Strong Approximations for Nonconventional Sums with Applications to Law of Iterated Logarithm and Almost Sure Central Limit Theorem

TL;DR

This paper establishes strong approximation results for nonconventional sums, leading to new versions of the law of iterated logarithm and almost sure central limit theorem, with applications in stochastic processes and dynamical systems.

Contribution

It introduces a strong invariance principle for nonconventional sums, enabling derivation of classical probabilistic limit laws in this context.

Findings

Established a strong invariance principle for nonconventional sums

Derived a version of the law of iterated logarithm for these sums

Proved an almost sure central limit theorem for nonconventional sums

Abstract

We obtain a strong invariance principle for nonconventional sums and applying this result we derive for them a version of the law of iterated logarithm, as well as an almost sure central limit theorem. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems.