Let be a sequence of i.i.d. real-valued random variables, and , . Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series , where , , and are taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980).

1. Introduction

Let be the set of all positive integer dimensional lattice points with coordinate-wise partial ordering, ; that is, for every , , if and only if , , where is a fixed integer. denote and means . Throughout the paper, are i.i.d. random variables with and . Let , and . we define

where are real numbers with . Further, set and .

Hartman and Wintner [1] studied the fundamental strong laws of classic Probability Theory for i.i.d random variables and obtained the following Hartman-Wintner law of the iterated logarithm (LIL).

Theorem 1.1. and if and only if

Afterward, the study of the estimate of the convergence rate in the above relation (1.2) has been attracting the attention of various researchers over the last few decades. Darling and Robbins [2], Davis [3], Gafurov [4], and Li [5] have obtained some good results on the estimate of convergence rate in (1.2). The best result is probably the one given by Li et al. [6]. For easy reference, we restate their result in Theorem 1.2.

Theorem 1.2. Let and be two positive real-valued functions on such that is nondecreasing, and as . For , let , and . Then the following results are equivalent: where is the reverse function of

Proof. Because by using Lemma 3.1, we have The last expression is finite if and only if This completes the proof of the lemma.

Lemma 4.2. implies (4.15).

Proof. We first prove the result for symmetric random variables. One may rearrange and obtain By Levy inequality, we set but Thus for large equation (4.15) follows from Lemma 4.1.
If are nonsymmetric random variables, then by using the standard symmetrization method, it is easy to prove that for some constant which implies that That is, (4.15) holds.

Proof of Theorem 2.5. Similar to the proof of Theorem 2.2, by Lemma 4.1, implies that Therefore, if , (2.12) is equivalent to which is equivalent to
On the other hand, (2.12) implies by Lemma 4.2. Hence, (2.11) and (2.12) are equivalent. If, in addition, (4.24) is equivalent to the following: Hence, (2.12) and (2.13) are equivalent.

Proof of Theorem 2.7. Like the proof of Theorem 2.1, and (2.17) imply (2.16). On the other hand, if (2.16) holds, then implies (2.17). The proof is complete.


The author is very grateful for referee's comments and suggestions which greatly improve the readability and quality of the current paper. The research is partly supported by the Natural Sciences and Engineering Research Council of Canada.