A prohorov bound for a poisson process and an arbitrary counting process with some applications

An upper bound for the Prohorov distance between a Poission process and an arbitrary couting process in terms of their compensators is examined. We discuss the sharpness of this bound and apply it to the known weak convergense results of Brown [1] and Kabanov, Liptser and Shiryayev [2]. We also use our bound to investigate empirical and renewal counting processes.