The asymptotic behavior of extrema of compound Cox processes with nonzero means

We give necessary and sufficient conditions for the weak convergence of one-dimensional distributions of extrema of compound Cox processes with jumps having finite variances and nonzero expectations. No moment-type restrictions are imposed on the controlling process. As corollaries we obtain criteria of the normal convergence of extrema of compound Cox processes with nonzero means. Convergence rate estimates are also presented. Supported by the Russian Foundation for Basic Research (grant Nos. 96-01-01919 and 97-01-00271), the Russian Humanitarian Sciences Foundation (grant No. 97-02-02235), and the Committee on Knowledge Extension Research (CKER) of the American Society of Actuaries. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.