scholarly journals On the Total Variation Distance of Semi-Markov Chains

2015 ◽  
pp. 185-199 ◽  
Author(s):  
Giorgio Bacci ◽  
Giovanni Bacci ◽  
Kim Guldstrand Larsen ◽  
Radu Mardare
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Total Variation Distance ◽  
Variation Distance

scholarly journals Complete Axiomatization for the Total Variation Distance of Markov Chains

2018 ◽  
Vol 336 ◽  
pp. 27-39
Author(s):  
Giorgio Bacci ◽  
Giovanni Bacci ◽  
Kim G. Larsen ◽  
Radu Mardare
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Total Variation Distance ◽  
Complete Axiomatization ◽  
Variation Distance

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.

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.

The average cost of Markov chains subject to total variation distance uncertainty

2018 ◽  
Vol 120 ◽  
pp. 29-35 ◽  
Author(s):  
A.A. Malikopoulos ◽  
C.D. Charalambous ◽  
I. Tzortzis
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Average Cost ◽  
Total Variation Distance ◽  
Variation Distance

A unified stability theory for classical and monotone Markov chains

2019 ◽  
Vol 56 (01) ◽  
pp. 1-22
Author(s):  
Takashi Kamihigashi ◽  
John Stachurski
Keyword(s):  
Markov Chains ◽  
Total Variation ◽  
Stochastic Dominance ◽  
Stability Theory ◽  
Total Variation Distance ◽  
Probability Measures ◽  
General State ◽  
Partially Ordered ◽  
Variation Distance ◽  
Borel Probability

AbstractIn this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.

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.

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