scholarly journals Error Bounds for Augmented Truncations of Discrete-Time Block-Monotone Markov Chains under Geometric Drift Conditions

2015 ◽  
Vol 47 (01) ◽  
pp. 83-105 ◽  
Author(s):  
Hiroyuki Masuyama
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Total Variation ◽  
Discrete Time ◽  
Error Bounds ◽  
Total Variation Distance ◽  
Time Block ◽  
Stationary Distributions ◽  
Variation Distance ◽  
Drift Conditions

In this paper we study the augmented truncation of discrete-time block-monotone Markov chains under geometric drift conditions. We first present a bound for the total variation distance between the stationary distributions of an original Markov chain and its augmented truncation. We also obtain such error bounds for more general cases, where an original Markov chain itself is not necessarily block monotone but is blockwise dominated by a block-monotone Markov chain. Finally, we discuss the application of our results to GI/G/1-type Markov chains.

scholarly journals Error Bounds for Augmented Truncations of Discrete-Time Block-Monotone Markov Chains under Geometric Drift Conditions

2015 ◽  
Vol 47 (1) ◽  
pp. 83-105 ◽  
Author(s):  
Hiroyuki Masuyama
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Total Variation ◽  
Discrete Time ◽  
Error Bounds ◽  
Total Variation Distance ◽  
Time Block ◽  
Stationary Distributions ◽  
Variation Distance ◽  
Drift Conditions

In this paper we study the augmented truncation of discrete-time block-monotone Markov chains under geometric drift conditions. We first present a bound for the total variation distance between the stationary distributions of an original Markov chain and its augmented truncation. We also obtain such error bounds for more general cases, where an original Markov chain itself is not necessarily block monotone but is blockwise dominated by a block-monotone Markov chain. Finally, we discuss the application of our results to GI/G/1-type Markov chains.

scholarly journals Comparison of Cutoffs Between Lazy Walks and Markovian Semigroups

2013 ◽  
Vol 50 (4) ◽  
pp. 943-959 ◽  
Author(s):  
Guan-Yu Chen ◽  
Laurent Saloff-Coste
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Discrete Time ◽  
Continuous Time ◽  
Transition Matrix ◽  
Mixing Time ◽  
Total Variation Distance ◽  
Variation Distance ◽  
Markovian Semigroups

We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.

Poisson Approximation for the Non-Overlapping Appearances of Several Words in Markov Chains

2001 ◽  
Vol 10 (4) ◽  
pp. 293-308 ◽  
Author(s):  
OURANIA CHRYSSAPHINOU ◽  
STAVROS PAPASTAVRIDIS ◽  
EUTICHIA VAGGELATOU
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Total Variation ◽  
Analogous Result ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Numerical Example ◽  
Stationary Markov Chain ◽  
Variation Distance

Let X1, …, Xn be a sequence of r.v.s produced by a stationary Markov chain with state space an alphabet Ω = {ω1, …, ωq}, q [ges ] 2. We consider a set of words {A1, …, Ar}, r [ges ] 2, with letters from the alphabet Ω. We allow the words to have self-overlaps as well as overlaps between them. Let [Escr ] denote the event of the appearance of a word from the set {A1, …, Ar} at a given position. Moreover, define by N the number of non-overlapping (competing renewal) appearances of [Escr ] in the sequence X1, …, Xn. We derive a bound on the total variation distance between the distribution of N and a Poisson distribution with parameter [ ]N. The Stein–Chen method and combinatorial arguments concerning the structure of words are employed. As a corollary, we obtain an analogous result for the i.i.d. case. Furthermore, we prove that, under quite general conditions, the r.v. N converges in distribution to a Poisson r.v. A numerical example is presented to illustrate the performance of the bound in the Markov case.

scholarly journals Comparison of Cutoffs Between Lazy Walks and Markovian Semigroups

2013 ◽  
Vol 50 (04) ◽  
pp. 943-959 ◽  
Author(s):  
Guan-Yu Chen ◽  
Laurent Saloff-Coste
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Discrete Time ◽  
Continuous Time ◽  
Transition Matrix ◽  
Mixing Time ◽  
Total Variation Distance ◽  
Variation Distance ◽  
Markovian Semigroups

We make a connection between the continuous time and lazy discrete time Markov chains through the comparison of cutoffs and mixing time in total variation distance. For illustration, we consider finite birth and death chains and provide a criterion on cutoffs using eigenvalues of the transition matrix.

Convergence to stationary state for a Markov move-to-front scheme

1995 ◽  
Vol 32 (03) ◽  
pp. 768-776 ◽  
Author(s):  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Stationary State ◽  
Higher Order ◽  
Total Variation Distance ◽  
Variation Distance ◽  
Coupling Time

This work considers items (e.g. books, files) arranged in an array (e.g. shelf, tape) with N positions and assumes that items are requested according to a Markov chain (possibly, of higher order). After use, the requested item is returned to the leftmost position of the array. Successive applications of the procedure above give rise to a Markov chain on permutations. For equally likely items, the number of requests that makes this Markov chain close to its stationary state is estimated. To achieve that, a coupling argument and the total variation distance are used. Finally, for non-equally likely items and so-called p-correlated requests, the coupling time is presented as a function of the coupling time when requests are independent.

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