Applied Mathematics
Vol.08 No.05(2017), Article ID:76331,8 pages
10.4236/am.2017.85053
On the Increments of Stable Subordinators
Abdelkader Bahram1,2, Bader Almohaimeed2
1Department of Mathematics, Djillali Liabes University, Sidi-Bel-Abbes, Algeria
2Department of Mathematics, Faculty of Science, Qassim University, Saudi Arabia
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: April 16, 2017; Accepted: May 20, 2017; Published: May 23, 2017
ABSTRACT
Let be a stable subordinator defined on a probability space and let for be a non-negative valued function. In this paper, it is shown that under varying conditions on , there exists a function such that
where , , and
Keywords:
Increments, Stable Subordinators, Iterated Logarithm Laws
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.
Cite this paper
Bahram, A. and Almohaimeed, B. (2017) On the Increments of Stable Subordinators. Applied Mathematics, 8, 663-670. https://doi.org/10.4236/am.2017.85053
References
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