Strong laws of large numbers for weakly correlated random variables (Q1825514)

Let \(\{X_ n:\) \(n\geq 1\}\) be a sequence of complex-valued r.v.'s. The following theorems, which include or improve previously known results, are proved \((\| \cdot \|_ p\) denotes the \(L^ p\) norm). (A) Let \(1<p\leq \infty\) and \(0<r\leq q\leq \infty\) with \(p^{-1}+rq^{- 1}\leq 1\). If \(\| X_ n\|_ p\leq 1\) and \[ \sum_{N}N^{-1}\| N^{-1}\sum_{n\leq N}X_ n\|^ r_ q<\infty \quad then\quad \lim_ NN^{-1}\sum_{n\leq N}X_ n=0\quad a.s.\quad (SLLN) \] (B) \(\sum_{N}N^{-1}\Phi_ 2(N)<\infty\), where \(\Phi_ 2(N):=\sup_ M \| N^{-1}\sum^{M+N}_{M+1}X_ n\|^ 2_ 2\), implies SLLN. Corollary: if \(\| X_ n\|_ 2\leq 1\) and Re(E \(X_ n\bar X_ m)\leq \Phi_ 1(| n-m|)\) with \(\Phi_ 1\geq 0\) satisfying \(\sum_{n}n^{-1}\Phi_ 1(n)<\infty\) then SLLN holds. Note that \(\Phi_ i\downarrow 0\) is not required in (B). These results are best possible. This is known for (B); the paper includes a construction of \textit{S. Szarek} which shows this for (A). For the proofs, two lemmas on numerical series [one of them extending a result of \textit{A. Dvoretzky}, Ann. Math. Statist., Baltimore Md. 20, 296-299 (1949; Zbl 0033.384)] and a maximal inequality are established.