Limit laws for maxima of a sequence of random variables defined on a Markov chain

Abstract

Consider the bivariate sequence of r.v.'s {(J n , X n ), n ≧ 0} with X 0 = - ∞ a.s. The marginal sequence {J n } is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s X n are conditionally independent given {J n }. Furthermore P{J n = j, X n ≦ x | J n − 1 = i} = p ij H i (x) = Q ij (x), where H 1(·), · · ·, H m (·) are c.d.f.'s. Setting M n = max {X 1, · · ·, X n }, we obtain P{J n = j, M n ≦ x | J 0 = i} = [Q n (x)] i, j , where Q(x) = {Q ij (x)}. The limiting behavior of this probability and the possible limit laws for M n are characterized. Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then: (a)ρ(x) is a c.d.f.; (b) if for a suitable normalization {Q ij n (a ijn x + b ijn )} converges completely to a matrix {U ij (x)} whose entries are non-degenerate distributions then U ij (x) = π j ρ U (x), where π j = lim n → ∞ p ij n and ρ U (x) is an extreme value distribution; (c) the normalizing constants need not depend on i, j; (d) ρ n (a n x + b n ) converges completely to ρ U (x); (e) the maximum M n has a non-trivial limit law ρ U (x) iff Q n (x) has a non-trivial limit matrix U(x) = {U ij (x)} = {π j ρ U (x)} or equivalently iff ρ(x) or the c.d.f. π i = 1 m H i π i(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {M n } are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.