Limit Theorems for an Extended Coupon Collector's Problem and for Successive Subsampling with Varying Probabilites*

1980 ◽  
Vol 29 (3-4) ◽  
pp. 113-132 ◽  
Author(s):  
Pranab Kumar Sen
Keyword(s):  
Asymptotic Normality ◽  
Limit Theorems ◽  
Finite Population ◽  
Waiting Times ◽  
Invariance Principles ◽  
Weak Invariance ◽  
Coupon Collector’S Problem ◽  
Asymptotic Distribution Theory ◽  
Multistage Sampling ◽  
Coupon Collector's Problem

Asymptotic normality as well as some weak invariance principles for bonus sums and waiting times in an extended coupon collector's problem are considered and incorporated in the study of the asymptotic distribution theory of estimators of (finite) population totals in successive sub-sampling (or multistage sampling) with varying probabilities (without replacement). Some applications of these theorems are also considered.

scholarly journals The Coupon Collector's Problem Revisited: Asymptotics of the Variance

2012 ◽  
Vol 44 (1) ◽  
pp. 166-195 ◽  
Author(s):  
Aristides V. Doumas ◽  
Vassilis G. Papanicolaou
Keyword(s):  
Large Class ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Gumbel Distribution ◽  
Coupon Collector’S Problem ◽  
Second Moments ◽  
General Limit ◽  
Coupon Collector's Problem

We develop techniques for computing the asymptotics of the first and second moments of the number TN of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[TN] of TN (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of TN (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

Dixie cups: sampling with replacement from a finite population

1994 ◽  
Vol 31 (04) ◽  
pp. 940-948 ◽  
Author(s):  
Chris A. J. Klaassen
Keyword(s):  
Sample Size ◽  
Random Sample ◽  
Finite Population ◽  
Asymptotic Properties ◽  
Random Sample Size ◽  
Coupon Collector’S Problem ◽  
Coupon Collector's Problem ◽  
Unequal Sampling

At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.

Dixie cups: sampling with replacement from a finite population

1994 ◽  
Vol 31 (4) ◽  
pp. 940-948 ◽  
Author(s):  
Chris A. J. Klaassen
Keyword(s):  
Sample Size ◽  
Random Sample ◽  
Finite Population ◽  
Asymptotic Properties ◽  
Random Sample Size ◽  
Coupon Collector’S Problem ◽  
Coupon Collector's Problem ◽  
Unequal Sampling

At which (random) sample size will every population element have been drawn at least m times? This special coupon collector's problem is often referred to as the Dixie cup problem. Some asymptotic properties of the Dixie cup problem with unequal sampling probabilities are described.

New limit results related to the coupon collector’s problem

2018 ◽  
Vol 55 (1) ◽  
pp. 115-140
Author(s):  
Lenka Glavaš ◽  
Pavle Mladenović
Keyword(s):  
State Space ◽  
Point Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Measure ◽  
Rates Of Convergence ◽  
The State ◽  
Coupon Collector’S Problem ◽  
Coupon Collector's Problem ◽  
Positive Integers

We study point processes associated with coupon collector’s problem, that are defined as follows. We draw with replacement from the set of the first n positive integers until all elements are sampled, assuming that all elements have equal probability of being drawn. The point process we are interested in is determined by ordinal numbers of drawing elements that didn’t appear before. The set of real numbers is considered as the state space. We prove that the point process obtained after a suitable linear transformation of the state space converges weakly to the limiting Poisson random measure whose mean measure is determined. We also consider rates of convergence in certain limit theorems for the problem of collecting pairs.

scholarly journals The Coupon Collector's Problem Revisited: Asymptotics of the Variance

2012 ◽  
Vol 44 (01) ◽  
pp. 166-195 ◽  
Author(s):  
Aristides V. Doumas ◽  
Vassilis G. Papanicolaou
Keyword(s):  
Large Class ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Gumbel Distribution ◽  
Coupon Collector’S Problem ◽  
Second Moments ◽  
General Limit ◽  
Coupon Collector's Problem

We develop techniques for computing the asymptotics of the first and second moments of the number T N of coupons that a collector has to buy in order to find all N existing different coupons as N → ∞. The probabilities (occurring frequencies) of the coupons can be quite arbitrary. From these asymptotics we obtain the leading behavior of the variance V[T N ] of T N (see Theorems 3.1 and 4.4). Then, we combine our results with the general limit theorems of Neal in order to derive the limit distribution of T N (appropriately normalized), which, for a large class of probabilities, turns out to be the standard Gumbel distribution. We also give various illustrative examples.

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