A form of the Borel-Cantelli lemma (Q1070649)

The main result of the paper is the Theorem: Let \(\{\) \({\mathcal F}_ n,n\geq 1\}\) be an increasing sequence of \(\sigma\)-fields and \(\{X_ n,n\geq 1\}\) an adapted sequence of random variables such that \(0\leq X_ n\leq 1\), \(S_ n=\sum^{n}_{i=1}X_ i\), \(s_ 1=0\), \(s_{n+1}=\sum^{n}_{i=1}E(X_{i+1}| {\mathcal F}_ i),\) \(n\geq 1\). Let \(f:[0,\infty)\to {\mathbb{R}}\) be a strictly positive and increasing function such that \(\int^{\infty}_{0}f^{-p}(x)dx<\infty\) for some \(p\in [1,2]\). Then \[ \lim_{n\to \infty}(S_ n-s_ n)/f(s_ n)<\infty \quad a.s. \] The limit is equal to zero on the set \(\{\) \(\lim_{n\to \infty}s_ n=+\infty \}.\) From this result the author deduces generalizations of the Borel- Cantelli-lemma and a convergence theorem for infinite series.