Convergence to stationarity in the Moran model

2000 ◽  
Vol 37 (03) ◽  
pp. 705-717 ◽  
Author(s):  
Peter Donnelly ◽  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Upper Bound ◽  
Separation Distance ◽  
Fixed Size ◽  
Moran Model ◽  
Variation Distance ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
Asymptotic Upper Bound

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A 1 and A 2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t)) t≥0 which counts the number of individuals of type A 1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t ∗ = Nγ-1logN + cN the separation distance between the law of X(t ∗) and its stationary distribution converges to 1 - exp(-γe-γc ) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

Convergence to stationarity in the Moran model

2000 ◽  
Vol 37 (3) ◽  
pp. 705-717 ◽  
Author(s):  
Peter Donnelly ◽  
Eliane R. Rodrigues
Keyword(s):  
Markov Chain ◽  
Total Variation ◽  
Upper Bound ◽  
Separation Distance ◽  
Fixed Size ◽  
Moran Model ◽  
Variation Distance ◽  
Lines Of Descent ◽  
Number Of Individuals ◽  
Asymptotic Upper Bound

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A1 and A2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t))t≥0 which counts the number of individuals of type A1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t∗ = Nγ-1logN + cN the separation distance between the law of X(t∗) and its stationary distribution converges to 1 - exp(-γe-γc) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

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 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.

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.

scholarly journals Poisson Approximation of the Number of Cliques in Random Intersection Graphs

2010 ◽  
Vol 47 (3) ◽  
pp. 826-840 ◽  
Author(s):  
Katarzyna Rybarczyk ◽  
Dudley Stark
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Upper Bound ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Intersection Graphs ◽  
Fixed Integer ◽  
Random Subset ◽  
Random Intersection Graph ◽  
Variation Distance

A random intersection graphG(n,m,p) is defined on a setVofnvertices. There is an auxiliary setWconsisting ofmobjects, and each vertexv∈Vis assigned a random subset of objectsWv⊆Wsuch thatw∈Wvwith probabilityp, independently for allv∈Vand allw∈W. Given two verticesv1,v2∈V, we setv1∼v2if and only ifWv1∩Wv2≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number ofh-cliques inG(n,m,p) and a related Poisson distribution for any fixed integerh.

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

1995 ◽  
Vol 32 (3) ◽  
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.

scholarly journals Poisson Approximation of the Number of Cliques in Random Intersection Graphs

2010 ◽  
Vol 47 (03) ◽  
pp. 826-840 ◽  
Author(s):  
Katarzyna Rybarczyk ◽  
Dudley Stark
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Upper Bound ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Intersection Graphs ◽  
Fixed Integer ◽  
Random Subset ◽  
Random Intersection Graph ◽  
Variation Distance

A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects W v ⊆ W such that w ∈ W v with probability p, independently for all v ∈ V and all w ∈ W . Given two vertices v 1, v 2 ∈ V , we set v 1 ∼ v 2 if and only if W v 1 ∩ W v 2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

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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