Asymptotic behaviour of Gaussian processes with integral representation. (Q1877537)

The asymptotic behaviour of \(a(t)X_t\), where \(a\) is a function vanishing at infinity and \(X\) is a Gaussian process with continuous paths defined by an integral representation, is studied. As an illustration of the general statement the author presents e.g. the following result: Let \(G\) be a locally compact Abelian group with unbounded translation invariant metric \(d\). Let \(T\subset G\) be a countable set ``uniformly discrete'', i.e. \(\sup _{x\in T}\,\text{Card}(B(x,s)\cap T)<\infty ,\) where \(B(x,s)\) denotes the ball centred at \(x\) and with the radius \(r\). If \((X_t)_{t\in G}\) is a stationary Gaussian process admitting a spectral density and with the variance \(\sigma ^2\), then \[ \limsup _{k\in T,\,d(0,k)\to \infty } a_kX_k =\inf \Bigl \{ \beta \geq 0\Bigm | \sum _{k\in T} a_k \exp \Bigl (-\frac {\beta ^2}{2\sigma ^2a_k^2}\Bigr ) <\infty \Bigr \}\quad \text{a.s.} \] for every positive sequence \(a_k\) vanishing at infinity.