Sufficient conditions for relative compactness of measures corresponding to two-parameter strong martingales. (English. Russian original) Zbl 0629.60052

Mishura, Yu. S.

Theory Probab. Math. Stat. 34, 117-125 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 104-111 (1986).

In this paper the author first establishes sufficient conditions for the relative compactness of the laws of a sequence of two-parameter random processes in terms of stopping times. The paths of these processes belong to the space of right-continuous functions on \(R^ 2_+\) vanishing on the axes, and without discontinuities of the second kind, equipped with the Skorohod topology. The conditions for relative compactness are similar to those given for one-parameter processes by R. Rebolledo in Bull. Soc. Math. Fr., Suppl., Mem. 62 (1979; Zbl 0425.60036).
As an application, the relative compactness of a sequence of square integrable strong martingales is obtained, assuming the weak convergence of the associated predictable increasing processes. This last result is based on a partial generalization of inequalities of E. Lenglart [Ann. Inst. Henri Poincaré, n. Sér., Sect. B 13, 171-179 (1977; Zbl 0373.60054)] to the two-parameter case.

Reviewer: D.Nualart

MSC:

60G44	Martingales with continuous parameter
60G60	Random fields
60G40	Stopping times; optimal stopping problems; gambling theory
60B10	Convergence of probability measures

Keywords:

maximal inequalities; relative compactness; two-parameter random processes; stopping times; Skorohod topology; strong martingales

Citations:

Zbl 0425.60036; Zbl 0373.60054