ALMOST SURE LIMIT THEOREM FOR THE MAXIMA OF STRONGLY DEPENDENT GAUSSIAN SEQUENCES

In this paper, we prove an almost sure limit theorem for the maxima of strongly dependent Gaussian sequences under some mild conditions. The result is an expansion of the weakly dependent result of E. Csáki and K. Gonchigdanzan.


Introduction and main result
In past decades, the almost sure central limit theorem (ASCLT) has been studied for independent and dependent random variables more and more profoundly.Cheng et al. [CPQ98], Fahrner and Stadtmüller [FS98] and Berkes and Csáki [BC01] considered the ASCLT for the maximum of i.i.d.random variables.An influential work is Csáki and Gonchigdanzan [CG02], which proved an almost sure limit theorem for the maximum of stationary weakly dependent sequence.Theorem A. Let X 1 , X 2 , • • • be a standardized stationary Gaussian sequence with r n = C ov(X 1 , X n+1 ) satisfying r n log n(log log n) 1+ǫ where I is indicator function.Shouquan Chen and Zhengyan Lin [CL06] extended the results in [CG02] to the non-stationary case.
Leadbetter et al [LLR83] showed the following theorem.
where and in the sequel φ is standard normal density.
In the paper, we consider the ASCLT version of (2).The theorem below is useful in our proof.Theorem C. [Leadbetter et al., 1983 with some positive constant K 1 depending only on δ.Throughout this paper, ξ 1 , ξ 2 , • • • is stationary dependent Gaussian sequence and ξ 1 , ξ 2 , • • • was called as dependent: weakly dependent for r = 0 and strongly dependent for r > 0. Let log n , r defined in (4). (5) In the paper, a very natural and mild assumption is We mainly consider the ASCLT of the maximum of stationary Gaussian sequence satisfying (4), under the mild condition (6), which is crucial to consider other versions of the ASCLT such as that of the maximum of non-stationary strongly dependent sequence and the function of the maximum.
In the sequel, a = O(b) is denoted by a ≪ b, C is a constant which may change from line to line.The main result is as follows.
Theorem.Let {ξ n } be a sequence of stationary standard Gaussian random variables with covariances Remark 1.When r = 0, clearly, Theorem induces Theorem A. When r > 0, ξ 1 , ξ 2 , • • • is strongly dependent.We mainly focus on the proof of Theorem 1 for this case.Remark 2. In the above definition of ρ n , when n = 1, the definition is incompatible.In the paper, we mainly consider the case of n → ∞.So here, n may be assumed in a neighborhood of +∞ and the incompatibility doesn't result in the invalidation of our argument.