Exact convergence rates in strong approximation laws for large increments of partial sums (Q1085875)

Consider partial sums \(S_ n\) of an i.i.d. sequence \(X_ 1,X_ 2,..\). of centered random variables having a finite moment generating function \(\phi\) in a neighborhood of zero. The asymptotic behaviour of \[ U_ n=\max _{0\leq k\leq n-b_ n}(S_{k+b_ n}-S_ k) \] is investigated, where \(1\leq b_ n\leq n\) denotes an integer sequence such that \(b_ n/\log n\to \infty\) as \(n\to \infty\). In particular, if \(b_ n=o(\log ^ pn)\) as \(n\to \infty\) for some \(p>1\), the exact convergence rate of \(U_ n/b_ n\alpha _ n=1+o(1)\) is determined, where \(\alpha _ n\) depends upon \(b_ n\) and the distribution of \(X_ 1\). In addition, a weak limit law for \(U_ n\) is derived. Finally, it is shown how strong invariance takes over if \(\lim _{n\to \infty} b_ n(\log \log n)^ 2/\log ^ 3n=\infty\).