On asymptotic behavior of a finite Markov chain

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1 1 On asymptotic behavior of a finite Markov chain Alina Nicolae Department of Mathematical Analysis Probability. University Transilvania of Braşov. Romania. Keywords: convergence, weak ergodicity, strong ergodicity. AMS: 60J10 Abstract We give in the finite case three sufficient conditions, namely, on convergence, weak strong ergodicity, respectively, of a nonhomogeneous Markov chain in terms of similar properties of a certain chain of smaller size. We apply our results to simulated annealing to provide new results about his asymptotic behavior. 1. Preliminaries Consider a finite Markov chain with state space S = {1,..., r} transition matrices (P n ) n 1. We shall refer to it as the finite Markov chain (P n ) n 1. For all integers m 0, n > m, define P m,n = P m+1 P m+2...p n = ((P m,n ) ij ) i,j S. Assume that the limit lim P n = P (1) exists that limit matrix P has p 1 irreducible aperiodic closed classes, perhaps transient states, we mean it takes the form S S P = S p 0, (2) L 1 L 2... L p T where S i are the transition matrices, i = 1, p, for the p irreducible aperiodic closed classes, T concerns the chain as long as it stays in the r p t=1 r t transient states L i concern transitions from the transient states into the ergodic sets S i, i = 1, p. Markov chains of this type occun simulated annealing, a stochastic algorithm for global optimization. We refer to van Laarhoven Aarts ([5]) for a general exposition historical background. Definition 1.1. We say that a probability distribution µ = (µ 1,..., µ r ) is invariant with respect to an r r stochastic matrix P if we have µp = µ.

2 2 We shall need the following result THEOREM 1.2. Consider a finite homogeneous Markov chain on space S having the transition matrix P of the form (2). Then where lim P n = Γ i = Γ Γ Γ p 0 Ω 1 Ω 2... Ω p , (3) is an strictly positive matrix; each row of the matrix Γ i represents the invariant probability vector = ( 1,..., µ(i) ) with respect to the matrix S i Ω i = 1 z r 1+r r p+1,i... z r1+r r p+1,i z r,i... z r,i is an (r p t=1 r t) matrix, where z ji = probability that the chain will enter thus, will be absorbed in S i given that the initial state is j T, j = p r t, r, with convention r 0 = 1, i = 1, p. Proof. For the form of Γ i, i = 1, p, see [2, p. 123] for Ω i, i = 1, p, see [4, p. 91]. Remark 1.3. Clearly, z ji 0, j = r t, r, i = 1, p, (4) z ji = 1, j = i=1 r t, r. (5) THEOREM 1.4 ([6]). Let A = I r + P with P of the form (2). Then exists a nonsingular r r complex matrix Q such that A = QJQ 1 (6)

3 3 where J is an r r Jordan matrix. Q reads Q =, z r1+r r p+1,1 z r1+r r p+1,2... z r1+r r p+1,p... z r,1 z r,2... z rp... where the first column contains 1 at the 1, r 1 rows, the next p 1 columns contains 1 at the 1 + 1, rows, i = 2, p, the last r p columns sts for complex numbers. For z ji, j = p r t, r, i = 1, p, we use the signification given in Theorem 1.2. The inverse Q 1 has the form Q 1 = µ (1) 1... µ (1) r µ (p) 1... µ r (p) p q p+1,1 q p+1,r q r,1 q rr where are the invariant probability vectors with respect to S i, i = 1, p, the last r p rows sts for complex numbers. Proof. See [6]. If A = (A ij ) is an m n matrix, then for M {1,..., m}, N {1,..., n}, M, N we define A M N = (A ij ) i M,j N., Definition 1.5 (see, e.g., [2, p. 217]). A sequence of stochastic matrices (P n ) n 1 is said to be weakly ergodic if only if, m 0 i, j, k S, lim [(P m,n) ik (P m,n ) jk ] = 0. Definition 1.6 (see, e.g., [2, p. 223]). A sequence of stochastic matrices (P n ) n 1 is said to be strongly ergodic if m 0, i, j S, the limit lim (P m,n) ij = (π m ) j,

4 4 exists does not depend on i. Remark 1.7 (see, e.g., [2, p. 223]). It is easy to prove that if the relation above holds, then the (π m ) j are also independent of m 0. Definition 1.8 (see, e.g., [6]). We say that a finite Markov chain (P n ) n 1 converge if m 0 the sequence (P m,n ) n>m converges. 2. Weak strong ergodicity results In this section is continued an earlier study of the author from [6]. We give sufficient conditions for convergence, weak strong ergodicity, respectively, of a class of nonhomogeneous Markov chains in terms of similar behavior of a certain Markov chain of smaller size. In the sequel, we shall consider (P n ) n 1 a nonhomogeneous Markov chain on space S such that P n P as n. Suppose that P has exactly p 1 recurrent aperiodic classes S i, i = 1, p,, possibly, transient states, i.e., P is of the form (2). Let be the invariant probability vectors with respect to S i z ji, j = p r t, r, i = 1, p, as in Theorem 1.2. Let P n = P + V n, n 1, where lim V n = 0. Let Q Q 1 as in Theorem 1.4. Set Ṽ n = Q 1 V n Q, n 1. (7) C n = I p + (Ṽn) M M, M = {1,..., p} n 1. (8) PROPOSITION 2.1. C n is a stochastic matrix, n 1. THEOREM 2.2 ([6]). In the above context, lim (P m,n) ij = 0, i S, j = uniformly with respect to m 0. If, moreover, r t, r, (Q 1 V n Q) (S\M) M <, (9) n=1 then the chain (P n ) n 1 is convergent. Proof. See ([6]). The main result of this papes THEOREM 2.3. Suppose that (Ṽn) (S\M) M <. (10) n=1

5 5 Then (i) If (C n ) n 1 is weakly ergodic, then (P n ) n 1 is weakly ergodic, i.e. the chain (P n ) n 1 is weakly ergodic ; (iii) If (C n ) n 1 is strongly ergodic, then (P n ) n 1 is strongly ergodic, i.e. the chain (P n ) n 1 is strongly ergodic. Remark 2.4. We can apply this results to simulated annealing to describe some properties about his asymptotic behavior. Acknowledgements: CNCSIS-AT Code Bibliography The research was supported in part by the Grant [1] Horn, R. Johnson, C. (1985). Matrix Analysis. Cambrige University Press, New York. [2] Iosifescu, M. (1980). Finite Markov Processes Their Applications. Wiley, Chichester Editura Tehnică, Bucureşti. [3] Isaacson, D. L. Madsen, R. W. (1976). Markov Chains: Theory Applications. Wiley, New York. [4] Karlin, S. Taylor, H. M. (1975). A First Course in Stochastic Processes. Academic Press, New York. [5] van Laarhoven, P. J. M. Aarts, E. H. L. (1987). Simulated Annealing: Theory Applications. D. Reidel Publishing Company, Dordrecht, Holl. [6] Nicolae, A. A sufficient condition for the convergence of a finite Markov chain. To appear.