A characterization of Chover-type law of iterated logarithm

作者：Li, Deli; Chen, Pingyan^*

来源：SpringerPlus, 2014, 3(1): 386.

DOI：10.1186/2193-1801-3-386

• (alpha, beta)-Chover-type law of the iterated logarithm
• Sums of i.i.d. random variables
• Symmetric stable distribution with exponent alpha

Let 0 < alpha <= 2 and -infinity < beta < infinity. Let {X-n; n >= 1} be a sequence of independent copies of a real-valued random variable X and set S-n = X-1 (. . .) X-n, n >= 1. We say X satisfies the (alpha, beta)-Chover-type law of the iterated logarithm (and write X is an element of CTLIL(alpha, beta)) if lim sup(n ->infinity) vertical bar S-n/n(1/alpha)vertical bar((log log n)-1) = e(beta) almost surely. This paper is devoted to a characterization of X is an element of CTLIL(alpha, beta). We obtain sets of necessary and sufficient conditions for X is an element of CTLIL(alpha, beta) for the five cases: alpha = 2 and 0 < beta < infinity, alpha = 2 and beta = 0, 1 < alpha < 2 and -infinity < beta < infinity, alpha = 1 and -infinity < beta < infinity, and 0 < alpha < 1 and -infinity < beta < infinity. As for the case where alpha = 2 and -infinity < beta < 0, it is shown that X is not an element of CTLIL(2, beta) for any real-valued random variable X. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm (i.e., X is an element of CTLIL(alpha, 1/alpha)) is given; that is, X is an element of CTLIL(alpha, 1/alpha) if and only if inf {b : E (vertical bar X vertical bar(alpha)/(log(rho V vertical bar X vertical bar))(b alpha)) < infinity} = 1/alpha where EX = 0 whenever 1 < alpha <= 2.
Mathematics Subject Classification (2000): Primary: 60F15; Secondary: 60G50

• 出版日期2014-7-28
• 单位暨南大学