Limit Distribution of a High Order Markov Chain

Abstract

The limit distribution of a Markov chain of order k > 1 is obtained under conditions weaker than those assumed by Raftery (1985). The results are shown to be valid also for arbitrary state space Markov sequences of order k > 1. XES such that n(y) > 0, L n(y) = 1. yES t

Investigating purchasing-sequence patterns for financial services using Markov, MTD and MTDg models

“…The MTD model has been developed for infinite lags (Le et al, 1996), for time series with missing data and for Markov chains with an infinite denumberable state space (Raftery, 1985b) or with an arbitrary state space, i.e., the GMTD model (Martin and Raftery, 1987;Adke and Deshmukh, 1988;Raftery, 1993;Le et al, 1996;Wong and Li, 2000). The MTD model has even been applied in a spatial context (Raftery and Banfield, 1991;Berchtold, 2001).…”

A Stochastic Characterization of a Fault-Tolerant Gossip Algorithm

“…By setting q (1) = 1− p (1) , the transition probability in (1) can thus be obtained by substituting q (1) for q, and the carefully chosen (n−mf )(1−q (1) ) nm for the adjustment coefficient ∆(n, m, f, q), in (2). Thus, for i ≥ j ≥ 1, we have…”

“…Specifically, we model the behavior of the gossip algorithm with a discrete stochastic process. As we will show later, this leads to a m-order Markov chain (cf., e.g., [1]) for m > 1 that we are unable to analytically resolve, thus we "approximate" it with a carefully designed 1-order Markov chain. The resulting model offers the following new insights that are useful in helping designers make the right decisions.…”

“…where the superscript "(1)" of τ (1) i,j is used to represent the setting of logical views. The physical views case.…”

“…where superscript "(1)" of p (1) is used to represent the setting of logical views. By setting q (1) = 1− p (1) , the transition probability in (1) can thus be obtained by substituting q (1) for q, and the carefully chosen (n−mf )(1−q (1) ) nm for the adjustment coefficient ∆(n, m, f, q), in (2).…”