Asymptotic expansions in local theorems for the maximum of sums of random variables with mean zero
The author considers a sequence \(\{\xi_ j\}\) of independent identically distributed random variables with mean value zero, variance \(\sigma^ 2\), moments \(\alpha_{\nu}\) and absolute moments \(\beta_{\nu}\) of order \(\nu\). He forms the related random variables \(S_ n=\sum^{n}_{j=1}\xi_ j\), \(\bar S_ n=\max_{1\leq k\leq n}S_ k\), \(\bar Z_ n=\bar S_ n/\sigma \sqrt{n}\). For the density of \(\bar Z_ n\) he gets an asymptotic expansion in form of a polynomial.
scientific article; zbMATH DE number 3895989