Strong limit theorems for blockwise \(m\)-dependent random variables and generalization of the conjectures of Móricz
Let \(B\) be a real separable Banach space, \(\{X_ n:\;n\geq 1\}\) a sequence of \(B\)-valued r.v.s, \(m\) a nonnegative integer, and \(\{a_ n:\;n\geq 1\}\) a sequence of positive real numbers such that \(a_ 1\geq 1\), \(a_{n+1}-a_ n\geq 1\) for \(n\geq 1\) and \((a_{n+1}-a_ n)\uparrow\infty\). \(\{X_ n\}\) is said to be blockwise \(m\)-dependent with respect to \(\{a_ n\}\) if for large enough \(n\), the two sets \(\{X_{[a_ n]+1},\dots,X_ k\}\) and \(\{X_ \ell,X_{\ell+1},\dots,X_{[a_{n+1}]}\}\) are independent provided \(a_ n<k\), \(\ell\leq a_{n+1}\), and \(\ell-k>m\). In the real-valued case, the reviewer introduced [Proc. Am. Math. Soc. 101, 709-715 (1987; Zbl 0632.60025)] another notion of blockwise \(m\)- dependence, which is a particular case of the above one when \(a_ n=2^ n\) for \(n\geq 1\). The authors prove strong laws of large numbers and the law of the iterated logarithm for such r.v.s. They also give positive answers to the conjectures raised by the reviewer.
scientific article; zbMATH DE number 11938