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} withX₀= - ∞ a.s. The marginal sequence {J_n} is an irreducible, aperiodic,m-state M.C.,m< ∞, and the r.v.'sX_nare conditionally independent given {J_n}. FurthermoreP{J_n=j, X_n≦x|J_{n− 1}=i} =p_ijH_i(x) =Q_ij(x), whereH₁(·), · · ·,H_m(·) are c.d.f.'s. SettingM_n= max {X₁, · · ·,X_n}, we obtainP{J_n=j, M_n≦x|J₀=i} = [Qⁿ(x)]_{i, j}, whereQ(x) = {Q_ij(x)}. The limiting behavior of this probability and the possible limit laws forM_nare characterized.Theorem.Let ρ(x) be the Perron-Frobenius eigenvalue ofQ(x) for realx; 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 thenU_ij(x) = π_jρ_U(x), where π_j= lim_{n→ ∞}p_ijⁿand ρ_U(x) is an extreme value distribution;(c) the normalizing constants need not depend oni, j;(d) ρⁿ(a_nx+b_n) converges completely to ρ_U(x);(e) the maximumM_nhas a non-trivial limit law ρ_U(x) iffQⁿ(x) has a non-trivial limit matrixU(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.