Asymptotic properties of the norm of extremum values of normal random elements in the space C[0, 1]

Abstract

We prove that $$\mathop {\lim }\limits_{n \to \infty } \left( {\left\| {Z_n } \right\| - (2 ln (n))^{1/2} \left\| \sigma \right\|} \right) = 0 a.s.,$$ where X is a normal random element in the space C [0,1], MX = 0, σ = {(M¦X(t)¦²)^1/2 t∈[0,1}, (X _n ) are independent copies of X, and $Z_n = \mathop {\max }\limits_{l \leqslant k \leqslant n} X_k$ . Under additional restrictions on the random element X, this equality can be strengthened.