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Some weak and strong laws of large numbers for \(D[0,1]\)-valued random variables. (English) Zbl 0639.60004

Authors’ abstract: Pointwise weak law of large numbers and weak law of large numbers in the norm topology of D[0,1] are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D[0,1]. Uniform convex tightness and uniform integrability conditions are jointly characterized.
Marcinkiewicz-Zygmund-Kolmogorov’s and Brunk-Chung’s strong laws of large numbers are derived in the setting of D[0,1]-space under uniform convex tightness and uniform integrability conditions.
Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology for sequences in D[0,1] is studied.
Reviewer: S.D.Chatterji

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
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References:

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