Theory of Probability and Mathematical Statistics
On the other LIL for variables without finite variance
R. P. Pakshirajan and M. Sreehari
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Abstract: In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables Xn satisfies the condition that limx→∞logH(x)/(logx)δ for some 0<δ<1/2, where H(x)=E(X2I(|X1|≤x)) is a slowly varying function. The condition above is not very restrictive.
Keywords: Chung’s LIL, domain of non-normal attraction, Skorohod embedding, slowly varying function, Karamata representation
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