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

Abstract

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.