A characterization of Chover-type law of iterated logarithm

• Published in: SpringerPlus Vol. 3; no. 1; pp. 386 - 7
• Main Authors: Li, Deli, Chen, Pingyan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 28.07.2014; Springer Nature B.V
• Subjects: Humanities and Social Sciences; multidisciplinary; Science; Science (multidisciplinary); Statistics; (; ,; Chover-type law of the iterated logarithm; Symmetric stable distribution with exponent; Sums of i.i.d. random variables; (α,β)-Chover-type law of the iterated logarithm; Symmetric stable distribution with exponent α
• ISSN: 2193-1801; 2193-1801
• DOI: 10.1186/2193-1801-3-386

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 limsup n → ∞ S n n 1 / α ( log log n ) − 1 = e β 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 inf b : E | X | α ( log ( e ∨ | X | ) ) bα < ∞ = 1 / α where EX = 0 whenever 1< α ≤ 2. Mathematics Subject Classification (2000) Primary: 60F15; Secondary: 60G50