Small deviations of modified sums of independent random variables (Q1037018)

Let \(X_1, X_2,\dots\) be i.i.d. random variables, and put \(S_0=0\) and \(S_n=X_1+\dots+X_n\). For a certain function \(d\) on \([0,1]\), the author sets \(S_k^*=S_k+d(k/n)S_n\), \(0\leq k\leq n\), and \(R_n=\max_{0\leq k\leq n}S_k^{*}-\min_{0\leq k\leq n}S_k^{*}\). He studies the asymptotic behavior, as \(x\to\infty\), of sums of the form \(\sum_{n\geq1}a_nP(R_n/B_n\leq x/b_n)\), where \(a_n\geq0\), \(\sum_{n\geq1}a_n=\infty\), \(B_n, n\geq1\), are some positive numbers, and \(b_n\nearrow\infty\). For instance, if \(EX_1=0\) and \(EX_1^2=1\), the following interesting corollary is obtained: \[ \sum_{n\geq1}(\frac{(\log n)^{s/2-1}}{n})P(\max_{1\leq k\leq n}|S_k|\leq\pi x\sqrt{\frac{n}{8\log n}})\sim x^{s}\frac{4}{\pi}\Gamma(\frac{s}{2})\sum_{n\geq0}(-1)^n(2n+1)^{-(s+1)} \] as \(x\to\infty\) for \(s>0\). Reviewer's remark. The translation of the paper from Russian into English is obviously defective in some places.