Symmetric sampling procedures, general epidemic processes and their threshold limit theorems

1986 ◽  
Vol 23 (02) ◽  
pp. 265-282 ◽  
Author(s):  
Anders Martin-Löf
Keyword(s):  
Limit Distribution ◽  
Limit Theorems ◽  
General Type ◽  
Finite Population ◽  
Binomial Sampling ◽  
Threshold Limit ◽  
Iterative Sampling ◽  
Factorial Series ◽  
Sampling Procedures ◽  
Symmetric Sampling

Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.

Symmetric sampling procedures, general epidemic processes and their threshold limit theorems

1986 ◽  
Vol 23 (2) ◽  
pp. 265-282 ◽  
Author(s):  
Anders Martin-Löf
Keyword(s):  
Limit Distribution ◽  
Limit Theorems ◽  
General Type ◽  
Finite Population ◽  
Binomial Sampling ◽  
Threshold Limit ◽  
Iterative Sampling ◽  
Factorial Series ◽  
Sampling Procedures ◽  
Symmetric Sampling

Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.

Threshold limit theorems for some epidemic processes

1980 ◽  
Vol 12 (02) ◽  
pp. 319-349 ◽  
Author(s):  
Bengt Von Bahr ◽  
Anders Martin-Löf
Keyword(s):  
Limit Distribution ◽  
Limit Theorems ◽  
Polymerization Process ◽  
Connected Components ◽  
Critical Threshold ◽  
Infected Person ◽  
Initial Number ◽  
Total Size ◽  
Frost Model ◽  
Healthy Persons

The Reed–Frost model for the spread of an infection is considered and limit theorems for the total size, T, of the epidemic are proved in the limit when n, the initial number of healthy persons, is large and the probability of an encounter between a healthy and an infected person per time unit, p, is λ/n. It is shown that there is a critical threshold λ = 1 in the following sense, when the initial number of infected persons, m, is finite: If λ ≦ 1, T remains finite and has a limit distribution which can be described. If λ > 1 this is still true with a probability σ m < 1, and with probability 1 – σ m T is close to n(1 – σ) and has an approximately Gaussian distribution around this value. When m → ∞ also, only the Gaussian part of the limit distribution is obtained. A randomized version of the Reed–Frost model is also considered, and this allows the same result to be proved for the Kermack–McKendrick model. It is also shown that the limit theorem can be used to study the number of connected components in a random graph, which can be considered as a crude description of a polymerization process. In this case polymerization takes place when λ > 1 and not when λ ≦ 1.

Equilibrium behavior of population genetic models with non-random mating. Part II: Pedigrees, Homozygosity and Stochastic Models

1968 ◽  
Vol 5 (3) ◽  
pp. 487-566 ◽  
Author(s):  
Samuel Karlin
Keyword(s):  
Stochastic Models ◽  
Limit Theorems ◽  
Finite Population ◽  
Random Mating ◽  
Bayes Rule ◽  
Identity By Descent ◽  
Equilibrium Behavior ◽  
Path Coefficients ◽  
Complete Study ◽  
Mating Part

Wright (1921) computed various correlations of relatives by a rather cumbersome procedure called the “method of path coefficients”. Wright's method is basically a disguised form of the use of Bayes' rule and the law of total probabilities. Malecot (1948) reorganized Wright's calculations by introducing the fundamental concept of identity by descent and exploiting its properties. The method of identity by descent has been perfected and developed by Malécot and his students, especially Gillois, Jauquard and Bouffette. Kempthorne (1957) has applied the concept of identity by descent to the study of quantitative inheritance. Kimura (1963) elegantly employed the ideas of identity by descent in determining rates of approach to homozygosity in certain mating situations with finite population size. Later in this chapter we will extend and refine the results of Kimura (1963) to give a more complete study of rates of approach to homozygosity. Ellison (1966) established several important limit theorems corresponding to polyploid, multi-locus random mating infinite populations by judicious enlargement of the concepts of identity by descent. Kesten (unpublished)) has recently refined the technique of Ellison.

Limit theorems for random mating in infinite populations

1966 ◽  
Vol 3 (01) ◽  
pp. 94-114 ◽  
Author(s):  
B. E. Ellison
Keyword(s):  
Limit Distribution ◽  
Limit Theorems ◽  
Random Mating ◽  
Limit Distributions ◽  
Infinite Population ◽  
The Law

This paper is concerned with the distribution of “types” of individuals in an infinite population after indefinitely many nonoverlapping generations of random mating. The absence of selection and mutation is assumed. The probabilistic law which governs the production of an offspring may be asymmetrical with respect to the “sexes” of the two parents, but the law is assumed to apply independently of the “sex” of the offspring. The question of the existence of a limit distribution of types, the rate at which a limit distribution is approached, and properties of limit distributions are treated.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (01) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes

1991 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
O. V. Seleznjev
Keyword(s):  
Gaussian Processes ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Point Processes ◽  
Random Processes ◽  
Trigonometric Polynomials ◽  
Stationary Processes ◽  
Periodic Processes

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

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.

Central limit theorems for a class of polynomial hypergroups

1990 ◽  
Vol 22 (01) ◽  
pp. 68-87 ◽  
Author(s):  
Michael Voit
Keyword(s):  
Random Walks ◽  
Central Limit ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Order Of Convergence ◽  
Transition Probabilities ◽  
Recursion Formula ◽  
Rayleigh Distribution ◽  
Central Limit Theorems ◽  
Polynomial Hypergroups

Central limit theorems are proved for random walks on the non-negative integers where the transition probabilities are homogeneous with respect to a sequence of orthogonal polynomials. Assuming some restrictions concerning the three-term recursion formula of these polynomials, one gets a Rayleigh distribution as limit distribution where bounds of the order of convergence can be computed explicitly. These central limit theorems are applied to generalized birth and death random walks and random walks on polynomial hypergroups. Finally some examples of polynomial hypergroups are discussed in view of the limit theorems above.

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 Limit distribution for a consecuttve-k-out-of-n: F system

1990 ◽  
Vol 22 (2) ◽  
pp. 491-493 ◽  
Author(s):  
Ourania Chryssaphinou ◽  
Stavros G. Papastavridis
Keyword(s):  
Weibull Distribution ◽  
Limit Distribution ◽  
Limit Theorems ◽  
F System ◽  
General Conditions

A consecutive-k-out-of-n: F system consists of n components ordered on a line. Each component, and the system as a whole, has two states: it is either functional or failed. The system will fail if and only if at least k consecutive components fail. The components are not necessarily equal and we assume that components' failures are stochastically independent. Using a result of Barbour and Eagleson (1984) we find a bound for the distance of the distribution of system's lifetime from the Weibull distribution. Subsequently, using this bound limit theorems are derived under quite general conditions.

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