Let be a stable subordinator defined on a probability space and let at for t>0 be a non-negative valued function. In this paper, it is shown that under varying conditions on at, there exists a function such that where , , and .
Let { X ( t ) , t ≥ 0 } be a stable ordinator with exponent α with 0 < α < 1 , defined on a probability space ( Ω , F , A ) . Let a t for t > 0 be a non-negative valued function and Y ( t ) = X ( t + a t ) − X ( t ) , t > 0 . Define
λ β ( t ) = θ α a t 1 α ( log t a t + β log log t + ( 1 − β ) log log a t ) α − 1 α ,
where 0 ≤ β ≤ 1 ,
θ α = ( B ( α ) ) 1 − α α and B ( α ) = ( 1 − α ) α α 1 − α ( cos ( π α 2 ) ) 1 α − 1 .
For any value of t, the characteristic function of X ( t ) is of the form
E ( e i u X ( t ) ) = exp ( − t | u | α ( 1 − u i | u | tan ( π α 2 ) ) ) , 0 < α < 1.
Limit theorems on the increments of stable subordinators have been investigated in various directions by many authors [
Theorem 1 ( [
lim inf t → ∞ θ α t − 1 α ( log log t ) 1 − α α X ( t ) = 1 almost surely ( a . s ) .
Theorem 2 ( [
(i) 0 < a t ≤ t for t > 0 ,
(ii) a t → ∞ as t → ∞ , and
(iii) a t / t is non-increasing. Then
lim inf t → ∞ ( X ( t + a t ) − X ( t ) ) ξ ( t ) = 1 a . s . , (1)
where ξ ( t ) = θ α a t 1 α ( log t a t + log log t ) α − 1 α .
In this paper, our aim is to investigate Liminf behaviors of the increments of Y. We establish that, under certain conditions on a t ,
lim inf t → ∞ Y ( t ) λ β ( t ) = 1 a . s . , where Y ( t ) = X ( t + a t ) − X ( t ) . (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 u ≥ 0 the functions log u = log ( max ( u , 1 ) ) and log log u = log log ( max ( u , 3 ) ) .
In this section, we reformulate the result obtained in Theorem 2 and establish our main result using λ β ( t ) with 0 ≤ β ≤ 1 instead of ξ ( t ) .
Theorem 3 Let a t , t > 0 , be a non-decreasing function of t such that
(i) 0 < a t ≤ t for t > 0 ,
(ii) a t → ∞ as t → ∞ , and
(iii) a t / t is non-increasing. Then
lim inf t → ∞ Y ( t ) λ β ( t ) = 1 a . s .
Remark 1 Let us mention some particular cases
1. For a t = t we obtain Fristedt’s iterated logarithm laws (see Thorem 1).
2. If β = 1 we have Vasudeva and Divanji theorem (see Theorem 2).
3. If β = 0 under assumptions (i), (ii) and (iii) of Theorem 3 we also have
lim inf t → ∞ Y ( t ) λ 0 ( t ) = 1 a . s .
In order to prove Theorem 3, we need the following Lemma
Lemma 1 (see [
E ( exp { i u X 1 } ) = exp { − | u | α ( 1 − i u | u | tan ( π α 2 ) ) } , 0 < α < 1.
Then, as x → 0 ,
P ( X 1 ≤ x ) ≃ x α 2 ( 1 − α ) 2 π α B ( α ) exp { − B ( α ) x α α − 1 }
where
B ( α ) = ( 1 − α ) α α − 1 α ( cos ( π α 2 ) ) 1 α − 1 .
Proof of Theorem 3. Firstly, we show that for any given ε > 0 , as t → ∞ ,
P ( Y ( t ) ≤ ( 1 + ε ) λ β ( t ) i . o ) = 1. (3)
Let u 1 be a number such that a u 1 > 1 . Define a sequence ( u k ) through u k + 1 = u k + a u k , for k = 1 , 2 , ⋯ . Now we show that
P ( Y ( u k ) ≤ ( 1 + ε ) λ β ( u k ) i . o ) = 1.
From Mijhneer [
P ( Y ( u k ) ≤ ( 1 + ε ) λ β ( u k ) ) = P ( X ( 1 ) ≤ ( 1 + ε ) λ β ( u k ) a u k 1 α ) . (4)
But
λ β ( u k ) a u k 1 α = θ α ( log u k a u k + β log log u k + ( 1 − β ) log log a u k ) α − 1 α .
