Operators of stable type
In the paper three non-equivalent notions of operators of stable type p are introduced for \(p\in]0,2[\). Let \(\{\Theta^ p_ i|\) \(i\geq 1\}\) be a sequence of independent copies of a symmetric p-stable real-valued random variable \(\Theta^ p\) defined on a probability space (\(\Omega\),\({\mathfrak A},P)\) and having \(\exp (-| t|^ p)\) as its characteristic function. Let B and F be real Banach spaces and F be separable. An operator \(u\in L(B, F)\) is called of \(S_ 1\)-type p if the convergence of a series \(\sum \{\| x_ i\|^ p_ B|\) \(i\geq 1\}\) implies the almost sure convergence of the series \(\sum \{\Theta^ p_ iu(x_ i)|\) \(i\geq 1\}\). An operator \(u\in L(B, F)\) is called of \(S_ 2\)-type p if there is a constant \(c>0\) such that for every finite family \(\{\xi_ i\}\) of independent symmetric B-valued random variables on (\(\Omega\),\({\mathfrak A},P)\) \[ \sup (t^ pP\{\| \sum u(\xi_ i)\| >t\}| \quad t>0)\leq c\sum \sup (t^ pP\{\| \xi_ i\| >t)| \quad t>0) \] is valid. The definition of an operator of \(S_ 3\)-type p is more technical. The class of all operators of \(S_ i\)-type p is denoted by \(S^ p_ i(B, F)\). The first class is appropriate for a description of stable measures, and the two others for domains of attraction of stable laws. Connections between the classes \(S^ p_ i(B,F)\) are investigated, some concrete examples of operators of stable type are given and some new results about random variables satisfying the central limit theorem with an index p are obtained.
scientific article; zbMATH DE number 3905521