A limit theorem for the moment of self-normalized sums
Consider a zero mean random walk \(\{S_n\}\). Hsu-Robbins-Spitzer-Baum-Katz type theorems concern finiteness of Borel-Cantelli sums related to deviations from the mean, that is, of \(P(|S_n|>\varepsilon g(n))\), where \(g(n)\) typically is a (suitable) power of \(n\) possibly with some logartihm or iterated logarithm added, depending on whether one considers LLN, LIL, and so on. Theorems called ``Precise estimates\dots'' concern, properly scaled, limits of such sums as \(\varepsilon\searrow\) (to \(0\) or \(\sigma\sqrt{2}\), etc., depending on the kind of problem at hand). Such results can be extended to moments, in which case probabilities are replaced by centered expectations, and, also, as in the present paper, to self-normalized sums. The usual objects of investigation in the context of precise asymptotics are asymptotics concerning probabilities. In this paper the authors establish corresponding results for expected values of the counting process associated with the sequence of record times, extending earlier ones for probabilities.
scientific article; zbMATH DE number 5624608