CHOVER-TYPE LAWS OF THE ITERATED LOGARITHM FOR WEIGHTED SUMS OF ρ ∗-MIXING SEQUENCES

We call this a Chover-type LIL (laws of the iterated logarithm). This type LIL has been established by Vasudeva and Divanji [13], Zinchenko [14] for delayed sums, by Chen and Huang [3] for geometric weighted sums, and by Chen [2] for weighted sums. Qi and Cheng [11] extended the Chover-type law of the iterated logarithm for the partial sums to the case where the underlying distribution is in the domain of attraction of a nonsymmetric stable distribution (see below for details). Let Lα denote a stable distribution with exponent α∈ (0,2). Recall that the distribution of X is said to be in the domain of attraction of Lα if there exist some constants An ∈ R


Introduction
Let {X i , i ≥ 1} be independent and identically distributed (i.i.d.) with symmetric stable distributions, which belong to the domain of normal attraction and nongeneration.So, their characteristic functions are of the forms: (1.1) Chover [4] has obtained that limsup n→∞ n i=1 X i n 1/α 1/ log logn = e 1/α a.s.(1.2) We call this a Chover-type LIL (laws of the iterated logarithm).This type LIL has been established by Vasudeva and Divanji [13], Zinchenko [14] for delayed sums, by Chen and Huang [3] for geometric weighted sums, and by Chen [2] for weighted sums.Qi and Cheng [11] extended the Chover-type law of the iterated logarithm for the partial sums to the case where the underlying distribution is in the domain of attraction of a nonsymmetric stable distribution (see below for details).
Let L α denote a stable distribution with exponent α ∈ (0,2).Recall that the distribution of X is said to be in the domain of attraction of L α if there exist some constants A n ∈ R 2 Chover-type LIL for weighted sums of mixing sequences and B n > 0 such that Under (1.3), Qi and Cheng [11] and Peng and Qi [10] showed that It is well known that (1.3) holds if and only if where, for x > 0, As for ρ * -mixing sequences of random variables, one can refer to Bryc and Smolenski [1], who established bounds for the moments of partial sums for a sequence of random variables satisfying Peligrad [7] established a CLT.Peligrad [8] established an invariance principle.Peligrad and Gut [9] established Rosenthal-type maximal inequalities and Baum-Katz-type results.Utev and Peligrad [12] established an invariance principle of nonstationary sequences.
To derive a Baum-Katz-type result, the main purpose of this paper is to establish a Chover-type law of the iterated logarithm for the weighted sums of ρ * -mixing and identically distributed random variables with a distribution in the domain of a stable law.Our result not only generalizes the main results of Peng and Qi [10] and Qi and Cheng [11] to ρ * -mixing sequences of random variables, but also improves them.
Throughout this paper, let h ∈ B[0,1] denote that the function h is bounded on [0,1].C will represent a positive constant though its value may change from one appearance to the next, and a n = O(b n ) will mean a n ≤ Cb n .

The main results
In order to prove our results, we need the following lemma and definition.Lemma 2.1 (Utev and Peligrad [12]).Let {X i , i ≥ 1} be a ρ * -mixing sequence of random variables, EX i = 0, E|X i | p < ∞ for some p ≥ 2 and for every i≥1.Then there exists Definition 2.2 (Lin and Lu [5]).A function f (x) > 0 (x > 0) is said to be quasimonotone nondecreasing, if Here are our main results.