Domains of operator semi-attraction of operator semi-stable probability measures
Let \(A\) be a linear operator acting in a finite-dimensional real vector space \(V\) and \(p\) be a probability measure on \(V\). The measure \(Ap\) is defined by \(Ap(E)=p(A^{-1}E)\) for each Borel subset \(E\subset V\), \(pq\) is the convolution of the measures \(p\) and \(q\), and \(\delta(x)\) denotes the measure concentrated at the point \(x\in V\). Moreover, \(\operatorname{Aut} V\) denotes the group of all non-singular operators on \(V\). If, for \(A_k\in\operatorname{Aut} V\), \(b_k\in V\), and a strictly increasing sequence of positive numbers such that \[ \lim_{k\to\infty}{n_k\over n_{k+1}}=r>0, \] \((A_kp^{n_k})\delta(b_k)\) converges weakly to \(q\) as \(k\to\infty\), then the measure \(q\) is called operator semi-stable, and \(p\) belong to the domain of operator semi-attraction of \(q\). In the case when \(n_k=k\), ``semi'' in above definition is omitted. The author presents new characterizations for operator semi-stable measures and for the domains of operator semi-attraction as well as of operator attraction. For example, if \(q\) is full, \((A_kp^{n_k})\delta(b_k)\) converges weakly to \(q\) and the intersection of \((0, 1)\) with the set of limit points of \({n_k\over n_{k+1}}\) is non-empty, then the measure \(q\) is operator semi-stable.
scientific article; zbMATH DE number 5780182