Between the LIL and the LSL

TL;DR

This paper investigates the behavior of delayed sums in a multiindex setting where the window size grows at a rate between the law of the iterated logarithm and the law of large numbers, extending previous work on the law of the single logarithm.

Contribution

It extends Lai's law of the single logarithm to a setting where the window size grows like n/log n, a rate between the LIL and the LLN, providing new asymptotic results.

Findings

Established the asymptotic behavior of delayed sums with window size n/log n.

Extended previous multiindex LIL results to higher growth rates.

Provided theoretical insights into the boundary between LIL and LLN regimes.

Abstract

In two earlier papers, two of the present authors (A.G. and U.S.) extended Lai's [Ann. Probab. 2 (1974) 432--440] law of the single logarithm for delayed sums to a multiindex setting in which the edges of the $\mathbf{n}$th window grow like $|\mathbf {n}|^{\alpha}$, or with different $\alpha$'s, where the $\alpha$'s belong to $(0,1)$. In this paper, the edge of the $n$th window typically grows like $n/\log n$, thus at a higher rate than any power less than one, but not quite at the LIL-rate.