Large deviations for functionals of some self-similar Gaussian processes

TL;DR

This paper establishes large deviation principles for functionals of self-similar Gaussian processes, including fractional, sub-fractional, and bi-fractional Brownian motions, with applications to exponential integrability.

Contribution

It provides the first large deviation results for a broad class of self-similar Gaussian process functionals with singular kernels.

Findings

Large deviation principles are proved for integrals involving self-similar Gaussian processes.

Results include fractional, sub-fractional, and bi-fractional Brownian motions.

Applications to exponential integrability of these functionals are discussed.

Abstract

We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional self-similar Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-\beta}$ with $\beta\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.