On the Paper “ Weak Convergence of Some Classes of Martingales with Jumps ”

This note extends some results of Nishiyama [Ann. Probab. 28 (2000) 685–712]. A maximal inequality for stochastic integrals with respect to integer-valued random measures which may have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion is obtained; if the so-called quadratic modulus is bounded in probability and if a certain entropy condition on the parameter space is satisfied, then the tightness follows. Our approach is based on the entropy techniques developed in the modern theory of empirical processes.