On the LSL for random fields

TL;DR

This paper extends Lai's law of the single logarithm for delayed sums to multi-dimensional random fields with regularly varying window sizes, providing new multiindex versions and specific case analyses.

Contribution

It introduces multiindex versions of Lai's law of the single logarithm for random fields with regularly varying window sizes, expanding previous one-dimensional results.

Findings

Established multiindex versions of Lai's law for random fields.

Analyzed cases with specific index sets in two dimensions.

Extended the law to include mixtures of expansion rates.

Abstract

In some earlier work we have considered extensions of Lai's (1974) law of the single logarithm for delayed sums to a multiindex setting with the same as well as different expansion rates in the various dimensions. A further generalization concerns window sizes that are regularly varying with index 1 (on the line). In the present paper we establish multiindex versions of the latter as well as for some mixtures of expansion rates. In order to keep things within reasonable size we confine ourselves to some special cases for the index set $\mathbb{Z}_+^2$.