Precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables (Q449022)

Let \(\{X_{n},n\geq 1\}\) be a sequence of i.i.d. random variables, set \( S_{n}=X_{1}+\dotsb+X_{n},\) \(n\geq 1,\) and let \(d,\beta ,\sigma >0.\) Under necessary and sufficient moment conditions, the authors compute NEWLINE\[NEWLINE \lim\limits_{\varepsilon \searrow 0}\varepsilon ^{2\beta/d}\sum\limits_{n\geq 3}a_{n}\operatorname{E}[\left| S_{n}\right| -b_{n}]_{+},NEWLINE\]NEWLINE where \(a_{n}=(\log \log n)^{\beta -1}/n^{3/2}\log n\) and \(b_{n}=\varepsilon \sigma \sqrt{n}(\log \log n)^{d/2}\).