Between local and global logarithmic averages
Let \(X,X_1,X_2, \dots\) be i.i.d. real-valued random variables with \(EX=0\), \(X^2=1\), and \(E|X|^\theta <\infty\) for some \(\theta>2\). Put \(S_k= \sum^k_{i=1} X_i\), \(k\geq 1\), and \[ T_n= \sum^n_{k=1} k^{-1} \bigl(\Phi(b_k) -\Phi(a_k)\bigr)^{-1} I(a_k\leq k^{-1/2} S_k\leq b_k), \quad n\geq 1, \] where \(\Phi\) stands for the standard normal distribution function and it is assumed that the sequences \((a_k)_{k\geq 1}\) and \((b_k)_{k\geq 1}\) satisfy the following conditions: (i) \(a_k<b_k\), \(k\geq 1\); (ii) \(a_k\to 0\) and \(b_k\to 0\) as \(k\to\infty\); (iii) \(k^{1/ \theta}|a_k|\to\infty\) and \(k^{1/ \theta} |b_k|\to\infty\) as \(k\to \infty\); (iv) \(\sum_{k\geq 1} k^{1/ \theta-3/2} (b_k-a_k)^{-1} <\infty\). It is proved that one can define a Wiener process \(\{W(t), t\geq 0\}\) such that \[ \bigl|T_n-\log n-W(\sigma^2_n) \bigr|= o\bigl((\log n)^{1/3+ \varepsilon} \bigr)\quad \text{a.s} \] for all \(\varepsilon>0\), where \((\sigma_n)_{n\geq 1}\) is a sequence satisfying \(\sigma^2_n/ \log n\to 4\log 2\) as \(n\to \infty\).
scientific article; zbMATH DE number 1031774
scientific article