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On convergence of random linear functionals

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  • Majumdar, Suman

Abstract

When is the pointwise limit of the characteristic functions of a sequence {Xn} of random elements taking values in a real separable Hilbert space H the characteristic function of a H valued random element? Uniform tightness of {[short parallel]Xn[short parallel]:n[greater-or-equal, slanted]1} is sufficient, but not necessary, whereas uniform tightness of { :n[greater-or-equal, slanted]1,[short parallel]y[short parallel][less-than-or-equals, slant]1} is necessary, but not sufficient. The sufficiency statement extends to a separable reflexive Banach space, whereas the necessity statement extends to a Banach space with separable dual.

Suggested Citation

  • Majumdar, Suman, 2005. "On convergence of random linear functionals," Statistics & Probability Letters, Elsevier, vol. 75(4), pages 249-255, December.
    • Handle: RePEc:eee:stapro:v:75:y:2005:i:4:p:249-255
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