No. 955

Title:

On tightness of $\ell ^\infty $-valued local martingales with infinitely
many jumps: metric and partitioning entropy approach

Author(s):

Nishiyama, Yoichi (The Institute of Statistical Mathematics)

Key words:

Tightness, weak convergence, martingale, multivariate point process, infinitely many jumps, metric entropy, partitioning entropy

Abstract:

    This note extends some results of Nishiyama ({Ann.\ Probab.}, 2000, {28}, 685-712). A maximal inequality for stochastic integrals with respect to multivariate point processes wihch have infinitely many jumps on compact time intervals is given. By using it, a tightness criterion for general $\ell ^\infty $-valued local martingales is presented; if the so-called {quadratic modulus} is bounded in probablity 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.