Local and uniform moduli of continuity of chi--square processes

Abstract

Let $\eta=\{\eta(t);t\in [0,1]\}$ be a mean zero continuous Gaussian process with covariance $U=\{U(s,t),s,t\in [ 0,1]\},$ with $U(0,0)>0$. Let $\{\eta_{i};i=1,\ldots, k\}$ be independent copies of $\eta$ and set $ Y_{k}(t)=\sum_{i=1}^{k} \eta^2_{i}(t), t\in [ 0,1].$ The stochastic process $Y_{k } =\{Y_{k }(t),t\in [ 0,1] \}$ is referred to as a chi--square process of order $k $ with kernel $U$. Let $\phi(t)$ be a positive function on $[0,\delta]$ for some $\delta>0$. If \[\limsup_{t\to 0}\frac{ \eta(t)-\eta(0)}{ \phi(t) }=1 \qquad a.s., \] then for all integers $k\ge 1$, \[ \limsup_{t\to 0} \frac{Y_{k }(t)-Y_{k }(0)} { \phi (t)} = 2 Y^{1/2}_{k}(0) \qquad a.s.\] Set \[ \sigma^2(u,v)=E(\eta(u)-\eta(v))^2\quad\text{and}\quad \widetilde\sigma^2(x)=\sup_{|u-v|\le x}\sigma^2(u,v).\] Assume that $\inf_{t\in [0,1]}U(t,t)>0$ and, \[ \lim_{x\to0}\widetilde\sigma^2(x)\log 1/x =0. \] Let $\varphi(t)$ be a positive function on $[0,1]$. Then if \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\in\Delta}}\frac{ \eta(u)-\eta(v)}{ \varphi(|u-v|) }=1 \qquad a.s.\] for all intervals $\Delta\subset [0,1]$, it follows that for all intervals $\Delta\subset [0,1]$ and all integers $k\ge 1$, \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\in\Delta}} \frac{Y_{k }(u)-Y_{k }(v) }{ \varphi (|u-v|)} = 2 \sup_{u\in\Delta}Y_{k }^{1/2}(u), \hspace{.2 in}a.s.\]