Study of the Convergence of the Increments of Gaussian Process ()
with
and
is an increasing sequence diverging to
.
Keywords:
1. Introduction
Let
be a standard Wiener process. Suppose that
is a nondecreasing function of t such that
with
is nonincreasing and
is an increasing sequence diverging to
. In [1] the following results are established.
i) If
, then
(1)
and
(2)
where
and
.
ii) If
, then
,
where
,
and
.
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
be an almost
surely continuous Gaussian process with
,
and
, where
is a function of
. Further we assume that
,
, is a nondecreasing continuous concave, regularly varying function at exponent
at
(e.g., if
is a standard Wiener pro- cess, then
).
Let
be a nondecreasing function of t with
. For large t, let us denote
![]()
where
and
is an increasing function of
.
We define two continuous parameter processes
and
by
![]()
and
.
2. 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.
Theorem 1. Let
be a nondecreasing function of t where
with the nonincreasing function
and let
be any increasing sequence diverging to
such that
, (3)
then
(4)
and
(5)
where
.
We note that
for large k in case of the Wiener process. It is interesting to compare (1) and (2) with (4) and (5) respectively.
Theorem 2. Let
be a nondecreasing function of
where
with the nonincreasing function
and let
be an increasing sequence diverging to
such that
, (6)
then
(7)
and
(8)
where
and
.
3. Proofs
In order to prove Theorems 1 and 2, we need to give the following lemmas.
Lemma 1. (See [3] ). For any small
there exists a positive
depending on
such that for all ![]()
,
where m is any large number and
is defined above.
Lemma 2. (See [4] ) Let
and
be centered Gaussian processes such that
for all
and
for all
. Then for any real number u
.
Proof of Theorem 1. Firstly, we prove that
(9)
For any
with the condition (3), we define an increasing sequence
by
.
For instance, let
for some
,
.
The condition (3) is satisfied, and for large k,
and
. By Lemma 1, we have, for any small
,
(10)
where k is large enough and
is a constant. By the definition of
,
.
We shall follow the similar proof process as in [5] . Set
.
Since
is an increasing sequence, the fact that
implies
. Consider the odd subse-
quence
of
and define the sequence of events
in the following form
.
By (10), for large k we have
![]()
where
is a constant. From the fact
, it is clear that
.
Since
, we get
. Also,
. (11)
Setting
![]()
and
,
we have
.
Let
,
and
.
Then, by (11) and the concavity of
we find that
![]()
This implies that
. Using Lemma 2, we obtain
![]()
where
. It follows from the Borel-Cantelli lemma that
![]()
Also, the same result for the even subsequence
of
is easily obtained. Therefore we have (9).
To finish the proof of Theorem 1, we need to prove
(12)
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
(13)
According to Lemma 1, we have
![]()
provided k is large enough, where
and
.
From the definition of
, it follows that
.
Thus, (13) is immediate by using Borel Cantelli lemma.
To finish the proof of Theorem 2 we need to prove
(14)
Let
.
Using the well known probability inequality
![]()
(see [6] ), one can find positive constants C and K such that, for all
,
![]()
where
and
. By the definition of
, we have
.
The condition (6) implies that there exists
such that
for all
. So, using Lemma 2 and the concavity of
, we obtain
,
where
and Borel-Cantelli lemma implies (14). If
, then Theorem 2 is immediate. Thus the proof of Theorem 2 is complete.
4. 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
be a stationary Gaussian sequence with
,
,
and
for all
We define
with
and set
.
Assume that
can be extended to a continuous function
with
which is nondecreasing and regularly varying with exponent
at
. Suppose that
is a nondecreasing sequence of positive integers such that
. For large n, we define
,
where
and
is an increasing function of n and also we define discrete time parameter processes by
![]()
and
,
respectively, where
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.
Theorem 3. Under the above statements of
,
and
, for
we have the following:
i) If
, then
![]()
ii) If
, then
![]()
where
.
Example. Let
be a fractional Brownian motion with the covariance function
. Then
.
Define random variables
,
![]()
and.
Then
![]()
and
is a stationary Gaussian sequence with
,
and
for all
. So we have Theorem 3.
In particular if
, then
is an i.i.d. Gaussian sequence with
and
.
5. 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.