On the asymptotic behaviour of stationary gaussian processes

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On the asymptotic behaviour of stationary Gaussian processes (*) by Rita Giuliano Antonini ( §) 0. Introduction Let be a real stationary gaussian process such that, for each t, X. is an N(0,1) distributed random variable; suppose moreover that has continuous paths and let r(t) be its covariance function. Under these assumptions, MARCUS [1]  The preceding remarks are due to G. LETTA.
In this paper we consider the process defined by Y t = X t 0 3 C 6 ( t ) , where (p is a function verifying suitable assumptions; under a condition on r(t) weaker than (0.2), we show that a. s. the limit set of (Yt) as t~ĩ s the interval and we characterize the number M in terms of (p. . We point out that our result can be easily extended to the multidimensional case, as is sketched in section 4. We are indebted to the referee for having simplified and improved the proofs: in particular we owe to him the use of theorem (2.3).

Assumptions and main theorem
Let real stationary gaussian process such that, for every t, Xĩ s N(0,1) distributed. Let r be the covariance function; we assume that (1.1) ~X~~o has continuous sample paths with probability one; (1.2) r(n) = 0 oo N) (mixing condition).
Then a. s. the limit set of (Yt) as t -~ is SM.
The two cases M=0 and M = oo will be discussed in section 3.
In the case 0M~, the proof will be carried out in the following two steps: ( 1.4) PROPOSITION. The limit set of (Yt) is a, s. contained in SM. ( 1.5) PROPOSITION. Each point of SM is a limit point of (Yt).
As to proposition (1.4), it is enough to notice that log °- 2. The proof of (1.5~. It is easy to prove the following result, which implies (1. such that lim sup Zn = c a. s., and almost surely the limit set of (Zn)n is n~ẽ qual to [c, c ] . . Indeed, in our setting, we have c = 2 M . This result can be found for example in M. TALAGRAND [4], where the fact that the limit set is not random is proved at the beginning of section III, and the rest in Lemma 7.
In this case we have lim sup log t 03C62(t) =o, and the desired result follows from (0.1).
That every yER is a limit point is a straightforward consequence of (2.3). Suppose that the following assumptions are verified (4.1) X. has continuous sample paths with probability one; Then as in the real case, a. s. S~ is the limit set of (Yt) as t -~ + oo.
Indeed, proposition (1.4) may be immediately extended by standard arguments.
As to Proposition (1.5), we already know that the limit set is not random and. a. s. contained in ~. Moreover it is closed and balanced a. s. by Lemma 7 of [4] (which holds in Rd); so it is enough to prove that, for every unit vector z = (z~...,Zd). we have