Asymptotics for the local time of a strongly dependent vector-valued Gaussian random field
The authors study the local time of normalized vector-valued Gaussian random fields \(X=(X_t; t\in\mathbb{R}^d)\), \(X_t\in\mathbb{R}^p\). Extending a result of \textit{S. M. Berman} [Stochastic Processes Appl. 12, 1-26 (1981; Zbl 0471.60082)] they obtain a series expansion of the local time \(\ell_X\) in terms of multidimensional Hermite polynomials. Moreover, for \(X\) stationary with a covariance matrix of long range dependence type, the authors apply a theorem of \textit{M. V. Sanchez de Naranjo} [J. Multivariate Anal. 44, No. 2, 227-255 (1993; Zbl 0770.60025)] to deduce, from this series expansion, the asymptotic behaviour of the local time \(\ell_X([0,\tau]^d,x)\) \((x\in\mathbb{R}^p)\) as \(\tau\to\infty\), in terms of random spectral measures associated to \(X\). Finally, the authors consider a sufficiently regular bijective transform \(Y\) of \(X\) (\(X\) stationary) and give series expansions of the kernel estimates for the marginal density of \(Y\). This extends a result of \textit{M. Rosenblatt} [NSF-CBMS Regional Conference Series in Probability and Statistics 3 (1991)].
scientific article; zbMATH DE number 878138