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 Kesten (unpublished)) has recently refined the technique of Ellison.

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

1968 ◽  
Vol 5 (03) ◽  
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.

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.

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 Order Book Dynamics with Liquidity Fluctuations: Limit Theorems and Large Deviations

2021 ◽  
Author(s):  
Helder Rojas ◽  
Anatoly Yambartsev ◽  
Artem Logachov
Keyword(s):  
Large Deviations ◽  
Stochastic Models ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Limit Order Book ◽  
Order Book ◽  
Limit Order ◽  
Bid Ask Spread ◽  
Large Numbers ◽  
The One

We propose a class of stochastic models for a dynamics of limit order book with different type of liquidities. Within this class of models we study the one where a spread decreases uniformly, belonging to the class of processes known as a population processes with uniform catastrophes. The law of large numbers (LLN), central limit theorem (CLT) and large deviations (LD) are proved for our model with uniform catastrophes. Our results allow us to satisfactorily explain the volatility and local trends in the prices, relevant empirical characteristics that are observed in this type of markets. Furthermore, it shows us how these local trends and volatility are determined by the typical values of the bid-ask spread. In addition, we use our model to show how large deviations occur in the spread and prices, such as those observed in flash crashes.

scholarly journals The time of detection of recessive visible genes in small populations

1978 ◽  
Vol 31 (3) ◽  
pp. 255-264 ◽  
Author(s):  
Alan Robertson
Keyword(s):  
Artificial Selection ◽  
Finite Population ◽  
Random Mating ◽  
Recessive Gene ◽  
New Mutation ◽  
Mating Population ◽  
Initial Generation ◽  
The Mean ◽  
Time To Detection ◽  
Mean Time

SUMMARYHomozygotes for recessive visible genes have often been discovered in lines under artificial selection, sometimes many generations from the start. As a help in the interpretation of this phenomenon, the distribution of the time to first detection as a homozygote of a recessive gene occurring only once in the initial generation has been obtained. Alternatively the results may be considered as referring to the time of first appearance as a homozygote of a new mutation occurring in a finite population. For a monoecious random mating population of size N with selfing permitted, the mean time to detection is very close to 2N⅓ over a range of N from 1 to 500 with a coefficient of variation of roughly 2/3 and a 95% upper limit about 2·5 times the mean. If selfing is prohibited, the mean time is increased by a little over 1 generation. The treatment is extended to cover the effects of artificial selection in favour of the heterozygote, of the frequency of occurrence in the initial generation and of the examination of more individuals each generation than are used as parents.

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