The exact approximation order of the random maximum Of cumulative sums Of independent variables

Abstract

We investigate the asymptotic behavior of the uniform distance between the distributions of the random maximum of cumulative sums and where W_t is the Wiener process. It is assumed here that the variates are independent and identically distributed. We show that, under some weak conditions on the random index of the maximum, the approximation order of the uniform distance is as sharp as in the Berry-Essen Inequality. The main tools of achieving this are the use of the Hausdorff-metric and some probabilistic arguments.

Keywords:

• Stopping rules
• Berry-Essen Inequality