  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.
