Large deviations for functionals of some self-similar Gaussian processes

ABSTRACT

We prove large deviation principles for ${\int }_0^t\gamma (X_s)\mathrm{d}s$, 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.

KEYWORDS:

• Self-similar Gaussian process
• fractional Brownian motion
• sub-fractional Brownian motion
• bi-fractional Brownian motion
• reproducing kernel Hilbert space
• local time
• large deviation principles

SUBJECT CLASSIFICATIONS:

• Primary 60G15
• 60G18
• 60G22
• 60J55
• 60F10

Disclosure statement

No potential conflict of interest was reported by the author(s).