An absolute moments condition for normed sums of independent variables (Q914234)

Let \(X_ n\), \(n\in N\), be i.i.d. symmetric random variables, \(1\leq p<q<2\). It is shown that the relation \[ 0<\inf | n^{- 1/q}\sum^{n}_{k=1}X_ k| \leq \sup | n^{- 1/q}\sum^{n}_{k=1}X_ k| <\infty \] is satisfied iff E \(X_ 1=0\) and there are constants \(a,b,c>0\) such that for \(x\geq c\) \[ ax^{-q}\leq P[| X_ 1| \geq x]\leq bx^{-q}. \] The case \(0<p<q<2\), \(p<2\), is also considered. The obtained assertions are refinements of the results of \textit{C. G. Esseen} and \textit{S. Janson}, Stochastic Processes Appl. 19, 173-182 (1985; Zbl 0554.60050).