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 X₀ = - ∞ 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_ijH_i(x) = Q_ij(x), where H₁(·), · · ·, H_m(·) are c.d.f.'s. Setting M_n = max {X₁, · · ·, X_n}, we obtain P{J_n = j, M_n ≦ x | J₀ = i} = [Qⁿ(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_ijⁿ(a_ijnx + b_ijn)} converges completely to a matrix {U_ij(x)} whose entries are non-degenerate distributions then U_ij(x) = π_jρ_U(x), where π_j = lim_{n → ∞}p_ijⁿ and ρ_U(x) is an extreme value distribution;

(c) the normalizing constants need not depend on i, j;

(d) ρⁿ(a_nx + b_n) converges completely to ρ_U(x);

(e) the maximum M_n has a non-trivial limit law ρ_U(x) iff Qⁿ(x) has a non-trivial limit matrix U(x) = {U_ij(x)} = {π_jρ_U(x)} or equivalently iff ρ(x) or the c.d.f. π_{i = 1}^mH_i^πi(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.