A continuous analogue of the invariance principle and its almost sure version

TL;DR

This paper extends the invariance principle to continuous analogues, demonstrating convergence of transformed processes to Wiener processes and establishing an almost sure version of this convergence.

Contribution

It introduces a continuous analogue of the invariance principle and proves an almost sure convergence result for these processes.

Findings

Processes converge in distribution to Wiener process

An integral type almost sure convergence theorem is established

The results extend classical invariance principles to continuous settings

Abstract

We deal with random processes obtained from a homogeneous random process with independent increments by replacement of the time scale and by multiplication by a norming constant. We prove the convergence in distribution of these processes to Wiener process in Skorokhod space endowed by the topology of uniform convergence. An integral type almost sure version of this theorem is obtained.