A thinning result for pure jump processes

A standard thinning procedure for point processes is extended to processes of pure jump type in which each jump is retained with probability p or deleted with probability 1 ? p, independently of everything else.Two theorems are proved, the first gives a sufficient condition for the existence of thinned pure jump processes, the second concerns the convergence of such processes to pure jump processes whose increments are generated by a Cox process. Some generalizations are discussed.