For a sequence of arbitrarily dependent random variables (Xn)n∈N and Borel sets (Bn)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.
Keywords:strong deviation theorem; likelihood ratio; delayed sums
Let (an)n∈N be a sequence of real numbers and let (kn)n∈N be 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 (Xn)n∈N 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)n∈N be a subsequence of positive integers, such that, for every ε > 0, .
Definition 1. The delayed likelihood ratio is defined by
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 (Xn)n∈N 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)n∈N is from being independent and how dependent they are. The closer approaches to 0, the smaller the deviation is.
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
Proof. Assume s > 0 to be a constant, and let
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 supn(an - bn) ≤ d ⇒ lim supn(an - cn) ≤ lim supn(bn - cn) + 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
Let D be a set of countable real numbers dense in the interval (1, +∞), and let A* = ∩s∈D A(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 κn(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
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. □
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 infn(an - bn) ≥ d ⇒ lim infn(an - cn) ≥ lim infn(bn - cn) + 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
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,
By (2.24), we obtain
(2.25) and (2.26) imply
Definition 2. (Generalized empirical distribution function) Let (Xn)n∈N 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)n∈N be i.i.d. random variables with common distribution function F, let Bn = (-∞, x], n = 1, 2,..., then
Corollary 2. Let (Xn)n∈N be independent random variables and (Bn)n∈N be as Theorem 1, then
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
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.
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.
Lai, TL: Limit theorems for delayed sums. Ann Probab. 2(3), 432–440 (1974). Publisher Full Text