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Lévy's Brownian motion as a set-indexed process and a related central limit theorem

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  • Ossiander, Mina
  • Pyke, Ronald

Abstract

The Brownian motion with multi-dimensional time parameter introduced by Paul Lévy can be viewed as a set-indexed Brownian process with independent increments. This is demonstrated in a way which yields a unified representation of Lévy's Brownian motion and the Brownian sheet. Lévy's Brownian motion, like Brownian sheet, is shown to be a special case of the additive set-indexed Gaussian process {Z(A): A [epsilon] A} with Cov(Z(A), Z(B) =[mu](A [intersection] B) for some measure [mu]. A particular family of spheres is seen to play the same basic role in this representation as the family of orthants plays for Brownian sheet. A related central limit theorem and invariance result are discussed for a natural family of empirical-like processes, indexed by large families of sets A.

Suggested Citation

  • Ossiander, Mina & Pyke, Ronald, 1985. "Lévy's Brownian motion as a set-indexed process and a related central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 21(1), pages 133-145, December.
    • Handle: RePEc:eee:spapps:v:21:y:1985:i:1:p:133-145
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    Cited by:

    1. Ziegler, Klaus, 1997. "Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions," Journal of Multivariate Analysis, Elsevier, vol. 62(2), pages 233-272, August.
    2. Zapala, August M., 2003. "A criterion for right continuity of filtrations generated by group-valued additive processes," Statistics & Probability Letters, Elsevier, vol. 63(1), pages 79-87, May.
    3. Xia Xu & Fengping Wu & Lina Zhang & Xin Gao, 2020. "Assessing the Effect of the Chinese River Chief Policy for Water Pollution Control under Uncertainty—Using Chaohu Lake as a Case," IJERPH, MDPI, vol. 17(9), pages 1-26, April.

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