More on P-Stable Convex Sets in Banach Spaces

Yu. Davydov, V. Paulauskas and A. Račkauskas

Journal of Theoretical Probability, 2000, vol. 13, issue 1, 39-64

Abstract: Abstract We study the asymptotic behavior and limit distributions for sums S n =bn -1 ∑i=1 n ξi,where ξ i, 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 S n(t) =bn -1 ∑i=1 [nt] ξi, t∈[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where ξ i are segments, the limit of S n is proved to be countable zonotope. Furthermore, if B = Rd, the singularity of distributions of two countable zonotopes Yp 1, σ1,Yp 2, σ2, corresponding to values of exponents p 1, p 2 and spectral measures σ 1, σ 2, is proved if either p 1 ≠ p 2 or σ 1 ≠ σ 2; (iv) Some new simple estimates of parameters of stable laws in Rd, based on these results are suggested.

Keywords: stable convex sets; LePage type representation; random zonotopes; invariance principle; Levy motion; stable laws; estimate of parameters (search for similar items in EconPapers)
Date: 2000
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DOI: 10.1023/A:1007726708227

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