Research

# Limit theorems for delayed sums of random sequence

Ding Fang-qing1 and Wang Zhong-zhi2*

Author Affiliations

1 Department of Mathematics & Physics, HeFei University, HeFei 230601, P. R. China

2 Faculty of Mathematics & Physics, AnHui University of Technology, Ma'anshan 243002, P. R. China

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Journal of Inequalities and Applications 2012, 2012:124 doi:10.1186/1029-242X-2012-124

 Received: 29 October 2011 Accepted: 31 May 2012 Published: 31 May 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

For a sequence of arbitrarily dependent random variables (Xn)nN and Borel sets (Bn)nN, on real line the strong limit theorems, represented by inequalities, i.e. the strong deviation theorems of the delayed average 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.

##### Keywords:
strong deviation theorem; likelihood ratio; delayed sums

### 1. Introduction

Let (an)nN be a sequence of real numbers and let (kn)nN be a sequence of positive integers. The numbers

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 (Xn)nN be a sequence of absolutely continuous random variables on a fixed probability space with the joint density function g1, n(x1,..., xn), n N, and fj(x), j = 1, 2,... be the the marginal density function of random variable Xj. (kn)nN be a subsequence of positive integers, such that, for every ε > 0, .

Definition 1. The delayed likelihood ratio is defined by

(1.1)

Let

(1.2)

is called asymptotic delayed log-likelihood ratio, where denotes the joint density function of random vector , ω is a sample point (with log 0 = -∞).

It will be shown in Lemma 1 that a.e. in any case.

Remark 1. It will be seen below that has the analogous properties of the likelihood ratio in [5], Although 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, , a.e. nN if (Xn)nN is independent. In view of the above discussion of the asymptotic logarithmic delayed likelihood ratio, it is natural for us to think of as a measure how far (the random deviation) of (Xn)nN is from being independent and how dependent they are. The closer approaches to 0, the smaller the deviation is.

Lemma 1. Let be define as above, then

(1.3)

Proof. Let Since

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

Hence

By Borel-Cantelli lemma, we have

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

### 2. Main results and proofs

Theorem 1. Let (Xn)nN, , be defined as above, (Bn)nN be a sequence of Borel sets of the real line. Let , and assume

(2.1)

then

(2.2)

where be the indicator function of Bn.

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

(2.3)

It is not difficult to see that , j = 1, 2,... Let

(2.4)

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

(2.5)

Since , by (2.3) we have

(2.6)

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

(2.7)

(2.5) and (2.7) yield

(2.8)

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

(2.9)

By (1.2), (2.9) and the property lim supn(an - bn) ≤ d ⇒ lim supn(an - cn) ≤ lim supn(bn - cn) + d, one gets

(2.10)

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

(2.11)

(2.11) and the inequality imply

(2.12)

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

(2.13)

Let c > 0, it easy to see that if , a.e., then, for fixed ω, as a function of s attains its smallest value 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 , take κn(ω) ∈ D, n = 1, 2,..., such that . We have by the continuity of with respect to s,

(2.14)

By (2.13), we obtain

(2.15)

(2.14) and (2.15) imply

(2.16)

If , (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),

(2.17)

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

Theorem 2. Let (Xn)nN, , , (Bn)nN, be defined as in Theorem 1 and assume

(2.18)

then, if a.e., then

(2.19)

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

(2.20)

By (1.2), (2.20) and the property lim infn(an - bn) ≥ d ⇒ lim infn(an - cn) ≥ lim infn(bn - cn) + d, one gets

(2.21)

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

(2.22)

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

(2.23)

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

(2.24)

Let c' > 0, it easy to see that if , a.e., then, for fixed ω, as a function of s attains its maximum value , 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 , take ln(ω) ∈ D', n = 1, 2,..., such that . We have by the continuity of with respect to s,

(2.25)

By (2.24), we obtain

(2.26)

(2.25) and (2.26) imply

(2.27)

If , (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 , a.e., we cannot get a better lower bound of . This motivates the following problem: under the conditions of Theorem 2, how to get a better lower bound of in case of , a.e.?

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

Fm, n = the observed frequency of values that are ≤ x from time m to m + n - 1. The F1,n is 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 (Xn)nN be i.i.d. random variables with common distribution function F, let Bn = (-∞, x], n = 1, 2,..., then

Corollary 2. Let (Xn)nN be independent random variables and (Bn)nN be as Theorem 1, then

(2.28)

Proof. Note that P (Xj Bj) ≤ 1, j = 1, 2,... and in this case, 0 ≤ c, c' ≤ 1, a.e., we have by (2.11)

(2.29)

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

(2.30)

Analogously as in the proof of Theorem 1, we obtain

(2.31)

Similarly, we have , a.e. hence (2.28) follows immediately. □

Remark 3. Let Bn = B, Corollary 2 implies that which gives the strong law of large numbers for the delayed arithmatic means.

### 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.

### 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.

### References

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

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

3. Lai, TL: Limit theorems for delayed sums. Ann Probab. 2(3), 432–440 (1974). Publisher Full Text

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

5. Liu, W: Strong deviation theorems and analytical method. Academic press, Beijing (2003) (in Chinese)