UC Berkeley

This thesis is composed of five chapters, regarding several models for dependence in stochastic processes. We first discuss the class $L$ of selfdecomposable laws, which is a subclass of the class of infinitely divisible laws and contains all stable laws. We show an example of selfdecomposable law whose selfdecomposability is related to path decomposition of planar Brownian motions. Then we introduce the family of self-similar additive processes, which is known to have a close relationship with the class $L$ of selfdecomposable laws. The discussion is suggested by the scale invariant Poisson spacings theorem, which arose in various contexts including records, extremal processes and random permutations. We are able to show that the range of a self-similar gamma process is a scale invariant Poisson point process $(\theta x^{-1} dx)$ and also conversely, this distribution of the range characterizes the gamma process among all self-similar additive processes. We then turn to a discussion of counting processes in discrete times. In particular, when the counting process is stationary $1$-dependent, its distribution is determined by the bivariate probability generating function in terms of run probability generating functions. A probabilistic explanation is provided, alongside with comparison to other known encodings including the determinantal representation and a combinatorial enumeration formula. We also compare the bivariate generating function for $1$-dependent sequences with similar generating functions derived from other dependence structures. Lastly, we discuss a positivity problem related to a bivariate probability generating function for renewal processes, allowing signed measures. Fascinating graphs and qualitative observational results are provided, as well as natural but challenging open problems to explain these facts.