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 $\{{\eta }_i;i=1,\dots ,k\}$ be independent copies of $\eta$ and set $Y_k(t)={\sum }_{i=1}^k{\eta }_i^2(t),t\in [0,1].$ 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 $\varphi (t)$ be a positive function on $[0,\delta ]$ for some $\delta >0$. If \[\limsup_{t\to 0}\frac{ η(t)-η(0)}{ ϕ(t) }=1 \qquad a.s., \] then for all integers $k\ge 1$, \[ \limsup_{t\to 0} \frac{Y_{k }(t)-Y_{k }(0)} { ϕ(t)} = 2 Y^{1/2}_{k}(0) \qquad a.s.\] Set \[ σ^2(u,v)=E(η(u)-η(v))^2\quad\text{and}\quad \widetildeσ^2(x)=\sup_{|u-v|\le x}σ^2(u,v).\] Assume that ${\inf }_{t\in [0,1]}U(t,t)>0$ and, \[ \lim_{x\to0}\widetildeσ^2(x)\log 1/x =0. \] Let $\phi (t)$ be a positive function on $[0,1]$. Then if \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}}\frac{ η(u)-η(v)}{ φ(|u-v|) }=1 \qquad a.s.\] for all intervals $\Delta \subset [0,1]$, it follows that for all intervals $\Delta \subset [0,1]$ and all integers $k\ge 1$, \[ \lim_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}} \frac{Y_{k }(u)-Y_{k }(v) }{ φ(|u-v|)} = 2 \sup_{u\inΔ}Y_{k }^{1/2}(u), \hspace{.2 in}a.s.\]

11 pages. arXiv admin note: text overlap with arXiv:2006.14457