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 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 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.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

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.

Poisson approximation of multivariate Poisson mixtures

2003 ◽  
Vol 40 (02) ◽  
pp. 376-390 ◽  
Author(s):  
Bero Roos
Keyword(s):  
Total Variation ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Poisson Mixtures ◽  
Poisson Distributions ◽  
Variation Distance

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (3) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Poisson approximation for some statistics based on exchangeable trials

1983 ◽  
Vol 15 (3) ◽  
pp. 585-600 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Total Variation ◽  
Limit Theorems ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Variation Distance ◽  
Image Position ◽  
Poisson Approximations

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of .An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

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