Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

Abstract

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn, where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and πn for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of πn to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r0 > 1 with , then the exact convergence rate of πn to π is characterized by r0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between πn and π based on the moments of A.

Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity

“…Although these matrices are very close to each other, we must stress that P; is not an augmented truncation of P, i.e. the assumptionpn(i,j)~p(i,j) for O~i~n and O~j~n is not always fulfilled for i = n. Therefore the established theory on the approximation of Markov chains relying on augmented truncation does not apply (Gibson and Seneta 1987, Simonot 1995, Simonot and Song 1996.…”

“…This property is crucial for the derivation of the convergence rate. Following Takacs (1962), the relationships between the TSD and the ASD are reviewed. In view of these results, it is enough to address the ASD.…”

“…In view of these results, it is enough to address the ASD. Section 3 is based on an approach introduced by Simonot (1995) and Simonot and Song (1996). A coupling tying the embedded Markov chains is built on, which permits an easy comparison to be made between the finite and infinite queueing systems.…”

Moments based approximation for the stationary distribution of a random walk in Z+ with an application to the M/GI/1/n queueing system

“…In this context, we must quote the work of Kalashnikov (see Kalashnikov and Rachev (1990), Chapter 8; Kalashnikov (1994), Chapter 8, and the bibliography therein for complete references) who addressed the problem of the convergence rate for uniform approximation of Markov chains. For stochastically monotone Markov chains, and in particular for the random walk in Z + , Simonot and Song (1996) derived exact upper bounds for the distance in total variation between the probability distributions π n and π, when the approximated chain is geometrically recurrent. Moreover, they established in Corollary 4.1 that the rate of convergence of π n to π is geometric if and only if the tail of the increment F(a) = P(A > a) is geometrically decreasing (i.e.…”

A note on the two-sided regulated random walk

“…more or less rapidly depending on the additional assumptions made on the increment A, this case has been investigated in many papers such as [3,[6][7][8][12][13][14]18,19].…”