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

TL;DR

This paper investigates the local and uniform moduli of continuity of chi-square processes derived from Gaussian processes, establishing almost sure limits based on the processes' local behavior and covariance structure.

Contribution

It provides new almost sure limit results for the moduli of continuity of chi-square processes, extending classical Gaussian process results to these non-Gaussian processes.

Findings

Almost sure limits for local moduli of chi-square processes.

Conditions under which the processes' increments behave regularly.

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