## 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

### Keywords:

limit set; stationary Gaussian process; covariance function
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### References:

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