Limit theorems for delayed sums of random sequence

For a sequence of arbitrarily dependent random variables (X_n)_n∈N and 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∈N be a sequence of real numbers and let (k_n)_n∈N 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 (X_n)_n∈N be a sequence of absolutely continuous random variables on a fixed probability space with the joint density function g^{1, 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∈N be a subsequence of positive integers, such that, for every ε > 0, .

Definition 1. The delayed likelihood ratio is defined by

Let

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. n∈ N if (X_n)_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 (X_n)_n∈N 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

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. □

Theorem 1. Let (X_n)_n∈N, , be defined as above, (B_n)_n∈N be a sequence of Borel sets of the real line. Let , and assume

then

where be 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∈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 lim_s→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, if a.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 lim_s→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∈N be 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 F_1,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 (X_n)_n∈N be i.i.d. random variables with common distribution function F, let B_n = (-∞, x], n = 1, 2,..., then

Corollary 2. Let (X_n)_n∈N be independent random variables and (B_n)_n∈N be 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.

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.

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)