Keywords:
1. Introduction
Let
be a stable ordinator with exponent
with
, defined on a probability space
. Let
for
be a non-negative valued function and
,
. Define
,
where
,
and
For any value of t, the characteristic function of
is of the form
Limit theorems on the increments of stable subordinators have been investigated in various directions by many authors [1] - [6] . Among the above many results, we are interested in Fristedt [4] and Vasudeva and Divanji [3] whose results are the following limit theorems on the increments of stable subordinators.
Theorem 1 ( [4] )
Theorem 2 ( [3] ) Let
for
, be a non-decreasing function of
such that
(i)
for
,
(ii)
as
, and
(iii)
is non-increasing. Then
(1)
where
In this paper, our aim is to investigate Liminf behaviors of the increments of Y. We establish that, under certain conditions on
,
(2)
Throughout the paper c and k (integer), with or without suffix, stand for positive constants. i.o. means infinitely often. We shall define for each
the functions
and
.
2. Main Result
In this section, we reformulate the result obtained in Theorem 2 and establish our main result using
with
instead of
.
Theorem 3 Let
,
, be a non-decreasing function of
such that
(i)
for
,
(ii)
as
, and
(iii)
is non-increasing. Then
Remark 1 Let us mention some particular cases
1. For
we obtain Fristedt’s iterated logarithm laws (see Thorem 1).
2. If
we have Vasudeva and Divanji theorem (see Theorem 2).
3. If
under assumptions (i), (ii) and (iii) of Theorem 3 we also have
In order to prove Theorem 3, we need the following Lemma
Lemma 1 (see [3] or [7] ) Let
be a positive stable random variable with characteristic function
Then, as
where
Proof of Theorem 3. Firstly, we show that for any given
, as
(3)
Let
be a number such that
. Define a sequence
through
, for
Now we show that
From Mijhneer [8] , we have
(4)
But
Applying Lemma 1 with
one can find a
such that, for all
,
where
is some positive constant. Notice that
Hence
Let
and
. Note that 1k is non-decreasing and
as
. In turn one finds a
such that
Therefore, for all
, we have
(5)
Observe that
(6)
From the fact that
and from (4), (5), and (6) one gets
Observe that
is a sequence of mutually independent random variables (for,
) and by applying Borel-Cantelli lemma, we get
which establishes (3).
Now we complete the proof by showing that, for any
,
(7)
Define a subsequence
, such that
(8)
and the events
and
as
and
Note that
Further, define
and observe that
Hence in order to prove (7) it is enough to show that
(9)
We have
and
The fact that
is non-increasing as
implies that
Hence for a given
satisfying
there exists a
such that
Let
. Then, for
,
From lemma 1, we can find a
such that for all
,
where
is a positive constant.
Let
,
Then, for all
,
Since
then from (8) and for all
, we have
Observe that
and
Hence
Now we get
, which in turn establishes (9) by applying to the Borel-Cantelli lemma. The proof of Theorem 3 is complete.
3. Conclusion
In this paper, we developed some limit theorems on increments of stable subordinators. We reformulated the result obtained by Vasudeva and Divanji [3] , and established our result by using
.
Acknowledgments
Our thanks to the experts who have contributed towards development of our paper.