D$
(2.13)

Let c > 0, it easy to see that if $L ω >0$, a.e., then, for fixed ω, $g s , L ω$as a function of s attains its smallest value $g 1 + L ω / c , L ω =2 c L ω +L ω +c$ on the interval (1, +∞), and g(s, 0) is increasing on the interval (1, +∞) and lims→1+ g(s, 0) = 0. For each ω A* ∩ A(1), if $L ω ≠∞$, take κ n (ω) D, n = 1, 2,..., such that $κ n ω →1+ L ω / c$. We have by the continuity of $g s , L ω$ with respect to s,

$lim n → + ∞ g κ n ω , L ω = L ω + c 2 ,$
(2.14)

By (2.13), we obtain

$lim sup n S n , k n ≤g κ n ω , L ω ,n=1,2,…$
(2.15)

(2.14) and (2.15) imply

$lim sup n S n , k n ω ≤ L ω + c 2 ,ω∈ A * ∩A 1$
(2.16)

If $L ω =∞$, (2.16) holds trivially. Since P (A* ∩ A(1)) = 1, (2.2) holds by (2.16), when c > 0.

When c = 0, we have by letting s = e in (2.11),

$lim sup n S n , k n ω ≤L ω ,ω∈A e$
(2.17)

since P (A(e)) = 1, (2.2) also holds by (2.17) when c = 0. □

Theorem 2. Let (X n )nN, $L n ω$, $L ω$, (B n )nN, $S n , k n ω$be defined as in Theorem 1 and assume

$c ′ = lim inf n 1 k n ∑ j = n n + k n - 1 P X j ∈ B j ,$
(2.18)

then, if$0≤L ω ≤ c ′$a.e., then

$lim inf n S n , k n ω ≥ c ′ - 2 c ′ L ω , a . e .$
(2.19)

Proof. Let 0 <s < 1, dividing the two sides of (2.8) by log s, we have

$lim inf n 1 k n ∑ j = n n + k n - 1 1 B j X j - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s + log L n ω log s ≥ 0 , ω ∈ A s$
(2.20)

By (1.2), (2.20) and the property lim inf n (a n - b n ) ≥ d lim inf n (a n - c n ) ≥ lim inf n (b n - c n ) + d, one gets

$lim inf n 1 k n ∑ j = n n + k n - 1 1 B j X j - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s ≥ L ω log s , ω ∈ A s$
(2.21)

By (2.21) and the property of the inferior above and the inequality log(1+ x) ≤ x(-1 <x ≤ 0), we obtain

$lim inf n S n , k n ω ≥ lim inf n 1 k n ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s + L ω log s ≥ lim n inf 1 k n ∑ j = n n + k n - 1 s - 1 P X j ∈ B j log s + L ω log s ≥ c ′ s - 1 log s + L ω log s , ω ∈ A s$
(2.22)

(2.22) and the inequality $1- 1 s < log s$ and log s < s - 1 (0 <s < 1) imply

$lim inf n S n , k n ω ≥ c ′ ⋅ s + L ω s - 1 , ω ∈ A s ∩ A 1$
(2.23)

Let D' be a set of countable real numbers dense in the interval (0, 1), and let $A * = ∩ s ∈ D ′ A s$, h(s, x) = c's + x/(s - 1), then we have by (2.23)

$lim inf n S n , k n ω ≥h s , L ω ,ω∈ A * ,s∈ D ′$
(2.24)

Let c' > 0, it easy to see that if $0, a.e., then, for fixed ω, $h s , L ω$ as a function of s attains its maximum value $h 1 - L ω / c ′ , L ω = c ′ -2 c ′ L ω$, on the interval (0, 1), and h(s, 0) is increasing on the interval (0, 1) and lims→1+ h(s, 0) = c'. For each ω A*A(1), if $L ω ≠∞$, take l n (ω) D', n = 1, 2,..., such that $l n ω →1- L ω / c ′$. We have by the continuity of $h s , L ω$ with respect to s,

$lim n → + ∞ h l n ω , L ω = c ′ - 2 c ′ L ω ,$
(2.25)

By (2.24), we obtain

$lim inf n S n , k n ≥h l n ω , L ω ,n=1,2,…$
(2.26)

(2.25) and (2.26) imply

$lim inf n S n , k n ω ≥ c ′ -2 c ′ L ω ,ω∈ A * ∩A 1$
(2.27)

If $L ω =∞$, (2.27) holds trivially. Since P (A*A(1)) = 1, (2.19) holds by (2.27), when c' > 0. (2.19) also holds trivially when c' = 0. □

Remark 2. In case $L ω > c ′ ≥0$, a.e., we cannot get a better lower bound of $lim inf n S n , k n ω$. This motivates the following problem: under the conditions of Theorem 2, how to get a better lower bound of $lim inf n S n , k n ω$ in case of $L ω > c ′ ≥0$, a.e.?

