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

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  • Schiopukratina, I.
  • Daffer, P.

Abstract

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 J1-topology of a sequence (xn) in D([0, 1]; E) does not imply convergence of a sequence (n) of averages. Convergence in the J1-topology of a sequence (n) of averages is shown, under the growth condition [short parallel] xn [short parallel] [infinity] = o(n), to be equivalent to the convergence of (n) in the uniform topology. Convergence of a sequence (xn,) is shown to imply convergence of the sequence (n) of averages in the M1 and M2 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.

Suggested Citation

  • Schiopukratina, I. & Daffer, P., 1995. "Convergence of Weighted Sums and Laws of Large Numbers in D([0,1]; E)," Journal of Multivariate Analysis, Elsevier, vol. 53(2), pages 279-292, May.
    • Handle: RePEc:eee:jmvana:v:53:y:1995:i:2:p:279-292
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