Optimal averaging procedures in almost sure central limit theory

Let \(X_1, X_2, \ldots\) be i.i.d. random variables with \(\mathbb{E}[X_1] = 0\), \(\mathbb{E}[X_1^2] = 1\), \(S_n = X_1 + \cdots + X_n\) and let \((d_k)\) be a positive numerical sequence. We investigate the a.s. convergence of the averages \[\frac{1}{D_N} \sum_{k = 1}^{N} d_k I \{S_k / \sqrt{k} \leq x\},\]where \(D_N = \sum_{k = 1}^{N} d_k\). In the case of \(d_k = 1/k\) we have logarithmic means and by the almost sure central limit theorem the above averages converge a.s. to \(\Phi(x)\), the standard normal distribution function. It is also known that the analogous convergence relation fails for \(d_k = 1\) (ordinary averages). In this paper we give a fairly complete solution of the problem for which weight sequences the above convergence relation and its refinements hold.