A b stra ct A spatial autoregressive process with two parameters is investigated in the stable and unstable cases. It is shown that the limiting distribution of the least squares estimator of these parameters is normal and the rate of convergence is n 3/2 if one of the parameters equals zero and n otherwise. In trod u ctionConsider the AR (1) The least squares estim ator <5" of a based on the observations {X k : k -1 , . . . , n} is -_ J2k=i X k -i X k n y 2 ' l^k=1 X k -1It is well known th a t in the stable (or, in other words, asym ptotically stationary) case when |a| < 1, the sequence (a n )n> 1 is asym ptotically normal (see M ann and Wald T h is research has been su p p o rte d by th e H ungarian Scientific Research Fund un d er G ra n t No. O T K A -F046061 /2004.A M S 2000 s u b je c t c la ssific a tio n s. P rim a ry 62M10; Secondary 62L12. K e y w o r d s a n d p h ra ses. S table a n d u n sta b le sp a tia l autoregressive m odels, a sy m p to tic norm ality, m artin g ale cen tral lim it theorem . 1[15] or Anderson [1]), namely, n 1/ 2 (3 n -a ) -^ N (0 , 1 -a 2).In the unstable (or, in other words, unit root) case when a -1, the sequence (a n )n> 1 is not asym ptotically normal but © JoW(t)dW (t) n (a n - The analysis of spatial models is of interest in many different fields such as ge ography, geology, biology and agriculture. One can tu rn to Basu and Reinsel [4] for a discussion on these applications. These authors considered a special case of the so called unilateral AR model having the form P1 P2 X k,ia i,j X k -i ,i -j + £k,i, aoio -0 .(1 .1 ) i= 0 j= 0 A special case of the above model is the so-called doubly geometric spatial autoreg ressive processintroduced by M artin [16]. This was the first spatial autoregressive model for which unstability has been studied. It is, in fact, the simplest spatial model, since the product structure y>(x, y) -x 2 -a x -3 y + a 3 -(x -a)(y -3 ) of its characteristic polynomial ensures th a t it can be considered as some kind of combination of two autoregressive processes on the line, and several properties can be derived by the analogy of one-dim ensional autoregressive processes. This model has been used by Jain [14] In the stable case when | | < 1 and | 3| < 1, asym ptotic norm ality of several estim ators ( 3 m,n ,3m ,n ) of (a , 3 ) based on the observations {X kjl : 1 ^ k ^ m and 1 ^ I ^ n} has been shown (e.g. Tj0 stheim [20,22] or Basu and Reinsel [3,4]), namely, as m, n with m /n ^ constant > 0 with some covariance m atrix . In the unstable case when a -3 -1, in contrast to the AR(1) model, the sequence of G auss-Newton estim ators (3 n,n ,3 n,n ) of (a, 3) has been shown to 2 be asym ptotically normal (see B hattacharyya, Khalil and Richardson [7] and Bhattacharyya, Richardson and Franklin [8 ]), namely, N (0, E) with some covariance m atrix E. In the unstable case a -1, |3| < 1 the least squares estim ator turns out to be asym ptotically normal again (B hattacharyya, Khalil and Richardson [7]).Baran, P ap and Zuijlen [2] discussed a speci...

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A sequence of Galton-Watson branching processes with immigration is investigated, when the offspring mean tends to its critical value one and the offspring variance tends to zero. It is shown that the fluctuation limit is an OrnsteinUhlenbeck type process. As a consequence, in contrast to the case where the offspring variance tends to a positive limit, the conditional least squares estimator of the offspring mean turns out to be asymptotically normal. The norming factor is n 3/2 , in contrast to the subcritical case where it is n 1/2 , and in contrast to the nearly critical case with positive limiting offspring variance, where it is n.

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2 , in contrast to both the subcritical case, in which it is n 1/2 , and the nearly critical case with positive limiting offspring variance, in which it is n.

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