Statistical Inference for Stochastic Processes 3: 199–223, 2000. c  2001 Kluwer Academic Publishers. Printed in the Netherlands. 199 Infill Asymptotics Inside Increasing Domains for the Least Squares Estimator in Linear Models ISTV ´ AN FAZEKAS 1,⋆ and ALEXANDER G. KUKUSH 2 1 University of Debrecen, Debrecen, Hungary 2 Kiev University, Kiev, Ukraine, e-mail: kuog@mechmat.univ.kiev.ua Abstract. A linear model observed in a spatial domain is considered. Consistency and asymptotic normality of the least squares estimator is proved when the observations become dense in a sequence of increasing domains and the error terms are weakly dependent. Similar statements are obtained for the linear errors-in-variables model. Mathematics Subject Classification (1991): 62M30, 60J05. Key words: α-mixing, asymptotic normality, consistency, errors-in-variables, infill asymptotics, least squares estimator, linear model, spatial observations 1. Introduction Analyzing statistical data one often encounters nonstandard problems. This paper deals with three problems: dependent observations, spatial data, and infill asymp- totics inside increasing domain. Consistency and asymptotic normality of the least squares estimator is well-known when the observations are taken from a sequence of increasing domains. However, consistency is not valid in the case of infill asymp- totics (see e.g. Lahiri [10], Fazekas et al. [6]). In this paper, we present conditions for consistency and asymptotic normality of the least squares estimator in a linear model when the observations become dense in a sequence of increasing domains. Consider the linear model z(x) = β ⊤ f(x) + ε(x), x ∈ T ∞ ⊂ R d , (1.1) where d is a fixed positive integer, z(x) is the observed random field, ε(x) is the nonobserved random error term, f(x) is the column vector of explanatory vari- ables, f is a known function, and β is the unknown parameter to be estimated. We suppose that β ∈ R p , where p is a fixed positive integer, f : T ∞ → R p is continuous, {ε(x), x ∈ T ∞ } is a mean zero random field. ⋆ Author for correspondence: Institute of Mathematics and Informatics, University of Debrecen, P.O. Box 12, H-4010 Debrecen, Hungary (e-mail: fazekasi@math.klte.hu)