Asymptotic behaviour of mean uniform norms for sequences of Gaussian processes and fields (Q881405)

Let \(X_{n}(t),\;t\in T_{n},\;n\geq1\), be a sequence of Gaussian zero mean random functions with continuous realizations. Denote by \(M_{n}(T_{n})=\max_{t\in T_{n}}| X_{n}(t)|\) the uniform norm of \(X_{n}(\cdot),\;n\geq1\). The author studies the asymptotic behavior of the mean uniform norms \(EM_{n}(T_{n})\) as \(n\to\infty\) provided that the normalized uniform norms \(M_{n}(T_{n})\) converge in distribution and for the \(m\)-cube \(T_{n}=[0,d_{n}]^{m},\;m\geq1, d_{n}\to\infty\) as \(n\to\infty\). It is proved that under some conditions \((EM_{n}^{p})^{1/p}\sim (2\log n)^{1/2}\) as \(n\to\infty\). The author considers some applications of the obtained results and demonstrates the rate of convergence for several Gaussian processes and fields.