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Limit theorems for delayed sums of random sequence
Journal of Inequalities and Applications volume 2012, Article number: 124 (2012)
For a sequence of arbitrarily dependent random variables (X n )n∈Nand Borel sets (B n )n∈N, 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.
Let (a n )n∈Nbe a sequence of real numbers and let (k n )n∈Nbe a sequence of positive integers. The numbers
are called the (forward) delayed first arithmetic means (See ). In , 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 , Lai studied the analogues of the law of the iterated logarithim for delayed sums of independent random variables. Recently, Chen  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 ) 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  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 )n∈Nbe a sequence of absolutely continuous random variables on a fixed probability space 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 )n∈Nbe a subsequence of positive integers, such that, for every ε > 0, .
Definition 1. The delayed likelihood ratio is defined by
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 , 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. n∈ N if (X n )n∈Nis 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 (X n )n∈Nis from being independent and how dependent they are. The closer approaches to 0, the smaller the deviation is.
Lemma 1. Letbe define as above, then
Proof. Let Since
From Markov inequality, for every ε > 0, we have
By Borel-Cantelli lemma, we have
for any ε > 0, (1.3) follows immediately. □
2. Main results and proofs
Theorem 1. Let (X n )n∈N, , be defined as above, (B n )n∈Nbe a sequence of Borel sets of the real line. Let , and assume
wherebe the indicator function of B n .
Proof. Assume s > 0 to be a constant, and let
It is not difficult to see that , j = 1, 2,... Let
From Lemma 1, there exists , P(A(s)) = 1, such that
Since , by (2.3) we have
It follows from (1.1), (2.4) and (2.6) that
(2.5) and (2.7) yield
Let s > 1, dividing the two sides of (2.8) by log s, we have
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
By (2.10) and the property of the superior above and the inequality 0 < log(1+x) ≤ x(x > 0), we obtain
(2.11) and the inequality imply
Let D be a set of countable real numbers dense in the interval (1, +∞), and let A* = ∩s∈DA(s), g(s, x) = cs + sx/(s - 1), then we have by (2.12)
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 κ n (ω) ∈ D, n = 1, 2,..., such that . We have by the continuity of with respect to s,
By (2.13), we obtain
(2.14) and (2.15) imply
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),
since P (A(e)) = 1, (2.2) also holds by (2.17) when c = 0. □
Theorem 2. Let (X n )n∈N, , , (B n )n∈N, be defined as in Theorem 1 and assume
then, ifa.e., then
Proof. Let 0 <s < 1, dividing the two sides of (2.8) by log s, we have
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
By (2.21) and the property of the inferior above and the inequality log(1+ x) ≤ x(-1 <x ≤ 0), we obtain
(2.22) and the inequality and log s < s - 1 (0 <s < 1) imply
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)
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 l n (ω) ∈ D', n = 1, 2,..., such that . We have by the continuity of with respect to s,
By (2.24), we obtain
(2.25) and (2.26) imply
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 (X n )n∈Nbe identically distribution with common distribution function F , for each m, n ∈ N, let
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 )n∈Nbe i.i.d. random variables with common distribution function F, let B n = (-∞, x], n = 1, 2,..., then
Corollary 2. Let (X n )n∈Nbe independent random variables and (B n )n∈Nbe as Theorem 1, then
Proof. Note that P (X j ∈ B j ) ≤ 1, j = 1, 2,... and in this case, 0 ≤ c, c' ≤ 1, a.e., we have by (2.11)
by (2.29) and the property of the superior above and the inequality 0 ≤ log(1+x) ≤ x(x > 0), we obtain
Analogously as in the proof of Theorem 1, we obtain
Similarly, we have , a.e. hence (2.28) follows immediately. □
Remark 3. Let B n = B, Corollary 2 implies that which gives the strong law of large numbers for the delayed arithmatic means.
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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.
The authors declare that they have no competing interests.
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
- strong deviation theorem
- likelihood ratio
- delayed sums