Graduate Thesis Or Dissertation
 

Densities and dependence for point processes

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  • Product densities have been widely used in the literature to give a concrete description of the distribution of a point process. A rigorous description of properties of product densities is presented with examples to show that in some sense these results are the best possible. Product densities are then used to discuss positive dependence properties of point processes. There are many ways of describing positive dependence. Two well known notions for Bernoulli random variables are the strong FKG inequalities and association, the strong FKG inequalities being much stronger. It is known, for example, from van den Berg and Burton, that the strong FKG inequalities are equivalent to all conditional distributions being associated, which is equivalent to all conditional distributions being positively correlated. In the case of point processes for which product densities exist, analogs of such positive dependence properties are given. Examples are presented to show that unlike the Bernoulli case none of these conditions are equivalent, although some are shown to be implied by others.
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