Asymptotics of Suprema of Weighted Gaussian Fields with Applications to Kernel Density Estimators

Abstract

For centered stationary Gaussian fields $X(s), s\in {\bf R}^d, d>1,$ we show that properly centered and normalized $\sup_{s\in {\bf R}^d}w(s/T)|X(s)|$ with a nonnegative weight function $w$ converges in distribution to a double exponential law as $T\to\infty$, provided that $w$ and the covariance function of $X$ satisfy certain smoothness and regularity conditions. This limit theorem extends the results for compact sets without weight functions obtained earlier in [V. Piterbarg, Asymptotics Methods in the Theory of Gaussian Processes and Fields, Amer. Math. Soc., Providence, RI, 1996]. This new result for Gaussian fields is then applied to obtain necessary and sufficient conditions for the properly centered and normalized sequence $\sup_{x\in {\bf R}^d}|\Psi(x)(\widehat{f}_n(x)-\e\,\widehat{f}_n(x))|$ to converge in distribution to a double exponential law under natural smoothness conditions, where $\widehat{f}_n$ denotes the kernel density estimator of the bounded continuous density $f$ on ${\bf R}^d$ based on the sample of size $n$, and $\Psi$ is a positive continuous function such that $\sup_{x\in {\bf R}^d}|\Psi(x) f(x)^\beta|<\infty$ for some $\beta\in (0, 1/2)$. This paper extends results of the paper [E. Giné, V. Koltchinskii, and L. Sakhanenko, Probab. Theory Related Fields, 130 (2004), pp. 167--198] to the case of $d>1$. It also extends the results of [E. Rio, Probab. Theory Related Fields, 98 (1994), pp. 21--45] for $\Psi(\cdot)=f^{-1/2}(\cdot)I_S(\cdot)$ with some compact set $S$ in ${\bf R}^d$ to a general class of functions $\Psi$. An example of an application completes this work.