2014/05/22 by Li, Deli, Chen, Pingyan
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1405.5619
Let 0 < α≤ 2 and - ∞ < β< ∞. Let \Xn; n ≥ 1 \ be a sequence of independent copies of a real-valued random variable X and set Sn = X1 + ⋯ + Xn, ~n ≥ 1. We say X satisfies the (α, β)-Chover-type law of the iterated logarithm (and write X ∈ CTLIL(α, β)) if \limsupn → ∞ | \fracSnn1/α |^(log log n)-1 = eβ 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 inf \b:~ 𝔼 (\frac|X|α(log (e \vee |X|))bα ) < ∞ \ = 1/α where 𝔼X = 0 whenever 1 < α≤ 2.