More on P-Stable Convex Sets in Banach Spaces
Tóm tắt
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.
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