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Convergence of elements in random normed spaces

Published online by Cambridge University Press:  17 April 2009

Robert Lee Taylor
Robert Lee Taylor
Affiliation:
Department of Mathematics and Computer Science, University of South Carolina, Columbia, South Carolina, USA.
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Abstract

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For a random normed space of mappings into a separable normed linear space, convergence of identically distributed elements in i the random norm (norm distribution) is shown to be equivalent to convergence in measure in the weak linear topology. Convergence in measure in each coordinate of a Schauder basis is also shown to be a necessary and sufficient condition for convergence in the random norm topology. These results have laws of large numbers for random elements in separable normed linear spaces as almost immediate corollaries and illustrate some of the recently obtained laws of large numbers for random elements. Similar results are also given for elements which need not have the same norm distributions, and the results are extended to linear metric spaces. Finally, applications of the results to stochastic processes are considered.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1975 , pp. 31 - 47
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Beck, Anatole, “On the strong law of large numbers”, Ergodic theory, 21–53 (Proc. Internat. Synrpos., Tulane University, New Orleans, 10, 1961. Academic Press, New York, London, 1963).Google Scholar
[2]Beck, Anatole and Giesy, Daniel P., “P-uniform convergence and a vector-valued strong law of large numbers”, Trans. Amer. Math. Soc. 147 (1970), 541–559.Google Scholar
[3]Beck, Anatole and Warren, Peter, “A strong law of large numbers for weakly orthogonal sequences of Banach-space valued random variables”, (MRC Technical Summary Report, #848. Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1968).Google Scholar
[4]Billingsley, Patrick, Convergence of probability measures (John Wiley & Sons, New York, London, Sydney, Toronto, 1968).Google Scholar
[5]Doob, J.L., “Probability in function space”, Bull. Amer. Math. Soc. 53 (1947), 15–30.Google Scholar
[6]Horváth, John, Topologioal vector spaces and distributions, Volume I (Addison-Wesley, Reading, Massachusetts; Palo Alto; London; Don Mills, Ontario; 1966).Google Scholar
[7]Mann, Henry B., “On the realization of stochastic processes by probability distributions in function spaces”, Sankhyā 11 (1951), 3–8.Google Scholar
[8]Marti, Jürg T., Introduction to the theory of bases (Springer Tracts in Natural Philosophy, 18. Springer-Verlag, Berlin, Heidelberg, New York, 1969).Google Scholar
[9]Mourier, Edith, “Eléments aléatoires dans un espace de Banach”, Ann. Inst. H. Poincaré 13 (1953), 16.Google Scholar
[10]Mourier, Edith, “L-random elements and L*-random elements in Banach spaces”, Proc. Third Berkeley Sympos. Math. Statist. Probability, Volume II, 231–242 (University of California Press, Berkeley and Los Angeles, 1956).Google Scholar
[11]Mushtari, D.Kh., “On almost sure convergence in linear spaces of random variables”, Theor. Probability Appl. 15 (1970), 337–342.Google Scholar
[12]Муштари, Д.Х., Шерстнев, A.H. [D.H. Muštari, A.M. Šerstnev], “О способах введения топологии в случайных метричвсних пространствах” [On methods of introducing a topology in random metric spaces], Izv. Vysš. Učebn. Zaved. Matematika 1966, no. 6 (55), 99–106.Google Scholar
[13]Nedoma, Jiři, “Note on generalized random variables”, Proa. First Prague Conf. Information Theory, Statistical Decision Functions, Random Processes, Liblice, 1956, 139–141 (Publishing House of the Czechoslovak Academy of Sciences, Prague, 1957).Google Scholar
[14]Padgett, W.J., Taylor, R.L., Lous of large numbers for normed linear spaces and certain Frechet spaces (Lecture Notes in Mathematics, 360. Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
[15]Prokhorov, Yu.V., “Convergence of random processes and limit theorems in probability theory”, Theor. Probability Appl. 1 (1956), 157–214.Google Scholar
[16]Шерстнвв, A.H. [A.M. Šerstnev], “О понятии случайного нормированного пространства” [On the concept of a stochastic normalized space], Dokl. Akad. Nauk SSSR 149 (1963), 280–283.Google Scholar
[17]Taylor, Robert Lee, “Weak laws of large numbers in normed linear spaces”, Ann. Math. Statist. 43 (1972), 1267–1274.Google Scholar
[18]Taylor, R.L. and Padgett, W.J., “Some laws of large numbers for normed linear spaces”, Sankhyā Ser. A (to appear).Google Scholar
[19]Wilansky, Albert, Functional analysis (Blaisdell [Ginn and Co.], New York, Toronto, London, 1964).Google Scholar