Definition 2. (Generalized empirical distribution function) Let (X n )nNbe identically distribution with common distribution function F , for each m, n N, let

$F m , n x = 1 n ∑ k = m m + n - 1 1 X k ≤ x .$

F m, n = the observed frequency of values that are ≤ x from time m to m + n - 1. The F1,nis the usual empirical distribution function, hence the name given above.

In particular, let B = (-∞, x], x R in Theorems 1 and 2, we can get a strong limit theorem for the generalized empirical distribution function.

Corollary 1. Let (X n )nNbe i.i.d. random variables with common distribution function F, let B n = (-∞, x], n = 1, 2,..., then

$lim n F n , n + k n - 1 x =F x ,a.e.$

Corollary 2. Let (X n )nNbe independent random variables and (B n )nNbe as Theorem 1, then

$lim n 1 k n ∑ j = n n + k n - 1 1 B j X j - P X j ∈ B j =0,a.e.$
(2.28)

Proof. Note that P (X j B j ) ≤ 1, j = 1, 2,... and in this case, 0 ≤ c, c' ≤ 1, $L ω =0$a.e., we have by (2.11)

$lim sup n 1 k n ∑ j = n n + k n - 1 1 B j X j - ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s ≤ 0 , ω ∈ A s$
(2.29)

by (2.29) and the property of the superior above and the inequality 0 ≤ log(1+x) ≤ x(x > 0), we obtain

$lim sup n 1 k n ∑ j = n n + k n - 1 1 B j X j - P X j ∈ B j ≤ lim sup n 1 k n ∑ j = n n + k n - 1 log 1 + s - 1 P X j ∈ B j log s - P X j ∈ B j ≤ lim sup n 1 k n ∑ j = n n + k n - 1 s - 1 P X j ∈ B j log s - P X j ∈ B j ≤ s - 1 log s - 1 , ω ∈ A s$
(2.30)

Analogously as in the proof of Theorem 1, we obtain

$lim sup n 1 k n ∑ j = n n + k n - 1 1 B j X j - P X j ∈ B j ≤0,a.e.$
(2.31)

Similarly, we have $lim inf n 1 k n ∑ j = n n + k n - 1 1 B j X j - P X j ∈ B j ≥0$, a.e. hence (2.28) follows immediately. □

Remark 3. Let B n = B, Corollary 2 implies that $lim S n , k n k n =P X 1 ∈ B$ which gives the strong law of large numbers for the delayed arithmatic means.

## References

1. 1.

Zygmund A: Trigonometric Series 1. Cambridge Universitiy Press, Cambridge; 1959.

2. 2.

Chow YS: Delayed sums and Borel summability for independent, identically distributed random variables. Bull Inst Math Academia Sinica 1972, 1: 207–220.

3. 3.

Lai TL: Limit theorems for delayed sums. Ann Probab 1974, 2(3):432–440. 10.1214/aop/1176996658

4. 4.

Chen PY: Limiting behavior of delayed sums under a non-identically distribution setup. Ann Braz Acad Sci 2008, 80(4):617–625.

5. 5.

Liu W: Strong deviation theorems and analytical method. Academic press, Beijing; 2003.

## Acknowledgements

This work is supported by The National Natural Science Foundation of China (Grant No. 11071104) and the An Hui University of Technology research grant: D2011025. The authors would like to thank two referees for their insightful comments which resulted in improving Theorems 1, 2 and Corollary 2 significantly.

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Correspondence to Wang Zhong-zhi.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

WZ and DF carried out the design of the study and performed the analysis. DF drafted the manuscript. All authors read and approved the final manuscript.

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Fang-qing, D., Zhong-zhi, W. Limit theorems for delayed sums of random sequence. J Inequal Appl 2012, 124 (2012). https://doi.org/10.1186/1029-242X-2012-124