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On the asymptotic behaviour of stationary Gaussian processes. (English) Zbl 0851.60035

Let \((X_t)_{t > 0}\) be a real stationary Gaussian process such that, for each \(t\), \(X_t\) is an \(N(0,1)\) distributed random variable; suppose moreover that \((X_t)_{t > 0} \) has continuous paths and let \(r(t)\) be its covariance function. One can easily show the following theorem: Assuming in addition that \[ \lim_{t \to \infty} r(t) \log t = 0, \tag{*} \] one has \[ \lim \sup_{t \to \infty} {X_t \over \sqrt {2 \log t}} = \lim \sup_{t \to \infty} {|X_t |\over \sqrt {2 \log t}} = 1 \quad \text{a.s.,} \] so that the limit set (as \(t \to \infty)\) of the process \(Y_t = X_t/ \sqrt {\log t}\) is the interval \(S = \{y \in \mathbb{R} : {1 \over 2} y^2 \leq 1\}\).
The author considers the process \((Y_t)_{t > 0}\) defined by \(Y_t = X_t/ \varphi (t)\), where \(\varphi\) is a function verifying suitable assumptions; under a condition on \(r(t)\) weaker than (*), it is proved that the limit set of \((Y_t)\) as \(t \to \infty\) is the interval \(S_M = \{y \in \mathbb{R} : {1 \over 2} y^2 \leq M\}\), and the number \(M\) is characterized in terms of \(\varphi\). More precisely, \(M\) is shown to be \[ M = \lim \sup_{n \to \infty} {\log (\sum^n_{k = 1} 1/ \varphi (k)) \over \varphi^2 (n)}. \] The preceding result can be easily extended to the multidimensional case, as is sketched in Section 4.

MSC:

60G10 Stationary stochastic processes
60G15 Gaussian processes
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References:

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