A convergence theorem for sums of dependent Hilbert space valued triangular arrays (Q1324597)

The following theorem is proved: Let the triangular array of separable Hilbert space valued random variables \(\varepsilon_{nm}\), \(m=1,2,\ldots,n\), \(n=1,2,\ldots\), be such that \(E\varepsilon_{nm}=0\), \(E\|\varepsilon_{nm}\|^ 2\leq Kn^{1-\delta}\), and \(E\|\varepsilon_{nm}-\varepsilon_{tm}\|^ 2\leq Kn^{-\delta} (n\leq t)\), \(| E\langle \varepsilon_{nm}, \varepsilon_{nl} \rangle |\leq K\cdot n^{-(1+\delta)}\) \((m\neq l)\), for fixed \(K\) and \(\delta>0\). Then \(n^{-1}\sum^ n_{m=1}\varepsilon_{nm}\to 0\) a.s. A theorem of \textit{S. K. Ahn} [Biometrika 75, No. 3, 590-593 (1988; Zbl 0651.62015)] and a theorem of \textit{J. D. Jobson} and \textit{W. A. Fuller} [J. Am. Stat. Assoc. 75, 176-181 (1980; Zbl 0437.62064)] can then be proved. Originally these two theorems were proved by a lemma of Jobson and Fuller, but there exists a counterexample for this lemma. The corollary from the theorem is a theorem of Chung (1974): Let the sequence of random variables \(X_ i\), \(i=1,2,\ldots\), be such that \(E(X_ iX_ j)=0\) for \(i\neq j\) and \(EX^ 2_ i=\sigma^ 2_ i\leq K\), where \(K\) is a fixed number. Then \(n^{-1}\sum^ n_{m=1} X_ m\to 0\) a.s.