Applying Lemma 1 with
x = ( 1 + ε ) θ α ( log u k a u k + β log log u k + ( 1 − β ) log log a u k ) α − 1 α ,
one can find a k 0 such that, for all k ≥ k 0 ,
P ( X ( 1 ) ≤ ( 1 + ε ) λ β ( u k ) a u k 1 α ) ≥ c 0 2 ( log ( u k ( log u k ) β ( log a u k ) 1 − β a u k ) ) 1 / 2 × exp { − ( 1 + ε ) α / ( α − 1 ) log ( u k ( log u k ) β ( log a u k ) 1 − β a u k ) } ,
where c 0 is some positive constant. Notice that
( 1 + ε ) α α − 1 = ( 1 − ε 1 ) < 1 for some ε 1 > 0.
Hence
P ( X ( 1 ) ≤ ( 1 + ε ) λ β ( u k ) a u k 1 α ) ≥ c 0 2 ( log ( u k ( log u k ) β ( log a u k ) 1 − β a u k ) ) 1 / 2 ( a u k u k ) × ( u k a u k ) ε 1 1 ( ( log u k ) β ( log a u k ) 1 − β ) ( 1 − ε 1 ) = c 0 2 ( log ( u k ( log u k ) β ( log a u k ) 1 − β a u k ) ) 1 / 2 ( u k + 1 − u k u k ) × ( u k a u k ) ε 1 1 ( ( log u k ) β ( log a u k ) 1 − β ) ( 1 − ε 1 ) .
Let 1 k = u k / a u k and m k = ( log u k ) β ( log a u k ) 1 − β . Note that 1k is non-decreasing and m k → ∞ as k → ∞ . In turn one finds a k 1 ≥ k 0 , such that
1 k ε 1 m k ε 1 ( l o g 1 k m k ) 1 / 2 ≥ 1, whenever k ≥ k 1 .
Therefore, for all k ≥ k 1 , we have
P ( X ( 1 ) ≤ ( 1 + ε ) λ β ( u k ) a u k 1 α ) ≥ c 0 ( u k + 1 − u k ) 2 u k ( log u k ) β ( log a u k ) 1 − β = c 0 ( u k + 1 − u k ) 2 u k ( log a u k log u k ) β 1 log a u k ≥ c 0 ( u k + 1 − u k ) 2 u k ( log a u k log u k ) 1 log a u k = c 0 ( u k + 1 − u k ) 2 u k log u k . (5)
Observe that
∫ k 1 ∞ d t t log t ≤ ∑ k = k 1 ∞ ( u k + 1 − u k ) u k log u k . (6)
From the fact that ∫ k 1 ∞ d t t log t = ∞ and from (4), (5), and (6) one gets
∑ k = 1 ∞ P ( Y ( u k ) ≤ ( 1 + ε ) λ β ( u k ) ) = ∞ .
Observe that ( Y ( u k ) ) is a sequence of mutually independent random variables (for, u k + 1 = u k + a u k ) and by applying Borel-Cantelli lemma, we get
P ( Y ( u k ) ≤ ( 1 + ε ) λ β ( u k ) i . o ) = 1
which establishes (3).
Now we complete the proof by showing that, for any ε ∈ ( 0,1 ) ,
P ( Y ( t ) ≤ ( 1 − ε ) λ β ( t k ) i . o ) = 0. (7)
Define a subsequence ( t k ) , such that
a t k = ( t k + 1 − t k ) / ( log t k ) ( 1 − β ) ( 1 + ε ) , k = 1 , 2 , ⋯ (8)
and the events A t and B k as
A t = { Y ( t ) ≤ ( 1 − ε ) λ β ( t ) }
and
B k = { inf t k ≤ t ≤ t k + 1 Y ( t ) ≤ ( 1 − ε ) λ β ( t k + 1 ) } , k = 1 , 2 , ⋯ .
Note that
( A t i . o . , t → ∞ ) ⊂ ( B k i . o . , k → ∞ ) .
Further, define
C k = { X ( t k + a t k ) − X ( t k + 1 ) ≤ ( 1 − ε ) λ β ( t k + 1 ) }
and observe that
( B k i . o . , k → ∞ ) ⊂ ( C k i . o . , k → ∞ ) .
