On the strong approximations of partial sums of f(nkx)

TL;DR

This paper establishes a strong invariance principle for sums of a smooth periodic function evaluated at a random increasing sequence, revealing precise asymptotic behavior of lacunary series with random gaps without gap size restrictions.

Contribution

It introduces a strong invariance principle for sums involving random lacunary sequences, extending classical results to cases with no assumptions on gap sizes.

Findings

Strong invariance principle proven for sums of f(n_k x) with random increasing sequences.

Asymptotic properties of lacunary series with random gaps are characterized precisely.

Results hold without restrictions on the size of the gaps.

Abstract

We prove a strong invariance principle for the sums PN k=1 f(nkx), where f is a smooth periodic function on R and (nk)k?1 is an increasing random sequence. Our results show that in contrast to the classical Salem-Zygmund theory, the asymptotic properties of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps.