Extreme values for stationary sequences

Abstract

Let (X_n) ^∞_{n = 1} be a strictly stationary real-valued sequence and let F(x) = P[X₁ ≤ x]. Let M_i,j = max(X_i+1,..., X_j) and M_j = max(X₁,...,X_j) = M_0,j. Let (c_n) ^∞_{n = 1} be a sequence of real numbers. The purpose of this article is to review the theory of the asymptotic behaviour of P[M_n ≤ c_n] as n → ∞. We mainly consider aspects which are not treated in Leadbetter, Lindgren and Rootzén [4]. To avoid trivialities we assume that F(c_n) < 1 for all n, F(c_n) → 1, and P[X₁ = sup{x: F(x) < 1}] = 0. All limits are “as n → ∞.”