Moderate deviations and Strassen's law for additive processes

TL;DR

This paper proves a moderate deviation principle for additive processes with independent increments and uses it to provide a new proof of Strassen's law of the iterated logarithm, including for square-integrable Lévy processes.

Contribution

It introduces a novel moderate deviation principle for additive processes and applies it to establish Strassen's law, including for Lévy processes.

Findings

Established a moderate deviation principle for additive processes.

Provided a new proof of Strassen's law of the iterated logarithm.

Showed that square-integrable Lévy processes satisfy Strassen's law.

Abstract

We establish a moderate deviation principle for processes with independent increments under certain growth conditions for the characteristics of the process. Using this moderate deviation principle, we give a new proof for Strassen's functional law of the iterated logarithm. In particular, we show that any square-integrable L\'evy process satisfies Strassen's law.