A Borel-Cantelli lemma and its applications (Q2862140)

The author proves the following version of the Borel-Cantelli lemma: Let \(X_{i}\) be nonnegative random variables and \(S_{n}=\sum_{i=1}^{n}X_i\). If \(\sup\operatorname{E}X_{i}<\infty\), \(\operatorname{E}S_{n}\rightarrow\infty\), and there exists \(\gamma>1\) such that NEWLINE\[NEWLINE \text{var}\left( S_{n}\right) =O\left( \frac{\left( \operatorname{E} S_{n}\right) ^{2}}{\left( \log\operatorname{E}S_{n}\right) \left( \log \log\operatorname{E}S_{n}\right) ^{\gamma}}\right) , NEWLINE\]NEWLINE then NEWLINE\[NEWLINE \frac{S_{n}}{\operatorname{E}S_{n}}\rightarrow1\quad\text{ a. s.} NEWLINE\]NEWLINE As applications, he obtains an almost sure local central limit theorem and a dynamical Borel-Cantelli lemma for systems with sufficiently fast decay of correlations with respect to Lipschitz observables.