Asymptotic behavior of trajectories of stationary Banach-valued Gaussian fields. III (Q1603267)

Let \(\xi(t)\), \(t\in R^d\), be a continuous stationary Gaussian field with values in a separable Banach space. Let \(T\) be an unbounded subset of \(R_+\), \(\psi\) a nondecreasing function on \(T\) and \(\{G_u\}_{u\in T}\) an increasing family of bounded sets in \(R^d\) such that \(\bigcup_{u\in T}G_u\) is unbounded. The author investigates the random sets \(\Xi_u\), \(u\in T\), being the closures of \(\{\xi(t)/ \psi(u): t\in G_u\}\) and their asymptotic behaviour under \(u\to \infty\). In the previous parts of the paper [J. Math. Sci., New York 88, No. 1, 76-80 (1998); translation from Zap. Nauchn. Semin. POMI 216, 110-116 (1994; Zbl 0873.60035) and ibid. 93, No. 3, 392-398 (1999) resp. 228, 220-228 (1996; Zbl 0944.60041)] the author found conditions on \(\xi\), \(\psi\) and \(\{G_u\}\) under which \(\limsup_{u\to\infty} \Xi_u\) is a.s. equal to the unit ball \(K\) of the reproducing kernel Hilbert space of the distribution of \(\xi(0)\). In the present part it is established the a.s. rate of convergence to zero of \(\text{dist} (\Xi_u,h)\) when \(u\to\infty\) for any \(h\in K\).