An Exponential Probability Bound for the Energy of a Type of Gaussian Process

Abstract

Real-valued stochastic processes of the form $x(t) = \int A(t, \lambda)Z(d\lambda)$ are considered, where $Z(\lambda)$ is a zero mean Gaussian process with independent increments and $\int\int |A(t, \lambda)|^2F(d\lambda) dt < \infty$, where $F(d\lambda) = E|Z(d\lambda)|^2$. It is shown that the energy of $x(t), \int x^2(t) dt$, is a well-defined random variable and an exponential bound for $P(\int x^2(t) dt - E\int x^2(t) dt \geqq \varepsilon)$ is derived. This bound is used to obtain an exponential bound for crossing probabilities $P(|y(t)| > a$ for some $t)$ where $y(t) = \int h(t - \tau)x(\tau) d\tau, \int h^2(t) dt < \infty$.