Weak and strong convergence of distributions of counting processes

The theme of the article is the convergence of distributions of counting processes. The paper contains several theorems connecting the convergence of predictable characteristics (compensators) with the convergence of distributions. If the limit process has independent (or conditionally independent) increments, we use the method of «strochastic exponentials»; by means of this method we obtain an estimate of the rate of convergence of finite-dimensional distributions to the corresponding distributions of the Poisson process. Techniques based on the compactness criterion in used to prove a weak convergence to a counting process with a (random) continuous compensator. We present also a criterion for the convergence in variation together with the estimates of the rate of convergence. As an illustration we investigate the strong convergence of conditionally Poisson processes with intensities depending on a Markov process. Another example is an estimate of the rate of convergence of counting processes connected with the empirical distribution functions to the Poisson process.