On integral functionals of extremal random functions. (Ukrainian, English) Zbl 1026.60027

Matsak, I. K.

Teor. Jmovirn. Mat. Stat. 65, 110-120 (2001); translation in Theory Probab. Math. Stat. 65, 122-134 (2002).

Let \(\{X(t),t\in T\}\) be a stochastic process. The author deals with processes of the form \(U_n=\{U_n(t)=b_n(t)(Z_n(t)-a_n(t)),t\in T\}\), where \(Z_n(t)=\max_{1\leq k\leq n}X_k(t)\), \(t\in T\), and \(X_k,k\geq 1\), are independent copies of the process \(\{X(t),t\in T\}\). He proposes conditions under which distributions of the integral functionals of processes \(U_{n}\) converge. This article may be considered as continuation of the paper of the author [Ukr. Math. J. 51, No. 9, 1352-1361 (1999); translation from Ukr. Mat. Zh. 51, No. 9, 1201-1209 (1999; Zbl 0957.60060)]. In this paper he weakens the moment conditions on the process \(\{X(t),t\in T\}\) and strengthens the assertion of the main theorem. For more details see the article by J. Pickands [Ann. Math. Stat. 39, 881-889 (1968; Zbl 0176.48804)] and the book by M. R. Leadbetter, G. Lindgren and H. Rootzen [“Extremes and related properties of random sequences and processes” (1983; Zbl 0518.60021)].

Reviewer: M.P.Moklyachuk (Kyiv)

MSC:

60F05	Central limit and other weak theorems
60F15	Strong limit theorems
60B12	Limit theorems for vector-valued random variables (infinite-dimensional case)
60G70	Extreme value theory; extremal stochastic processes

Citations:

Zbl 0957.60060; Zbl 0176.48804; Zbl 0518.60021