On optimal condition numbers for Markov chains |
| |
Authors: | Stephen J Kirkland Michael Neumann Nung-Sing Sze |
| |
Institution: | (1) Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada;(2) Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA |
| |
Abstract: | Let T and be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)–norm pair if . Here π and are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect
to the (1, ∞) and (∞, ∞)-norm pairs have been suggested in the literature. They were ranked according to their size by Cho
and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient
, where e is the n-vector of all 1’s and A
# is the group generalized inverse of A = I − T, is the smallest condition number of Markov chains with respect to the (p, ∞)-norm pair. We use this result to identify the smallest condition number of Markov chains among the (∞, ∞) and (1, ∞)-norm
pairs. These are, respectively, κ
3 and κ
6 in the Cho–Meyer list of 8 condition numbers. Kirkland has studied κ
3(T). He has shown that and he has characterized transition matrices for which equality holds. We prove here again that 2κ
3(T) ≤ κ(6) which appears in the Cho–Meyer paper and we characterize the transition matrices T for which . There is actually only one such matrix: T = (J
n
− I)/(n − 1), where J
n
is the n × n matrix of all 1’s.
This research was supported in part by NSERC under Grant OGP0138251 and NSA Grant No. 06G–232. |
| |
Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 65F35 60J10 15A51 15A12 15A18 |
本文献已被 SpringerLink 等数据库收录! |
|