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 $k\ge 1$, \[ \lim_{ h\to 0}\sup_{ \stackrel{ |u-v|\le h }{ u,v\in\Delta}} \frac{ Y_{ k }(u)-Y_{ k }(v) }{ \varphi (|u-v|)} = 2 \sup_{ u\in\Delta}Y_{ k }^{ 1/2}(u), \hspace{ .2 in}a.s.\]
Categories: math.PR