A divergent, two-parameter, bounded martingale

Dubins, Lester E.; Pitman, Jim

doi:10.1090/S0002-9939-1980-0553386-9

A divergent, two-parameter, bounded martingale
HTML articles powered by AMS MathViewer

by Lester E. Dubins and Jim Pitman

Proc. Amer. Math. Soc. 78 (1980), 414-416

DOI: https://doi.org/10.1090/S0002-9939-1980-0553386-9

PDF | Request permission

Abstract:

An example is given of a divergent, uniformly bounded martingale $X = \{ {X_t}:t \in T\}$ where the index t ranges over the set T of pairs of positive integers with the usual coordinatewise ordering.

References

R. Cairoli, Une inégalité pour martingales à indices multiples et ses applications, Séminaire de Probabilités, IV (Univ. Strasbourg, 1968/69) Lecture Notes in Mathematics, Vol. 124, Springer, Berlin, 1970, pp. 1–27 (French). MR 0270424
R. Cairoli and John B. Walsh, Stochastic integrals in the plane, Acta Math. 134 (1975), 111–183. MR 420845, DOI 10.1007/BF02392100
Jean Dieudonné, Sur un théorème de Jessen, Fund. Math. 37 (1950), 242–248 (French). MR 43176, DOI 10.4064/fm-37-1-242-248
Lester E. Dubins and Jim Pitman, A pointwise ergodic theorem for the group of rational rotations, Trans. Amer. Math. Soc. 251 (1979), 299–308. MR 531981, DOI 10.1090/S0002-9947-1979-0531981-7
Allan Gut, Convergence of reversed martingales with multidimensional indices, Duke Math. J. 43 (1976), no. 2, 269–275. MR 407969
K. Krickeberg, Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1956), 313–337. MR 91328, DOI 10.1090/S0002-9947-1956-0091328-4
J. Neveu, Discrete-parameter martingales, Revised edition, North-Holland Mathematical Library, Vol. 10, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Translated from the French by T. P. Speed. MR 0402915
R. T. Smythe, Sums of independent random variables on partially ordered sets, Ann. Probability 2 (1974), 906–917. MR 358973, DOI 10.1214/aop/1176996556