Two poisson limit theorems for the coupon collector’s problem with group drawings

2021 ◽  
Vol 58 (4) ◽  
pp. 966-977
Author(s):  
Judith Schilling ◽  
Norbert Henze
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Limit Theorems ◽  
Limit Distributions ◽  
Different Types ◽  
Complete Series ◽  
Uniform Case ◽  
Poisson Limit ◽  
Coupon Collector's Problem ◽  
Special Case

AbstractIn the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most $c-1$ times, where $c \ge 1$ is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case $c=1$ . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.

scholarly journals Poisson Approximation in a Poisson Limit Theorem Inspired by Coupon Collecting

2009 ◽  
Vol 46 (02) ◽  
pp. 585-592
Author(s):  
Anna Pósfai
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Random Variables ◽  
Poisson Approximation ◽  
Poisson Law ◽  
Coupon Collector’S Problem ◽  
Poisson Limit ◽  
Coupon Collector's Problem ◽  
Error Order

In this paper we refine a Poisson limit theorem of Gnedenko and Kolmogorov (1954): we determine the error order of a Poisson approximation for sums of asymptotically negligible integer-valued random variables that converge in distribution to the Poisson law. As an application of our results, we investigate the case of the coupon collector's problem when the distribution of the collector's waiting time is asymptotically Poisson.

scholarly journals Poisson Approximation in a Poisson Limit Theorem Inspired by Coupon Collecting

2009 ◽  
Vol 46 (2) ◽  
pp. 585-592
Author(s):  
Anna Pósfai
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Random Variables ◽  
Poisson Approximation ◽  
Poisson Law ◽  
Coupon Collector’S Problem ◽  
Poisson Limit ◽  
Coupon Collector's Problem ◽  
Error Order

In this paper we refine a Poisson limit theorem of Gnedenko and Kolmogorov (1954): we determine the error order of a Poisson approximation for sums of asymptotically negligible integer-valued random variables that converge in distribution to the Poisson law. As an application of our results, we investigate the case of the coupon collector's problem when the distribution of the collector's waiting time is asymptotically Poisson.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

A Poisson limit theorem for incomplete symmetric statistics

1983 ◽  
Vol 20 (01) ◽  
pp. 47-60 ◽  
Author(s):  
M. Berman ◽  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Poisson Limit ◽  
Symmetric Statistics ◽  
Limit Result ◽  
Independent Identically Distributed

Silverman and Brown (1978) have derived Poisson limit theorems for certain sequences of symmetric statistics, based on a sample of independent identically distributed random variables. In this paper an incomplete version of these statistics is considered and a Poisson limit result shown to hold. The powers of some tests based on the incomplete statistic are investigated and the main results of the paper are used to simplify the derivations of the asymptotic distributions of some statistics previously published in the literature.

Rates of Poisson convergence for U-statistics

1979 ◽  
Vol 16 (2) ◽  
pp. 428-432 ◽  
Author(s):  
T. C. Brown ◽  
B. W. Silverman
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Plane Region ◽  
U Statistics ◽  
Poisson Limit ◽  
Special Case

Poisson limit theorems for U-statistics are studied. A general rate of convergence is obtained; this rate is improved for the special case where the U-statistic arises from the consideration of distances between uniformly distributed points in a well-behaved plane region.

Compound Poisson limit theorems for Markov chains

1997 ◽  
Vol 34 (1) ◽  
pp. 24-34 ◽  
Author(s):  
Shoou-Ren Hsiau
Keyword(s):  
Markov Chain ◽  
Limit Theorem ◽  
Markov Chains ◽  
Limit Theorems ◽  
Compound Poisson ◽  
Poisson Limit

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.

Rates of Poisson convergence for U-statistics

1979 ◽  
Vol 16 (02) ◽  
pp. 428-432 ◽  
Author(s):  
T. C. Brown ◽  
B. W. Silverman
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Plane Region ◽  
U Statistics ◽  
Poisson Limit ◽  
Special Case

Poisson limit theorems for U-statistics are studied. A general rate of convergence is obtained; this rate is improved for the special case where the U-statistic arises from the consideration of distances between uniformly distributed points in a well-behaved plane region.

Almost sure limit theorems for random allocations

2005 ◽  
Vol 42 (2) ◽  
pp. 173-194
Author(s):  
István Fazekas ◽  
Alexey Chuprunov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Domain ◽  
Almost Sure Limit Theorem ◽  
The Central Limit Theorem ◽  
Almost Sure Limit Theorems ◽  
Poisson Limit

Almost sure limit theorems are presented for random allocations. A general almost sure limit theorem is proved for arrays of random variables. It is applied to obtain almost sure versions of the central limit theorem for the number of empty boxes when the parameters are in the central domain. Almost sure versions of the Poisson limit theorem in the left domain are also proved.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (1) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n2). The general case is discussed in terms of operator semigroups.

A Poisson limit theorem for weakly exchangeable events

1979 ◽  
Vol 16 (04) ◽  
pp. 794-802 ◽  
Author(s):  
G. K. Eagleson
Keyword(s):  
Limit Theorem ◽  
Spatial Data ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Random Variable ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Poisson Limit ◽  
Image Position ◽  
Independent Identically Distributed

Let Y 1, Y2 , · ·· be a sequence of independent, identically distributed random variables, g some symmetric 0–1 function of m variables and set Silverman and Brown (1978) have shown that under certain conditions the statistic is asymptotically distributed as a Poisson random variable. They then use this result to derive limit distributions for various statistics, useful in the analysis of spatial data. In this paper, it is shown that Silverman and Brown's theorem holds under much weaker assumptions; assumptions which involve only the symmetry of the joint distributions of the X il…i m .

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