On the limiting behaviors of increments of sums of random variables without moment conditions
Let \(\{S_ n,n \geq 1\}\) be the partial sums of a sequence of independent random variables. Let \(\{a_ n\}\) be a nondecreasing sequence of positive integers and \(\{B_{n,N}, N \geq 1, 0 \leq n \leq N\}\) a positive double sequence such that \(B_{n,N} \uparrow \infty\) uniformly in \(n\) as \(N \to \infty\). Upper and lower almost-sure bounds are presented, under certain hypotheses, for \(\limsup_{N \to \infty} | S_{N+a_ N}- S_ N |/ (B_{N,N+a_ N} b_ N)\), where \(b^ 2_ N=2\{2 \log (B_{N+a_ N}/B_ N)+\log \log B^ 2_ N\}\). These results generalize earlier theorems of the authors [Sci. Sin. 30, 464-477 (1990)] and are reminiscent of work of \textit{M. J. Klass} and \textit{R. J. Tomkins} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 68, 107-120 (1984; Zbl 0552.60026)] on generalized laws of the iterated logarithm for \(\{S_ n\}\).
sums of a sequence of independent random variables
scientific article; zbMATH DE number 495844