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 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.

Poisson approximation for some statistics based on exchangeable trials

1983 ◽  
Vol 15 (03) ◽  
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.

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.

Improved Poisson approximations for word patterns

1993 ◽  
Vol 25 (02) ◽  
pp. 334-347 ◽  
Author(s):  
Anant P. Godbole ◽  
Andrew A. Schaffner
Keyword(s):  
Total Variation ◽  
Random Variable ◽  
Total Variation Distance ◽  
Stationary Probability ◽  
Poisson Random Variable ◽  
Eigenvalue Bounds ◽  
Letter Word ◽  
Letter Alphabet ◽  
Variation Distance ◽  
Poisson Approximations

Let X 1, X 2, · ··, Xn be a sequence of n random variables taking values in the ξ -letter alphabet . We consider the number N = N(n, k) of non-overlapping occurrences of a fixed k-letter word under (a) i.i.d. and (b) stationary Markovian hypotheses on the sequence , and use the Stein–Chen method to obtain Poisson approximations for the same. In each case, results and couplings from Barbour et al. (1992) are used to show that the total variation distance between the distribution of N and that of an appropriate Poisson random variable is of order (roughly) O(kS (k)), where S (k) denotes the stationary probability of the word in question. These results vastly improve on the approximations obtained in Godbole (1991). In the Markov case, we also make use of recently obtained eigenvalue bounds on convergence to stationarity due to Diaconis and Stroock (1991) and Fill (1991).

Improved Poisson approximations for word patterns

1993 ◽  
Vol 25 (2) ◽  
pp. 334-347 ◽  
Author(s):  
Anant P. Godbole ◽  
Andrew A. Schaffner
Keyword(s):  
Total Variation ◽  
Random Variable ◽  
Total Variation Distance ◽  
Stationary Probability ◽  
Poisson Random Variable ◽  
Eigenvalue Bounds ◽  
Letter Word ◽  
Letter Alphabet ◽  
Variation Distance ◽  
Poisson Approximations

Let X1, X2, · ··, Xn be a sequence of n random variables taking values in the ξ -letter alphabet . We consider the number N = N(n, k) of non-overlapping occurrences of a fixed k-letter word under (a) i.i.d. and (b) stationary Markovian hypotheses on the sequence , and use the Stein–Chen method to obtain Poisson approximations for the same. In each case, results and couplings from Barbour et al. (1992) are used to show that the total variation distance between the distribution of N and that of an appropriate Poisson random variable is of order (roughly) O(kS(k)), where S(k) denotes the stationary probability of the word in question. These results vastly improve on the approximations obtained in Godbole (1991). In the Markov case, we also make use of recently obtained eigenvalue bounds on convergence to stationarity due to Diaconis and Stroock (1991) and Fill (1991).

scholarly journals Universality for Random Surfaces in Unconstrained Genus

10.37236/8623 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Thomas Budzinski ◽  
Nicolas Curien ◽  
Bram Petri
Keyword(s):  
Total Variation ◽  
Degree Sequence ◽  
Random Variable ◽  
Total Variation Distance ◽  
Alternative Proof ◽  
Arbitrary Sequence ◽  
Weak Version ◽  
Variation Distance ◽  
Planar Maps ◽  
Dirichlet Partition

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

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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