Asymptotic expansions for the distribution of the maximum of Gaussian random fields (Q1424677)

For a Gaussian random field \(X(t)\), \(t\in S\subset R^N\), with smooth enough trajectories and compact \(S\) the asymptotic expansions are derived of \(G(x)=\Pr\{\max_{t\in S} X(t)\geq u\}\) as \(u\to\infty\). For example, the following theorem is proved: \[ G(u)\sim\lambda(S)\sum_{j=0}^{[N/2]} k_{2j}\psi_{2j}(u), \] where \(\lambda(S)\) is the Lebesgue measure of \(S\), \[ k_{2j}=(-1)^j(\det\hbox{Var} X'(t))^{1/2}\binom{N}{2j}{(2j)!\over 2^j j!}, \] \[ \psi_{2j}(u)={1\over 2^{1+j}\pi^{(N+1)/2}} \int_{(u^2)/2}^{+\infty} x^{(N-1-2j)/2}e^{-x}dx. \] The remainders in these expansions are exponentially small. They are obtained by a Rice method. An application to the sea waves is considered.