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A contribution to the theory of asymptotic martingales

Published online by Cambridge University Press:  18 May 2009

Allan Gut
Affiliation:
Uppsala University, Department of Mathematics, S-752 38 Uppsala, Sweden
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During the last few years several articles on asymptotic martingales (amarts) have appeared. The first unified treatment was given by Edgar and Sucheston in [7], where further references can be found. The purpose of this paper is to add some further results to the theory of amarts.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1982

References

REFERENCES

1.Austin, D. G., Edgar, G. A., and Tulcea, A. Ionescu, Pointwise convergence in terms of expectations, Z. Wahrsch. Verw. Gebiete 30 (1974), 1726.CrossRefGoogle Scholar
2.Baxter, J. R., Pointwise in terms of weak convergence, Proc. Amer. Math. Soc. 46 (1974), 395398.CrossRefGoogle Scholar
3.Bellow, A., Stability properties of the class of asymptotic martingales, Bull. Amer. Math. Soc. 82 (1976), 338340.CrossRefGoogle Scholar
4.Bellow, A., Several stability properties of the class of asymptotic martingales, Z. Wahrsch. Verw. Gebiete 37 (1977), 275290.CrossRefGoogle Scholar
5.Blackwell, D. and Dubins, L. E., A converse to the dominated convergence theorem, Illinois J. Math. 7 (1963), 508514.CrossRefGoogle Scholar
6.Doob, J. L., Stochastic processes (Wiley, 1953).Google Scholar
7.Edgar, G. A. and Sucheston, L., Amarts: A class of asymptotic martingales. A. Discrete parameter, J. Multivariate Anal. 6 (1976), 193221.CrossRefGoogle Scholar
8.Gut, A., Moments of the maximum of normed partial sums of random variables with multidimensional indices, Z. Wahrsch. Verw. Gebiete 46 (1979), 205220.CrossRefGoogle Scholar
9.Klass, M. J., On stopping rules and the expected supremum of S n/an and |Sn|/ an, Ann. Probab. 2 (1974), 889905.CrossRefGoogle Scholar
10.Krengel, U. and Sucheston, L., On semiamarts, amarts and processes with finite value, Advances in Probability 4 (1978), 197266.Google Scholar
11.Loève, M., Probability theory, 3rd edition (Van Nostrand, 1963).Google Scholar
12.Mertens, J. F., Théorie des processus stochastiques genéraux applications aux surmartingales, Z. Wahrsch. Verw. Gebiete 22 (1972), 4568.CrossRefGoogle Scholar
13.Meyer, P. A., Probabilityés et Potentiel (Hermann, 1966).Google Scholar
14.Neveu, J., Discrete-parameter martingales (North-Holland, 1975).Google Scholar
15.Sudderth, W., A “Fatou equation” for randomly stopped variables, Ann. Math. Statist. 42 (1971), 21432146.CrossRefGoogle Scholar