Operator stable laws: Series representations and domains of normal attraction
Let \(X_ 1,X_ 2,..\). be independent identically \({\mathbb{R}}^ d\)-valued random vectors, \(A_ 1,A_ 2,..\). matrices on \({\mathbb{R}}^ d\) and \(a_ 1,a_ 2,..\). vectors from \({\mathbb{R}}^ d\). A measure \(\mu\) is called operator-stable if it is a weak limit of the following sequence \[ (1)\quad A_ n(X_ 1+X_ 2+...+X_ n)+a_ n,\quad n\in {\mathbb{N}}. \] Of course, operator-stable measures are a natural extension to \({\mathbb{R}}^ d\) of the stable laws on \({\mathbb{R}}\). Further, each \(\mu\) (a limit of (1)) possesses an operator (matrix) exponent B. If \(A_ n\) in (1) are of the form \(n^{-B}\) then we say that the probability distribution of \(X_ 1\) belongs to a generalized domain of normal attraction of \(\mu\) (GDONA(\(\mu)\)). The present paper gives a series representation for an operator-stable measure (it generalizes Le Page's result for multidimensional stable laws) and provides a characterization of GDONA(\(\mu)\) in terms of order statistics. The latter is based on polar coordinates associated with the one-parameter group \(t^ B\), \(t>0\), on \({\mathbb{R}}^ d\).
series representation for an operator-stable measure
scientific article; zbMATH DE number 4094482
scientific article