On the convergence of series of pairwise independent random variables (Q1063931)

Let \(\{X_ n,n=1,2,...\}\) be pairwise independent random variables and \((0=)\) \(N_ 0<N_ 1<..\). be integers. The authors prove that the series \(\sum^{\infty}_{n=1}(X_ n-EX_ n)\) converges almost surely if the sequence \(\{X_ n\}\) satisfies one of the following pairs of conditions: \[ (1)\quad \sum^{\infty}_{m=0}\sqrt{\sum^{N_{m+1}}_{n=N_ m+1}D^ 2(X_ n)}<\infty \quad and\quad \sum^{N_{m+1}}_{n=N_ m+1}E| X_ n-EX_ n| =0(1)\quad as\quad m\to \infty \] \[ (2)\quad \sum^{\infty}_{m=1}(\sum^{N_{m+1}}_{n=N_ m+1}D^ 2(X_ n))\log^ 2m<\infty \quad and\quad \sum^{N_{m+1}}_{n=N_ m+1}E| X_ n-EX_ n| =0(1)\quad as\quad m\to \infty. \] The authors also point out that the conditions (1) and (2), respectively, are best possible in some sense, and that the pairwise independence cannot be relaxed to orthogonality.