LMR lmr v.2004/01/22 Prn:4/04/2007; 13:32 F:LMR129.tex; (gm) p. 1 Liet. matem. rink., 47, No. 1, 2007, 1–14 A NOTE ON SELF-NORMALIZATION FOR A SIMPLE SPATIAL AUTOREGRESSIVE MODEL 1 V. Paulauskas and R. Zov˙ e Vilnius University, Department of Mathematics and Informatics and Institute of Mathematics and Informatics Akademijos 4, 08663 Vilnius, Lithuania (e-mail: vygantas.paulauskas@maf.vu.lt) Abstract. In this paper, we consider the problem of self-normalization for one rather simple autoregressive model X t,s = aX t −1,s + bX t,s−1 + ε t,s on a two-dimensionallattice. We show that there is some similarity between this problem and the corresponding problem for AR(1) time series model. Keywords: autoregressive models, self-normalization,random fields. Received 02 01 2007 1. INTRODUCTION AND FORMULATION OF RESULTS Nearly a hundred years passed from the famous paper by Gosset [5] written under the pseudonym “Student,” and now self-normalization is widely used in probability and mathematical statistics. Let X i ,i  1, be a stationary mean-zero sequence. Then one considers the sum S n = ∑ n i =1 X i normalized by the square root of the sum of squares V 2 n = ∑ n i =1 X 2 i . In the case of independent and identically distributed (i.i.d.) random variables, it is justified by the fact that usually good normalization is achieved by the square root of variance of S n which is nEX 2 1 . Due to the Law of Large Num- bers, V 2 n is a good approximation for the last quantity. At present, limit behavior (limit theorems with rates of convergence and asymptotic expansions, large deviations) of self-normalized sequence V −1 n S n in the case of i.i.d. is deeply investigated. There is a large amount of literature, and we refer to [3],[4], [9], [10]. The situation becomes more complicated for sequences of dependent random variables, and solution of the problem for general stationary sequences is far from being completed. Some new ef- fects comparing with i.i.d. case were noticed in [7] for exchangeable random variables. Recently, in [8], a specific form of dependence was considered, precisely, it was sup- posed that X i ,i ∈ Z, is AR(1) process obtained as a solution of the equation X i = ρX i −1 + ε i , where |ρ | < 1 and ε i ,i ∈ Z, is a sequence of i.i.d. random variables with Eε 1 = 0 and finite variance. It is well known that, in this case, X i ,i ∈ Z, is a stationary sequence 1 The research was supported by the bilateral France-Lithuania scientific project Gilibert.