Marjoram, Paul; (1992) Correlation structures in applied probability. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis examines consequences of correlation structure in three areas of applied probability: mathematical population genetics, birth processes, and "exchangeable" measures on distributive lattices. The first three chapters concern probabilistic models in genetics. Initially we generalize the Moran model to allow more than one individual to reproduce per generation, investigating the effect of this on the behaviour of the model. The conclusion is that while things apparently happen faster, the basic properties are the same. This model also serves to unify conventional neutral theory, as it links the Moran model to the Wright-Fisher model. We then consider aspects of the neutral theory. Commonly a neutral model is supposed in which successive generations behave independently. This may well be unrealistic. Here we take the Moran model and adapt it to allow for correlations in offspring numbers between generations. An analysis of the model shows that the conditional distribution of allele frequencies is unchanged, although the expected number of alleles present decreases. Similar results are also obtained when correlation is introduced to the more general model with more than one reproducer per generation. In each case the approach involves a detailed study of the genealogy of the models. Next we consider the effect of correlation in Markov Birth Processes. We show that if the birth rates form a super(sub) linear sequence then the sizes of its families are positively(negatively) correlated. From this we prove a conjecture of Faddy which says that if the birth rates of a process X(t) are super(sub)-linear then the variance ratio V (t) (defined as VarX(t)/(EX(t)[EX(t)/X(0)-1])) is greater than (less than) 1. Finally we study correlation inequalities. The FKG Inequality is a well known result giving sufficient conditions for positive correlations in probability measures on distributive lattices. There are few analogous results concerning negative correlation. We give sufficient conditions for a particular form of negative correlation when the underlying distributions possess a certain exchangeability property.

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