p-variation and Chung's LIL of sub-bifractional Brownian motion and applications

Theorem 2.1

Let T > 0 , a > 0 , and v n = n a . Then, for any p ≥ 1 , we have, as n → ∞ ,

where [ x ] denotes the integer part of x > 0 , and Γ ( x ) ≔ ∫ 0 ∞ t x − 1 e − t d t for x > 0 , which is a Gamma function.

Corollary 2.2

Let T > 0 , a > 0 , and v n = n a . Then, for any p ≥ 1 , we have, as n → ∞ ,

Theorem 2.3

Let T > 0 , a > 0 , and v n = n a . Then, we have, as n → ∞ ,

Lemma 2.1

For any ε > 0 , there exists a positive constant c 2,1 = c 2,1 ( ε ) > 0 such that

for any T ≥ 0 , a > 0 , and x ≥ x 0 > 0 with some x 0 > 0 .

Proof

By (1.2) and the inequality for the normal distribution function Φ ( x ) : 1 − Φ ( x ) ≤ 1 x e − x 2 2 for all x > 0 , we obtain that

for any t ≥ 0 , h > 0 , and x ≥ x * > 0 with some x * > 0 . Therefore, by Lemmas 2.1 and 2.2 in [8] (when applied to σ 1 ( h ) = h H K and σ 2 ( ⋅ ) ≡ 0 ), we obtain (2.4) immediately.□

Lemma 2.2

Let X = { X ( t ) , t ∈ R } be a centered Gaussian process in R and let F ⊂ R be a closed set equipped with the canonical metric defined by

Then, there exists a positive constant c 2,3 such that for all u > 0 ,

where N d ( F , ε ) denotes the smallest number of open d-balls of radius ε needed to cover F and where D = sup { d ( s , t ) : s , t ∈ F } is the diameter of F.

Lemma 2.3

If X ( t ) and Y ( t ) are a.s. bounded, centered Gaussian processes on Λ such that E ( X 2 ( t ) ) = E ( Y 2 ( t ) ) for all t ∈ Λ , and

then for all real λ ,

and

Proof

It is Slepian’s inequality (see, p. 49 in [13]).□

Lemma 2.4

There exist positive constants c 2,5 and c 2,6 such that for u > 1 ,

and

Proof of Theorem 2.1

Without loss of generality, we suppose T v n is an integer. For integers n and j ≥ 1 , we take a n , j = ( j n ) β , where β > 0 is a constant. Define two Gaussian processes

and

Clearly, by (2.13), we have

It is important to note that for a fixed n , the Gaussian processes X n , j ( 1 ) ( t ) , j = 1 , 2 , … , are independent; moreover, for every j ≥ 1 , X n , j ( 1 ) ( t ) and X n , j ( 2 ) ( t ) are also independent. Since

In the following, we will show that the terms I 1 and I 3 almost surely converge to zero, I 2 and I 4 converge to zero, as n → ∞ , respectively.

First, we prove for a > 0 , T > 0 , and p ≥ 1 , as n → ∞ ,

In fact,

where

and where we use the fact

We have

For Y n , j , by Lemmas 2.2 and 2.4, elementary calculus can show that there exists n 0 such that for any n ≥ n 0 , for every 1 ≤ j ≤ T v n and for any t > 0 ,

Thus, for any ε > 0 , we obtain

Taking β > 0 large enough such that 2 β H K − 2 a − 2 a ( p − 1 ) H K > 0 , by the Borel-Cantelli lemma, we have

Combining (2.18) and (2.22), we prove that (2.17) holds.

Second, we prove for a > 0 , T > 0 , and p ≥ 1 , as n → ∞ ,

In fact, for any 1 ≤ j ≤ T v n and r > 2 , by (2.21) and Hölder’s inequality, we obtain

where we use by letting s 2 r = t , then

Hence,

Taking first β > 0 large enough and then taking r > 2 large enough such that a ( p − 1 ) H K − β H K + a + a + β r < 0 . Therefore, we obtain

Similar to (2.24), by using (2.4), for any p > 1 , we have that

Since, for any j ≥ 1 , t ≥ 0 , h > 0 , we have

Then, (2.4) remains true for X n , j ( 1 ) . Thus, similar to (2.26), we obtain for any 1 ≤ j ≤ T v n ,

Hence, combining (2.20) and (2.25), we have that (2.23) holds.

Third, we prove that for a > 0 , T > 0 , and p ≥ 1 , as n → ∞ ,

In fact, since for a fixed n , the processes { X n , j ( 1 ) ( t ) , t ≥ 0 } , j = 1 , 2 , … , T v n are independent and so are X n , j ( 1 ) j v n − X n , j ( 1 ) j − 1 v n p , j = 1 , 2 , … , T v n . For any ε > 0 and r > 1 , by Markov inequality and the moment inequality of partial sums of independent random variables, we have

Since

and