Journal of Theoretical Probability

, Volume 13, Issue 1, pp 39–64 | Cite as

More on P-Stable Convex Sets in Banach Spaces

  • Yu. Davydov
  • V. Paulauskas
  • A. Račkauskas


We study the asymptotic behavior and limit distributions for sums Sn =bn-1i=1n ξi,where ξi, i ≥ 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes Sn(t) =bn-1i=1[nt] ξi, t∈[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where ξi are segments, the limit of Sn is proved to be countable zonotope. Furthermore, if B = Rd, the singularity of distributions of two countable zonotopes Yp1, σ1,Yp2, σ2, corresponding to values of exponents p1, p2 and spectral measures σ1, σ2, is proved if either p1p2 or σ1σ2; (iv) Some new simple estimates of parameters of stable laws in Rd, based on these results are suggested.

stable convex sets LePage type representation random zonotopes invariance principle Levy motion stable laws estimate of parameters 


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© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Yu. Davydov
  • V. Paulauskas
  • A. Račkauskas

There are no affiliations available

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