Logarithmic moving averages (Q743237)

Let \((s_{n},n\geq 0)\) denote a sequence of real numbers and for \(x\geq 0\) let \(U(x)=\sum_{0\leq i\leq x}s_{i}/(i+1)\). The sequence is summable to \(s\) by the logarithmic \(l\)-method if \( U(n)/\log n\rightarrow s\) as \(n\rightarrow \infty \). For \(\nu<1\), the sequence \((s_{n})\) is summable by the logarithmic moving average method if \((U(n)-U(n^{\nu }))/\log n\rightarrow (1-\nu )s\). The authors prove that these summability methods are equivalent. They also relate these methods to other summability methods such as the Riesz mean and the random-walk method. Finally, the authors link these and other summability methods to the strong law of large numbers.