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Operator geometric stable laws. (English) Zbl 1069.60017
Operator geometric stable laws (OGS) on a vector space \(\mathbb{R}^d\) are limits of randomized affine normalized sums \(A_p\sum_1^{N_p}(X_i + b_b)\), \(p\to 0\), where \(X_i\) are iid and \(N_p\) is a geometrically distributed random variable (with mean \(1/p\)) which is independent from \(X_i, i\in \mathbb{N}\). If \(Y\) is OGS, then the array \(X_i\) is said to belong to the domain of (geometric stable) attraction of \(Y\). First the paper collects some general properties of OGS and operator stable laws which are partially known in more general situations as long as strict stability is assumed, i.e. if the shift terms \(b_p\) vanish. Section 4 is concerned in particular with geometric stable laws, i.e. with scalar normalisations \(A_n =a_n I\), and their marginals and densities; in particular laws with Linnik and Laplace marginals. These distributions turn out to have interesting applications in mathematical finance. The proofs of the main results and some auxiliary results are found in an Appendix.

MSC:
60E07 Infinitely divisible distributions; stable distributions
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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