A characterization of Chover-type law of iterated logarithm

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∈C T L I L(α,β)) if [Formula: see text] almost surely. This paper is devoted to a characterization of X ∈C T L I L(α,β). We obtain sets of necessary and sufficient conditions for X∈C T L I L(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∉C T L I L(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∈C T L I L(α,1/α)) is given; that is, X∈C T L I L(α,1/α) if and only if [Formula: see text] where [Formula: see text] whenever 1< α ≤ 2.

Primary: 60F15; Secondary: 60G50.