Sojourn in an elliptical domain
Let \(X(t)=(X_ 1(t),X_ 2(t))\), \(t\geq 0\), be a two-dimensional stationary Gaussian process with zero mean. \(X_ 1(t)\) and \(X_ 2(t)\) are correlated with long-range dependence and the cross-relations between \(X_ 1(t)\) and \(X_ 2(t)\) are allowed to have the same order of magnitude as the auto-correlations. Let \(D=\{(x_ 1,x_ 2):\) \(a^ 2x^ 2_ 1+b^ 2x^ 2_ 2\leq 1\}\), \(0<a\leq b\), be an elliptical domain. Define \(M(t)=\int^{t}_{0}I[X(s)\in D]ds\), \(t\geq 0\), which denotes the amount of time that the process X(\(\cdot)\) has spent in the domain D by time t. The main theorem concerns the asymptotic behaviour as \(t\to \infty\) of the standardized process \[ Z_ t(\tau)=\{M(t\tau)- EM(t\tau)\}/[Var M(t)]^{1/2},\quad \tau \geq 0. \] It is shown that as \(t\to \infty\) the process \(Z_ t(\tau)\) converges weakly in C[0,\(\infty)\) to a limiting process \(\bar Z(\tau)\) which admits a Wiener- Itô double integral representation.
two-dimensional stationary Gaussian process
Wiener-Itô double integral representation
scientific article; zbMATH DE number 3965093