Probabilities of large excursions of nearly Gaussian stochastic processes (Q757981)

A continuously differentiable random process \(\xi\) is considered. Moreover, uniform bounds are assumed for the cumulants of every fixed number of variables from the union of the original and the derived processes, namely \(| cum\{Y_ 1,...,Y_ k\}| \leq (k!)^{\beta}/\Lambda^{k-2}\) for every \(\{Y_ 1,...,Y_ k\}\in \cup_{0\leq t\leq T}(\xi (t),\xi '(t))\). The aim of the paper is to study the probability of the trajectories on the interval [0,T] lying between two continuously differentiable functions -v(t), u(t). An asymptotic expression is derived for the case when the upper and the lower functions are uniformly tending to plus infinity and minus infinity, respectively, and the higher order cumulants are vanishing. More precisely, if \(\min \min_{0\leq t\leq T}(v(t),u(t))\to \infty\) and \(\Lambda\to \infty\), then \[ P\{-v(t)\leq \xi (t)\leq u(t);\quad 0\leq t\leq T\}-G=o(1-G), \] where \[ G=[\int^{u(0)}_{-v(0)}\phi (x)dx]\exp \{- \sum_{g=u,v}\int^{T}_{0}\phi (g(t))\phi (g'(t)+z)z dz\}, \] and \[ \phi (x)=e^{-x^ 2/2}/\sqrt{2\pi}. \]