Inequalities for random fields

Abstract

This paper gives extensions of the Doob and Burkholder inequalities for certain classes of random fields. The Brennan-Doob inequality for V-quasimartingales is extended to the case p > 1 and is shown to hold for the class of decomposable processes satisfying the Doob inequality of Wong and Zakai [10]. A Doob inequality for the class of i-martingales having finite quadratic variation in the non-martingale coordinate is shown. For the class of quasi martingales having independent increments two Burkholder-type inequalities are derived