LOGLOG LAW FOR THE PERIODOGRAM

We consider linear process X_n=\sum_j=1^\infty c_j W_n-y_0 where \left\W_n\right\ satisfies: \left\W_n\right\ be indeinendent, E W_n=0, E W_n^2=1 \exists r. v. Y, constant \alpha such that E Y^2< \infty and P\left(\left|W_n\right|>\right. r) \&it P(|Y|>x) for arbitrary x and \mathbfn; or, \left\W_n\right\ be strictly stationary, E\left(W_n+1 \mid \mathscrF_n\right)= 0 \mathrma. \mathrms\left(\mathscrF_\mathrmn\right. is the \sigma-algebra generated by \left.\left\W_j: j \leqslant n\right\\right), E W_1^2 \log ^+\left|W_1\right|< \infty, E\left(W_1^2 \mid \mathscrF_-\infty\right) \sim A H_i^\prime 2-1 a.s. \left\c_j\right\ satisfies: \sum_j=1^\infty j\left|c_j\right|< \infty. Using a theorem of Heyde and Scott (1974), we olitain the strong invariance principle for the periodogram of \left\X_n\right\, prove the follow ing result: \limsup _r \rightarrow-\infty(N \log \log N)^-1 / 2\left|\sum_n=1^N h(n / N) X_n e^-i n \lambda\right|=\left(\pi c(\lambda) g(\lambda) \int_0^1 h^2(r) d r\right)^1 / 2 \text a.s. where h is an arbitrary continuous real function with bounded derevatives, g(\lambda)=\left.(2 \pi)^-1 \sum_j=0^\infty c_j t^-i j \lambda\right|^2 is the spectral density of \left\X_n\right\, c(\lambda)=2 for \lambda \neq 0, \pi, c(\lambda)=4 for \lambda=0, \pi.