A characterization of Chover-type law of iterated logarithm

ABSTRACT: Let 0 < α ≤ 2 and − ∞ <β <∞. 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 (α,β)-Chover-type law of the iterated logarithm (and write X∈CTLIL(α,β)) if [Image: see text] almost surely. This paper is devoted to a characterization of X ∈CTLIL(α,β). We obtain sets of necessary and sufficient conditions for X∈CTLIL(α,β) for the five cases: α = 2 and 0 < β <∞, α = 2 and β = 0, 1<α<2 and −∞<β<∞, α = 1 and −∞ <β <∞, and 0 < α <1 and −∞ <β <∞. As for the case where α = 2 and −∞ <β <0, it is shown that X∉CTLIL(2,β) 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∈CTLIL(α,1/α)) is given; that is, X∈CTLIL(α,1/α) if and only if [Image: see text] where [Image: see text] whenever 1< α ≤ 2. MATHEMATICS SUBJECT CLASSIFICATION (2000): Primary: 60F15; Secondary: 60G50