Between the LIL and the LSL

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 $|\mathbf{n}|^α$, or with different $α$’s, where the $α$’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.