Large excursion probabilities for random fields close to Gaussian ones

Abstract

This work continues the research started by Rudzkis (Soviet Math Dokl, 45(1), 226–228, 1992), Rudzkis and Bakshaev (Lithuanian Mathematical Journal, 52(2), 196–213, 2012) and extends it to the case of random fields close to Gaussian ones. Let $\{\xi (t), t \in \mathbb {R}^{m} \}$ be a differentiable (in the mean square sense) random field with $\mathbb {E} \xi (t)\equiv 0, \mathbb {D} \xi (t)\equiv 1$ and continuous trajectories. The paper is devoted to the problem of large excursions of the random field ξ. Let T be an m-dimensional interval and u(t) be a continuously differentiable function. We investigate the asymptotic properties of the probability $P=\mathbb {P}\{ \xi (t)< u(t), t \in T \}$ as $\inf _{t \in \mathbb {R}^{m}} u(t) \rightarrow \infty$ and the mixed cumulants of the random field ξ and its partial derivatives tend to zero, i.e. the scheme of series is considered. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then $\frac {1-P}{1-G}=1+o(1)$, where G is a constructive functional depending on u, T and a matrix function $R(t) = cov(\xi ^{\prime }(t),\xi ^{\prime }(t))$, $\xi ^{\prime }(t) = \left (\frac {\partial \xi (t)}{\partial t_{1}},...,\frac {\partial \xi (t)}{\partial t_{m}} \right )$.