Complex Stable Sums of Complex Stable Random Variables

Complex random vectors and ICA models: identifiability, uniqueness, and separability

“…The second-order structure of complex r.vc.s has been studied in [11]- [13], [15], and a general framework for higher-order statistics can be found from [23]. Some research has been conducted on complex elliptically symmetric distributions [24] and on complex stable distributions [25]. Polya's theorem to complex case is presented in [26].…”

Generalized Random Energy Model at Complex Temperatures

“…, z d are not necessarily real), then the term m z1 is real and hence, (z d − 1)ζ P (z) (which is considered as a random vector with values in C ≡ R 2 ) has a two-dimensional stable distribution (which need not be isotropic); see [42,Chapter 2]. In general, for z ∈ 1 2 D without any additional assumptions on the components, the distribution of the random variable (z d − 1)ζ P (z) (again considered as a random vector with values in C ≡ R 2 ) is strictly complex stable in the sense of Hudson and Veeh [21]. A random variable with values in C is called strictly complex stable, see [21], if for every m ∈ N the sum of m independent copies of this random variable, after dividing it by an appropriate complex number, has the same law as the original random variable.…”

“…In general, for z ∈ 1 2 D without any additional assumptions on the components, the distribution of the random variable (z d − 1)ζ P (z) (again considered as a random vector with values in C ≡ R 2 ) is strictly complex stable in the sense of Hudson and Veeh [21]. A random variable with values in C is called strictly complex stable, see [21], if for every m ∈ N the sum of m independent copies of this random variable, after dividing it by an appropriate complex number, has the same law as the original random variable. More generally, all finite-dimensional distributions of the stochastic process {(z d − 1)ζ P (z) : z ∈ 1 2 D} are strictly operator stable (and hence, infinitely divisible).…”