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.

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 &gt; 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.

Compound Poisson approximation for long increasing sequences

2001 ◽  
Vol 38 (2) ◽  
pp. 449-463 ◽  
Author(s):  
Ourania Chryssaphinou ◽  
Eutichia Vaggelatou
Keyword(s):  
Asymptotic Behaviour ◽  
Poisson Distribution ◽  
Limit Distribution ◽  
Continuous Distribution ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Increasing Trend ◽  
Increasing Sequences

Consider a sequence X1,…,Xn of independent random variables with the same continuous distribution and the event Xi-r+1 < ⋯ < Xi of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

Compound Poisson approximation for long increasing sequences

2001 ◽  
Vol 38 (02) ◽  
pp. 449-463
Author(s):  
Ourania Chryssaphinou ◽  
Eutichia Vaggelatou
Keyword(s):  
Asymptotic Behaviour ◽  
Poisson Distribution ◽  
Limit Distribution ◽  
Continuous Distribution ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Increasing Trend ◽  
Increasing Sequences

Consider a sequence X 1,…,X n of independent random variables with the same continuous distribution and the event X i-r+1 &lt; ⋯ &lt; X i of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

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.

Compound Poisson approximation and the clustering of random points

2000 ◽  
Vol 32 (1) ◽  
pp. 19-38 ◽  
Author(s):  
A. D. Barbour ◽  
Marianne Månsson
Keyword(s):  
Poisson Distribution ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Compound Poisson Distribution ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Unit Square

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλi be decreasing.

Compound Poisson approximation and the clustering of random points

2000 ◽  
Vol 32 (01) ◽  
pp. 19-38 ◽  
Author(s):  
A. D. Barbour ◽  
Marianne Månsson
Keyword(s):  
Poisson Distribution ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Compound Poisson Distribution ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Unit Square

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλ i be decreasing.

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.

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