Asymptotic expansions for the probabilities of large runs of nonstationary Gaussian processes (Q802200)

Let X(t), \(t\in (\)-\(\infty,\infty)\), be a real-valued nonstationary Gaussian process with \(EX(t)=m(t)\), \(E(X(t)\)-m(t))(X(s)-m(s))\(=r(t,s)\), and let \(X^{(n)}(t)\), \(n\geq 2\), be mean square derivatives such that \(M(X^{(n)}(t)\)-X\({}^{(n)}(s))^ 2\leq c| t\)-s\(|^{\alpha}\), \(\alpha\geq 0\). Under some conditions of smoothness the asymptotic expansions \[ U^{1/\ell}\exp (u^ 2/2\sigma^ 2(t_ 0))P(\sup_{- \infty <t<\infty}X(t)>u)=\sum^{n-2\ell}_{k=0}c_ ku^{-k}+O(u^{- n+2\ell -\alpha}),\quad u\to \infty, \] are given, where \(t_ 0\) is the unique point such that sup \(\sigma\) \({}^ 2(t)=\sigma^ 2(t_ 0)\), the integer \(\ell >0\) such that \(\sigma^{(r)}(t_ 0)=0\), \(r<2\ell\), \(\underline{\lim \sup}_{t\to \infty}\sigma (t)<\sigma (t_ 0)\), \(\sigma^ 2(t)=r(t,t)\), and \(c_ k=c_ k(X^{(j)}(t_ 0))\), \(k\leq n\)-2\(\ell\).