On the other LIL for variables without finite variance

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 X n X_n satisfies the condition that lim x → ∞ log ⁡ H ( x ) ( log ⁡ x ) δ = 0 \lim _{ x\rightarrow \infty } \frac {\log H(x)}{(\log x)^\delta } = 0 for some 0 > δ > 1 / 2 0 >\delta > 1/2 , where H ( x ) = E ( X 1 2 I ( | X 1 | ≤ x ) ) H(x)=\mathsf E\left (X_1^2 I(|X_1|\le x)\right ) is a slowly varying function. The condition above is not very restrictive.