Elliptically contoured measures on infinite-dimensional Banach spaces

Crawford, John J.

doi:10.4064/sm-60-1-15-32

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Generalized Weak Convolution1

Lemma1

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2006

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Cited by 19 publications

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“…Let us recall that the random vector X ∈ R n is rotationally invariant (spherically symmetric) if L(X) d = X for every unitary linear operator L : R n → R n . It is known (see [5,14] for the details) that the following conditions are equivalent…”

Section: Generalized Weak Convolutionmentioning

confidence: 99%

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Weak stability and generalized weak convolution for random vectors and stochastic processes

Misiewicz¹

2006

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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A random vector X is weakly stable iff for all a, b ∈ R there exists a random variable Θ such that aX + bX ′ d = XΘ. This is equivalent (see [11]) with the condition that for all random variables Q 1 , Q 2 there exists a random variable Θ such thatwhere X, X ′ , Q 1 , Q 2 , Θ are independent. In this paper we define generalized convolution of measures defined by the formulaif the equation ( * ) holds for X, Q 1 , Q 2 , Θ and µ = L(Θ). We study here basic properties of this convolution, basic properties of ⊕µ-infinitely divisible distributions, ⊕µ-stable distributions and give a series of examples.

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Section: Generalized Weak Convolutionmentioning

confidence: 99%

“…It is known (see [5,14] for the details) that the following conditions are equivalent (a) X ∈ R n is rotationally invariant…”

Section: Lemmamentioning

confidence: 99%