Compound Poisson approximation for the Johnson-Mehl model

2000 ◽  
Vol 37 (01) ◽  
pp. 101-117
Author(s):  
Torkel Erhardsson
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Random Intervals ◽  
Variation Distance ◽  
The One

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Compound Poisson approximation for the Johnson-Mehl model

2000 ◽  
Vol 37 (1) ◽  
pp. 101-117 ◽  
Author(s):  
Torkel Erhardsson
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Random Intervals ◽  
Variation Distance ◽  
The One

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

On Compound Poisson Approximation for Sequence Matching

2000 ◽  
Vol 9 (6) ◽  
pp. 529-548 ◽  
Author(s):  
MARIANNE MÅNSSON
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Rates Of Convergence ◽  
Poisson Approximation ◽  
Finite Alphabet ◽  
Independent Random Variables ◽  
Sequence Matching ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Variation Distance

Consider sequences {Xi}mi=1 and {Yj}nj=1 of independent random variables, taking values in a finite alphabet, and assume that the variables X1, X2, … and Y1, Y2, … follow the distributions μ and v, respectively. Two variables Xi and Yj are said to match if Xi = Yj. Let the number of matching subsequences of length k between the two sequences, when r, 0 [les ] r < k, mismatches are allowed, be denoted by W.In this paper we use Stein's method to bound the total variation distance between the distribution of W and a suitably chosen compound Poisson distribution. To derive rates of convergence, the case where E[W] stays bounded away from infinity, and the case where E[W] → ∞ as m, n → ∞, have to be treated separately. Under the assumption that ln n/ln(mn) → ρ ∈ (0, 1), we give conditions on the rate at which k → ∞, and on the distributions μ and v, for which the variation distance tends to zero.

scholarly journals Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

2007 ◽  
Vol 39 (01) ◽  
pp. 128-140 ◽  
Author(s):  
Etienne Roquain ◽  
Sophie Schbath
Keyword(s):  
Markov Chain ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Rare Event ◽  
Poisson Approximation ◽  
Rare Word ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Method ◽  
Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

1999 ◽  
Vol 36 (01) ◽  
pp. 97-104 ◽  
Author(s):  
Michael Weba
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Medium Size ◽  
Bernoulli Random Variables ◽  
Variation Distance ◽  
Approximation Problems

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p 1,p 2,…, p n is often approximated by a Poisson distribution with parameter λ = p 1 + p 2 + p n . Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p 1,p 2,…,p n . Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

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.

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.

scholarly journals Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

2007 ◽  
Vol 39 (1) ◽  
pp. 128-140 ◽  
Author(s):  
Etienne Roquain ◽  
Sophie Schbath
Keyword(s):  
Markov Chain ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Rare Event ◽  
Poisson Approximation ◽  
Rare Word ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Method ◽  
Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

1999 ◽  
Vol 36 (1) ◽  
pp. 97-104 ◽  
Author(s):  
Michael Weba
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Medium Size ◽  
Bernoulli Random Variables ◽  
Variation Distance ◽  
Approximation Problems

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p1,p2,…, pn is often approximated by a Poisson distribution with parameter λ = p1 + p2 + pn. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p1,p2,…,pn.Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

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.

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.

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