Study of the Convergence of the Increments of Gaussian Process

Let ( ) { } X t t ; 0 ≥ be a Gaussian process with stationary increments ( ) ( ) { } ( ) 2 + − E X t s X t s 2 = σ . Let ( ) ≥ t a t 0 be a nondecreasing function of t with t a t 0 ≤ ≤ . This paper aims to study the almost sure behaviour of ( ) ( ) ( ) → + k tk k k t s a k X t s X t , 0 lim sup sup ≤ ≤ ∞ − α β where ( ) ( ) ( ) ( ) ( ) , −   + +   2 1 2 2 log log log 1 log log k k k k t k t k t t a t a t a α β σ α α = − with 0 1 α ≤ ≤ and { } k t is an increasing sequence diverging to ∞ .

is an increasing sequence diverging to ∞ .In [1] the following results are established.
i) If ( ) where 0 1 In this paper the limit theorems on increments of a Wiener process due to [1] are developed to the case of a Gaussian process.This can be considered also as an extension of the results to Gaussian processes obtained in [2].Throughout this paper, we shall always assume the following statements: Let ( )

{ }
; 0 X t t ≥ be an almost surely continuous Gaussian process with ( )

Main Results
In this section we provide the following two theorems which are the main results.We concern here with the development of the limit theorems of a Wiener process to the case of a Gaussian process under consideration the above given assumptions. and ( ) We note that ( ) ( ) be an increasing sequence diverging to ∞ such that ( ) and where 0 1

Proofs
In order to prove Theorems 1 and 2, we need to give the following lemmas.Lemma 1. (See [3]).For any small 0 , where m is any large number and Proof of Theorem 1. Firstly, we prove that For any { } k t with the condition (3), we define an increasing sequence { } k u by for some 1 The condition (3) is satisfied, and for large k, where k is large enough and C′ is a constant.By the definition of k u a , ( ) ( ) We shall follow the similar proof process as in [5].Set and define the sequence of events { } k A in the following form By (10), for large k we have where C′′ is a constant.From the fact 2 1 2 1 ) Then, by (11) and the concavity of ( ) , 0 Cov X X ≤ .Using Lemma 2, we obtain where k l ≠ .It follows from the Borel-Cantelli lemma that Also, the same result for the even subsequence { } 2k t of { } k t is easily obtained.Therefore we have (9).To finish the proof of Theorem 1, we need to prove The proof of ( 12) is similar to the provided proof in [1].Thus the proof of Theorem 1 is complete.Proof of Theorem 2. Firstly, we prove that According to Lemma 1, we have Thus, (13) is immediate by using Borel Cantelli lemma.
To finish the proof of Theorem 2 we need to prove Using the well known probability inequality (see [6]), one can find positive constants C and K such that, for all k K ≥ , . By the definition of ** ε , we have ( ) The condition (6) implies that there exists 0 K > such that where k l ≠ and Borel-Cantelli lemma implies (14).If ** 0 ε = , then Theorem 2 is immediate.Thus the proof of Theorem 2 is complete.

Some Results for Partial Sums of Stationary Gaussian Sequence
In this section we obtain similar results as Theorems 1 and 2 for the case of partial sums of a stationary Gaussian sequence.Let { } n X be a stationary Gaussian sequence with 0 0 with ( )  ( ) is a nondecreasing sequence of positive integers such that 0 n a n ≤ ≤ .For large n, we define is an increasing function of n and also we define discrete time parameter processes by ( ) ( ) ( ) is an increasing sequence of positive integers diverging to ∞ .By the same way as in the proofs of Theorems 1 and 2, we obtain the following results.   1 E X = .

Conclusion
In this paper, we developed some limit theorems on increments of a Wiener process to the case of a Gaussian process.Moreover, we obtained similar results of these limit theorems for the case of partial sums of a stationary Gaussian sequence.Some obtained results can be considered as extensions of some previous given results to Gaussian processes.

Theorem 3 .
Under the above statements of { }