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.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (02) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Rarefactions of compound point processes

1984 ◽  
Vol 21 (04) ◽  
pp. 710-719
Author(s):  
Richard F. Serfozo
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Classical Result ◽  
Point Processes ◽  
Rare Events ◽  
Random Measure ◽  
Bernoulli Trials ◽  
Random Measures ◽  
Infinitely Divisible ◽  
Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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.

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.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Estimating a State-Space Model from Point Process Observations

2003 ◽  
Vol 15 (5) ◽  
pp. 965-991 ◽  
Author(s):  
Anne C. Smith ◽  
Emery N. Brown
Keyword(s):  
State Space ◽  
Point Process ◽  
State Space Model ◽  
The State ◽  
Conditional Intensity ◽  
Space Model ◽  
Neural Spiking ◽  
Latent Process ◽  
Conditional Intensity Function ◽  
Spiking Activity

A widely used signal processing paradigm is the state-space model. The state-space model is defined by two equations: an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time. Inspired by neurophysiology experiments in which neural spiking activity is induced by an implicit (latent) stimulus, we develop an algorithm to estimate a state-space model observed through point process measurements. We represent the latent process modulating the neural spiking activity as a gaussian autoregressive model driven by an external stimulus. Given the latent process, neural spiking activity is characterized as a general point process defined by its conditional intensity function. We develop an approximate expectation-maximization (EM) algorithm to estimate the unobservable state-space process, its parameters, and the parameters of the point process. The EM algorithm combines a point process recursive nonlinear filter algorithm, the fixed interval smoothing algorithm, and the state-space covariance algorithm to compute the complete data log likelihood efficiently. We use a Kolmogorov-Smirnov test based on the time-rescaling theorem to evaluate agreement between the model and point process data. We illustrate the model with two simulated data examples: an ensemble of Poisson neurons driven by a common stimulus and a single neuron whose conditional intensity function is approximated as a local Bernoulli process.

Detecting the filamentary structure of 3D point clouds: Proposal of a wavelet-based method

2017 ◽  
Vol 08 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Roberto D’Ercole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Three Dimensional ◽  
Random Measure ◽  
Point Clouds ◽  
Filamentary Structure ◽  
Point Patterns ◽  
3D Point Clouds ◽  
Mexican Hat ◽  
Dirac Measures

The analysis of the filamentary structure of the cosmo as well as that of the internal structure of the polar ice suggests the development of models based on three-dimensional (3D) point processes. A point process, regarded as a random measure, can be expressed as a sum of Delta Dirac measures concentrated at some random points. The integration with respect to the point process leads the continuous wavelet transform of the process itself. As possible mother wavelets, we propose the application of the Mexican hat and the Morlet wavelet in order to implement the scale-angle energy density of the process, depending on the dilation parameter and on the three angles which define the direction in the Euclidean space. Such indicator proves to be a sensitive detector of any variation in the direction and it can be successfully implemented to study the isotropy or the filamentary structure in 3D point patterns.

A CONTINUOUS WAVELET-BASED APPROACH TO DETECT ANISOTROPIC PROPERTIES IN SPATIAL POINT PROCESSES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
ROBERTO D'ERCOLE ◽  
JORGE MATEU
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Unitary Operator ◽  
Random Measure ◽  
Continuous Wavelet ◽  
Two Dimensional ◽  
Mother Wavelet ◽  
Spatial Point Processes ◽  
Stochastic Point Process ◽  
Dirac Measures

A two-dimensional stochastic point process can be regarded as a random measure and thus represented as a (countable) sum of Delta Dirac measures concentrated at some points. Integration with respect to the point process itself leads to the concept of the continuous wavelet transform of a point process. Applying then suitable translation, rotation and dilation operations through a non unitary operator, we obtain a transformed point process which highlights main properties of the original point process. The choice of the mother wavelet is relevant and we thus conduct a detailed analysis proposing three two-dimensional mother wavelets. We use this approach to detect main directions present in the point process, and to test for anisotropy.

scholarly journals Limits of compound and thinned point processes

1975 ◽  
Vol 12 (2) ◽  
pp. 269-278 ◽  
Author(s):  
Olav Kallenberg
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Random Measure ◽  
Poisson Processes ◽  
Limiting Behaviour ◽  
Mutually Independent

Let η =Σjδτj be a point process on some space S and let β,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σjβjδτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

scholarly journals On the conditional distributions of spatial point processes

2011 ◽  
Vol 43 (2) ◽  
pp. 301-307 ◽  
Author(s):  
François Caron ◽  
Pierre Del Moral ◽  
Arnaud Doucet ◽  
Michele Pace
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Random Measure ◽  
Theoretic Approach ◽  
Conditional Distributions ◽  
Spatial Point Processes ◽  
Spatial Point ◽  
Observation Process ◽  
Original Analysis ◽  
Latent Process

We consider the problem of estimating a latent point process, given the realization of another point process. We establish an expression for the conditional distribution of a latent Poisson point process given the observation process when the transformation from the latent process to the observed process includes displacement, thinning, and augmentation with extra points. Our original analysis is based on an elementary and self-contained random measure theoretic approach. This simplifies and complements previous derivations given in Mahler (2003), and Singh, Vo, Baddeley and Zuyev (2009).

Sign in / Sign up

Export Citation Format

Share Document