Let be a Gaussian process with stationary increments . Let be a nondecreasing function of t with . This paper aims to study the almost sure behaviour of where with and is an increasing sequence diverging to .
Let
i) If
and
where
ii) If
where
In this paper the limit theorems on increments of a Wiener process due to [
surely continuous Gaussian process with
Let
where
We define two continuous parameter processes
and
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.
Theorem 1. Let
then
and
where
We note that
Theorem 2. Let
then
and
where
In order to prove Theorems 1 and 2, we need to give the following lemmas.
Lemma 1. (See [
where m is any large number and
Lemma 2. (See [
Proof of Theorem 1. Firstly, we prove that
For any
For instance, let
The condition (3) is satisfied, and for large k,
where k is large enough and
We shall follow the similar proof process as in [
Since
quence
By (10), for large k we have
where
Since
Setting
and
we have
Let
and
Then, by (11) and the concavity of
This implies that
where
Also, the same result for the even subsequence
To finish the proof of Theorem 1, we need to prove
The proof of (12) is similar to the provided proof in [
Proof of Theorem 2. Firstly, we prove that
According to Lemma 1, we have
provided k is large enough, where
From the definition of
Thus, (13) is immediate by using Borel Cantelli lemma.
To finish the proof of Theorem 2 we need to prove
Let
Using the well known probability inequality
(see [
where
The condition (6) implies that there exists
where
In this section we obtain similar results as Theorems 1 and 2 for the case of partial sums of a stationary Gaussian sequence. Let
Assume that
where
and
respectively, where
Theorem 3. Under the above statements of
i) If
ii) If
where
Example. Let
Define random variables
Then
and
In particular if
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.