where is some positive constant. Notice that
Let and . Note that 1k is non-decreasing and as . In turn one finds a such that
Therefore, for all , we have
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 ,
Define a subsequence , such that
and the events and as
and observe that
Hence in order to prove (7) it is enough to show that
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 ,
then from (8) and for all , we have
Now we get , which in turn establishes (9) by applying to the Borel-Cantelli lemma. The proof of Theorem 3 is complete.
In this paper, we developed some limit theorems on increments of stable subordinators. We reformulated the result obtained by Vasudeva and Divanji  , and established our result by using .
Our thanks to the experts who have contributed towards development of our paper.
Conflicts of Interest
The authors declare no conflicts of interest.
Cite this paper
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