Extremal Processes with One Jump

Convergence of a sequence of deterministic functions in the Skorohod topology implies convergence of the jumps. For processes with independent additive increments the fixed discontinuities converge. In this paper it will be shown that this is not true for processes with independent max-increments. The limit in of a sequence of stochastically continuous extremal processes may have fixed discontinuities. Our construction makes use of stochastically continuous extremal processes whose sample functions have only one jump.