More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S n =bn -1 (sigma)i=1 n (xi) i,where (xi)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 (sigma) i=1 [nt] (xi)i, t (element of) [0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where (xi)i are segments, the limit of S n is proved to be countable zonotope. Furthermore, if B = R d , the singularity of distributions of two countable zonotopes Yp 1, (sigma) 1,Yp 2, (sigma)2, corresponding to values of exponents p 1, p 2 and spectral measures (sigma)1, (sigma)2, is proved if either p 1 (not equal of) p 2 or (sigma)1 (not equal of) (sigma) 2; (iv) Some new simple estimates of parameters of stable laws in R d , based on these results are suggested.