Local and uniform moduli of continuity of chi–square processes

Abstract

Let ${\{{\mathrm{\eta }}_i(t),t\in [0,1]\}}_{i=1}^k$ be independent copies of $\mathrm{\eta }=\{\mathrm{\eta }(t),t\in [0,1]\}$, a mean zero continuous Gaussian process. Let

$Y_k:=Y_k(t)={\sum }_{i=1}^k{\mathrm{\eta }}_i^2(t),t\in [0,1].$

This paper shows how exact local (at 0) and uniform moduli of continuity (on [0,1]) of $Y_k$ can be obtained from the exact local and uniform moduli of continuity of η.