Persistence probabilities of weighted sums of stationary Gaussian sequences

Abstract

With $\{{\xi }_i{\}}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $\rho (i):=\mathbb{E}[{\xi }_0{\xi }_i]$ and $\{\sigma (i){\}}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ${\sum }_{i=1}^{\ell }\sigma (i){\xi }_i$, $\ell \ge 1$. For summable correlations $\rho$, we show that the persistence exponent is universal. On the contrary, for non-summable $\rho$, even for polynomial weight functions $\sigma (i)\sim i^p$ the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter $H$) and on the polynomial rate $p$ of $\sigma$. In this case, we show existence of the persistence exponent $\theta (H,p)$ and study its properties as a function of $(p,H)$. During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations -- e.g.\ a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero -- that might be of independent interest.

Minor changes, accepted in SPA

Cited by in corpus (1)

• Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion