On a set of transformations of Gaussian random functions

TL;DR

This paper explores transformations of Gaussian random functions, establishing a link between their small ball asymptotics and deriving explicit Karhunen-Loève expansions for certain Gaussian processes.

Contribution

It introduces a connection between small ball asymptotics of original and transformed Gaussian functions and provides explicit Karhunen-Loève expansions for a class of Gaussian processes.

Findings

Established a link between small ball asymptotics of original and transformed functions

Derived explicit Karhunen-Loève expansions for specific Gaussian processes

Provided theoretical insights into transformations of Gaussian random functions

Abstract

We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit Karhunen -- Lo\'eve expansion is obtained for a proper class of Gaussian processes.