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Convergence of Weighted Sums and Laws of Large Numbers in D([0,1]; E)

✍ Scribed by I. Schiopukratina; P. Daffer

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
426 KB
Volume
53
Category
Article
ISSN
0047-259X

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✦ Synopsis

Convergence properties of weighted sums of functions in (D([0,1] ; E)(E) a Banach space) are investigated. We show that convergence in the Skorokhod (J_{1})-topology of a sequence (\left(x_{n}\right)) in (D([0,1] ; E)) does not imply convergence of a sequence (\left(\bar{x}{n}\right)) of averages. Convergence in the (J{1})-topology of a sequence (\left(\bar{x}{n}\right)) of averages is shown, under the growth condition (\left|x{n}\right|{\infty}=o(n)), to be equivalent to the convergence of (\left(\bar{x}{n}\right)) in the uniform topology. Convergence of a sequence (\left(x_{n}\right)) is shown to imply convergence of the sequence (\left(\bar{x}{n}\right)) of averages in the (M{1}) and (M_{2}) topologies. The strong law of large numbers in (D[0,1]) is considered and an example is constructed to show that different definitions of the strong law of large numbers are nonequivalent. 1995 Academic Press, Inc.

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