Moment inequalities for spatial processes

Abstract

This paper establishes a general moment inequality for spatial processes satisfying the α-mixing condition [cf., Tran, 1990. Kernel density estimation on random fields. J. Multivariate Analy. 34, 37–53]. Such a general moment inequality is a nontrivial extension of the corresponding result established in Cox and Kim [1995. Moment bounds for mixing random variables useful in nonparametric function estimation. Stochastic Process. Appl. 56, 151–158] for the time series case. As is the case for the Cox–Kim inequality for nonparametric estimation of time series, the new inequality is useful in nonparametric kernel estimation of spatial processes.