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Let {Xn, n ≥ 1} be independent identically distributed random variables with a common non-degenerate distribution function F. For each n ≥ 1, denote Mn = max {X1,…, Xn}. Under certain conditions on F, there exist constants an > 0 and bn ∈ R such that 
. In this paper, we shall show that {(Mn – bn)/an} exhibits ergodic behaviour under additional conditions of F.
The existence of ordinary and exponential moments of a point process with conditional intensity of the form
is deduced from a Markov chain representation for t – ρN(t). These results form an application of recent theorems of Tweedie (1983a, b) and are used to obtain laws of large numbers for a range of functionals of the process.