Hence in order to prove (7) it is enough to show that
P ( C k i . o . ) = 0. (9)
We have
P ( X ( t k + a t k ) − X ( t k + 1 ) ≤ ( 1 − ε ) λ β ( t k + 1 ) ) = P ( X ( 1 ) ≤ ( 1 − ε ) λ β ( t k + 1 ) ( a t k + t k − t k + 1 ) 1 / α )
and
( 1 − ε ) λ β ( t k + 1 ) ( a t k + t k − t k + 1 ) 1 / α ≃ ( 1 − ε ) θ α ( a t k + 1 a t k ) 1 / α ( log ( t k + 1 ( log t k + 1 ) β ( log a t k ) 1 − β a t k ) ) ( α − 1 ) / α .
The fact that a t / t is non-increasing as t → ∞ implies that
a t k + 1 t k + 1 ≤ a t k t k or a t k + 1 a t k ≤ t k + 1 t k .
Hence for a given ε 1 > 0 satisfying ( 1 − ε ) ( 1 + ε 1 ) 1 / α < 1 , there exists a k 2 such that
a t k + 1 / a t k ≤ ( 1 + ε 1 ) , for all k ≥ k 2 .
Let ( 1 − ε ) ) ( 1 + ε 1 ) 1 / α = ( 1 − ε 2 ) . Then, for k ≥ k 2 ,
P ( C k ) ≤ P ( X ( 1 ) ≤ ( 1 − ε 2 ) θ α ( log t k + 1 a t k + 1 ( log t k + 1 ) β ( log a t k + 1 ) 1 − β ) ( α − 1 ) / α ) .
From lemma 1, we can find a k 3 ( ≥ k 2 ) such that for all k ≥ k 3 ,
P ( C k ) ≤ c 1 ( log t k + 1 a t k + 1 ( log t k + 1 ) β ( log a t k + 1 ) 1 − β ) − 1 2 × exp { ( 1 − ε 2 ) α / ( α − 1 ) ( log t k + 1 a t k ( log t k + 1 ) β ( log a t k + 1 ) 1 − β ) } ,
where c 1 is a positive constant.
Let ( 1 − ε 2 ) α / ( α − 1 ) = ( 1 + ε 3 ) , ε 3 > 0. Then, for all k ≥ k 3 ,
P ( C k ) ≤ c 1 ( log t k + 1 a t k ( log t k + 1 ) β ( log a t k + 1 ) 1 − β ) − 1 / 2 ( a t k + 1 t k ) ( 1 + ε 3 ) ( ( log t k + 1 ) β ( log a t k + 1 ) 1 − β ) − ( 1 + ε 3 ) .
Since
( a t k + 1 / t k + 1 ) ( 1 + ε 3 ) ≤ ( a t k / t k ) ( 1 + ε 3 ) ≤ a t k / t k ,
then from (8) and for all k ≥ k 3 , we have
P ( C k ) ≤ c 1 ( l o g t k a t k ( l o g t k ) β ( l o g a t k ) 1 − β ) − 1 / 2 ( a t k t k ) ( ( l o g t k ) β ( l o g a t k ) 1 − β ) − ( 1 + ε 3 ) .
P ( C k ) ≤ c 1 ( log t k a t k ( log t k ) β ( log a t k ) 1 − β ) − 1 / 2 ( t k + 1 − t k t k ) × 1 ( log t k ) 1 + ε 3 1 ( log a t k + 1 ) ( 1 − β ) ( 1 + ε 3 ) ≤ c 1 ( t k + 1 − t k t k ) 1 ( log t k ) ( 1 + ε 3 ) .
Observe that
∫ k 4 ∞ d t t ( log t ) ( 1 + ε 3 ) ≥ ∑ k = k 4 ∞ ( t k + 1 − t k ) t k + 1 ( log t k + 1 ) ( 1 + ε 3 ) ,
and
( t k + 1 − t k ) t k + 1 ( log t k + 1 ) ( 1 + ε 3 ) ≃ ( t k + 1 − t k ) t k ( log t k ) ( 1 + ε 3 ) .
Hence
∑ k = k 4 ∞ ( t k + 1 − t k ) t k ( log t k ) ( 1 + ε 3 ) < ∞ .
Now we get ∑ k = k 4 ∞ P ( C k ) < ∞ , 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 [
Our thanks to the experts who have contributed towards development of our 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