An almost sure limit theorem for the product of partial sums with stable distribution

On the functional limits for partial sums under stable law

“…Our main result in this note is to establish an almost sure version of the result by Kosiński [8] that can generalize the early results by Gonchigdanzan and Rempa la [4] and Gonchigdanzan [5,6].…”

“…Proof of Theorem 2. We first show the equivalence of (4) and (5) under the conditions of Theorem 2 setting d n := a n . In fact (3) is a direct consequence of Theorem 6.2 in DeAcosta and Giné [3].…”

Moderate deviations principle for products of sums of random variables

“…e √ 2Φ . Gonchigdanzan [5] proved an almost sure limit theorem for the product of partial sums of random variables in the domain of attraction of an α-stable law when 1 < α < 2. In [9,10], the first author established some CLT and ASCLT for the products of some general partial sums.…”