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A central limit theorem for quadratic forms in strongly dependent linear (or moving average) variables is proved, generalizing the results of Avram [-1] and Fox and Taqqu [-3] for Gaussian variables. The theorem is applied to prove asymptotical normality of Whittle's estimate of the parameter of strongly dependent linear sequences.

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Summary.Conditions for the CLT for non-linear functionals of stationary Gaussian sequences are discussed, with special references to the borderline between the CLT and the non-CLT. Examples of the non-CLT for such functionals with the norming factor t/N are given. O. IntroductionIn the study of limit theorems for sums of dependent random variables, a particular role has been played by the case when the summands are (non-linear) functionals of a stationary Gaussian process. It was this case which was considered by M. Rosenblatt in his famous example of a non-Gaussian limit law [18]. More recently, the non-central limit theorem (non-CLT) for functionals of Gaussian process was the object of studies by Dobrushin and Major [5], Gordeckii [8], Major [12], Rosenblatt [19,20] The aim of the present paper is to study the CLT for functionals of Gaussian processes 'in the vicinity of non-CLT'. In order to do that, we also prove some new non-CLT with the norming factor I/N. To be more explicit, let ~t= ~ ~ (p,(x)e.(x;t) ~ [(o,~[21(x:lxl +...+x,lr Acknowledgment. The authors are grateful to the referee for the careful reading and many helpful criticisms of the first version of this paper.

We consider the long memory and leverage properties of a model for the conditional variance of an observable stationary sequence, where the conditional variance is the square of an inhomogeneous linear combination of past values of the observable sequence, with square summable weights. This model, which we call linear ARCH (LARCH), specializes to the asymmetric ARCH model of Engle (1990), and to a version of the quadratic ARCH model of Sentana (1995), these authors having discussed leverage potential in such models. The model which we consider was suggested by Robinson (1991), for use as a possibly long memory conditionally heteroscedastic alternative to i.i.d. behaviour, and further studied by Giraitis, Robinson and Surgailis (2000), who showed that integer powers, of degree at least 2, can have long memory autocorrelation. We establish conditions under which the cross-autovariance function between volatility and levels decays in the manner of moving average weights of long memory processes. We also establish a leverage property and conditions for finiteness of third and higher moments.