No. 989

Title:

Additions to the paper ``Weak convergence of some classes of martingales with jumps''

Author(s):

Nishiyama, Yoichi (The Institute of Statistical Mathematics)

Key words:

Maximal inequality; weak convergence; martingale; integer-valued random measure; entropy

Abstract:

    This note extends some results of Nishiyama ({Ann.\ Probab.}, 2000, {28}, 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.