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

Abstract

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn 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 {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.