This dissertation aims to investigate several aspects of the Poisson convergence: Poisson approximation, multivariate Poisson approximation, Poisson process approximation and weak convergence to a Poisson process.
The size-bias coupling is a powerful tool that, when combined with the Chen-Stein method, leads to many general results on Poisson approximation. We define an approximate size-bias coupling for integer-valued random variables by introducing error terms, and we combine it with the Chen-Stein method to compare the distributions of integer-valued random variables and Poisson random variables. In particular, we provide explicit bounds on the pointwise difference between the cumulative distribution functions. By these findings, we show approximation results in the Kolmogorov distance for minimal circumscribed radii and maximal inradii of stationary Poisson-Voronoi tessellations. Moreover, we compare the distributions of Poisson random variables and U-statistics with underlying Poisson processes or binomial point processes, which, in particular, allows us to approximate the rescaled minimum Euclidean distance between pairs of points of a Poisson process with midpoint in an observation window by an exponentially distributed random variable using the Kolmogorov distance.
A multivariate version of the size-bias coupling is employed to investigate the Gaussian approximation for random vectors by L. Goldstein and Y. Rinott. We extend the notion of approximate size-bias coupling for random variables to random vectors, and we combine it with the Chen-Stein method to investigate the multivariate Poisson approximation in the Wasserstein distance and the Poisson process approximation in a new metric defined herein. As an application, we obtain a bound on the Wasserstein distance between the sum of m-dependent Bernoulli random vectors and a Poisson random vector. Moreover, we consider point processes of U-statistic structure, that is, point processes that, once evaluated on a measurable set, become U-statistics. For point processes of U-statistic structure with an underlying Poisson process, we establish a Poisson process approximation result that is the analogue of the one shown by L. Decreusefond, M. Schulte, and C. ThÃ¤le with the Kantorovich-Rubinstein distance replaced by our new metric.
General criteria for the weak convergence of locally finite point processes to a Poisson process are derived from the relation between probabilities of two consecutive values of a Poisson random variable. P. Calka and N. Chenavier studied the limiting behavior of characteristic radii of homogeneous Poisson-Voronoi tessellations. By our general results, we extend and improve their findings by showing Poisson process convergence for point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations.