Large deviations for subordinated fractional Brownian motion and applications

Accepted Manuscript Large deviations for subordinated fractional Brownian motion and applications

Weigang Wang, Zhenlong Chen

PII: DOI: Reference:

S0022-247X(17)30934-4 https://doi.org/10.1016/j.jmaa.2017.10.035 YJMAA 21753

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Journal of Mathematical Analysis and Applications

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15 August 2017

Please cite this article in press as: W. Wang, Z. Chen, Large deviations for subordinated fractional Brownian motion and applications, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2017.10.035

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LARGE DEVIATIONS FOR SUBORDINATED FRACTIONAL BROWNIAN MOTION AND APPLICATIONS WEIGANG WANG*, ZHENLONG CHEN Abstract. Let W H = {W H (t), t ∈ R} be a real valued fractional Brownian motion with Hurst index H ∈ (0, 1) and let T = {Tt , t ≥ 0} be an inverse α-stable subordinator independent of W H . The inverse stable subordintor fractional Brownian motion Z H = {Z H (t), t ≥ 0} is defined by Z H (t) = W H (Tt ), which may arise as scaling limit of CTRW or random walk in a random environment. In this paper we establish large deviation results for the process Z H and its supremum process. And we also give asymptotic properties of the tail probability of the supremum process.

1. Introduction and statement of results Fractional Brownian motion (fBm) is a centered Gaussian process W H = {W H (t), t ∈ R} with W (0) = 0 and covariance function    1  2H E W H (s)W H (t) = |s| + |t|2H − |s − t|2H , 2 H

where H ∈ (0, 1) is a constant. It is known that W H is self-similar with index H (i.e., for all constants c > 0, the processes {W H (ct), t ∈ R} and {cH W H (t), t ∈ R} have the same finite-dimensional distributions) and has stationary increments. When H = 1/2, W H is a two-sided Brownian motion, which will be written as W . FBm is an improtant example of self-similar processes which arise naturally in limit theorems of random walks and other stochastic processes, and it has been applied to model various phenomena in a wide range of scientific ares including telecommunications, turbulence, image processing and finance. In this paper, we consider a class of iterated self-similar processes which is related to continuoustime random walks considered in [3, 13]. Let X = {Xt , t ≥ 0} be a real-valued α-stable subordinator, where 0 < α < 1. We assume that X is independent of W H , T = {Tt , t ≥ 0} be the inverse process of X, i.e Tt = inf{τ ; Xτ > t}. Let Z H = {Z H (t), t ≥ 0} be the real-valued stochastic process defined by Z H (t) = W H (Tt ) for all t ≥ 0. This iterated process will be called subordinated fractional Brownian motion. When H = 1/2, Z H (t) will be written as Z, which is also called fractional kinetic process [2, 12]. References [8, 11] proved the large deviations for subordinated fractional Brownian motion under the condition of 2H(1 − α) < 1. In this paper, we will establish large deviations for the subordinated fractional Brownian motion Z H and supremum sup0≤s≤t Z H (s) without the exact condition. And we also give asymptotic properties of the tail probability of the supremum process in Section 4. The following are our main results. Theorem 1.1. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion. Then for every β > 0 such that 0 < β( 12 + H − αH) < 1, every function a(t) with limt→∞ a(t) = ∞, and every Borel set D ⊆ R,   |Z H (t)|β 1 lim sup log P ∈ D ≤ − inf Λ∗1 (x) (1) 1 1 1 ¯ x∈D t→∞ a(t)1/[1−β( 2 +H−αH)] a(t)β( 2 +H−αH)/[1−β( 2 +H−αH)] tαβH 2000 Mathematics Subject Classification. 60G20. Key words and phrases. Large deviation; Tail probability; Inverse of α-stable subordinator; Fractional Brownian motion. *Corresponding author. Email: wwgys [email protected]. 1

2

WEIGANG WANG, ZHENLONG CHEN

and lim inf t→∞

1 1

a(t)1/[1−β( 2 +H−αH)]

 log P

|Z H (t)|β 1

1

a(t)β( 2 +H−αH)/[1−β( 2 +H−αH)] tαβH

 ∈D

≥ − inf o Λ∗1 (x), (2) x∈D

o

¯ and D denote respectively the closure and interior of D and where D ⎧ 2(1−α)H 2 2αH ⎨( 1 + H − αH)α 1+2H−2αH H − 1+2H−2αH x β(1+2H−2αH) , Λ∗1 (x) = 2 ⎩∞,

if x > 0, if x ≤ 0.

(3)

For the supremum of Z H (t), we have the following theorem. Theorem 1.2. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion. For every function a(t) with limt→∞ a(t) = ∞, we have the following statements hold (i) If H ∈ (0, 1/2), then

 x γ˜1  x α˜ 1  P sup Z H (s) > x ∼ c˜1 αH as x → ∞, (4) exp −β˜1 αH t t 0≤s≤t (ii) If H = 1/2, then

  1 γ ˜ α ˜ H P αH sup Z (s) > x ∼ c˜2 (a(t)x) 2 exp −β˜2 (a(t)x) 2 t a(t) 0≤s≤t (iii) If H > 1/2, then

  1 γ ˜ α ˜ P αH sup Z H (s) > x ∼ c˜3 (a(t)x) 3 exp −β˜3 (a(t)x) 3 t a(t) 0≤s≤t

as x → ∞,

(5)

as x → ∞,

(6)

in the above ˜2 = α ˜3 = α ˜1 = α

2 , 1 + 2H(1 − α)

β˜1 = β˜2 = β˜3 2H(1−α)   1+2H(1−α)   2H(1−α) −1 1 − = (1 − α)αα(1−α) 2 1+2H(1−α) (2H(1 − α)) 1+2H(1−α) + (2H(1 − α)) 1+2H(1−α) , (1/H) − 3 , 1 + 2H(1 − α) −1 γ˜2 = γ˜3 = 1 + 2H(1 − α)

1/2

(3/2)−α−2H(1−α) 1+2H(1−α) H(1 − α) 1 α(1−α) Aα −1/(2H) 2 α , c˜1 = H 1 + 2H(1 − α) H

1/[2+4H(1−α)] 1 α(1−α) α , c˜2 = 2Aα H

1/[2+4H(1−α)] 1 α(1−α) α c˜3 = Aα H −1/2  . and Aα = 2π(1 − α)αα/(2(1−α)) γ˜1 =

(7)

2. Moment estimates and Lemmas A real-valued L´evy process X = {Xt , t ≥ 0} is called stable subordinator of index α ∈ (0, 1) if its Laplace exponent is given by  ∞   c 1 − e−λx 1+α dx, Φ(λ) = (8) x 0 where c > 0 is a constant. The inverse α-stable subordinator Tt is defined as Tt = inf{τ ; Xτ > t}.

(9)

LARGE DEVIATIONS FOR SUBORDINATED F-BM AND APPLICATIONS

3

For more information on subordinators and more general L´evy process, we refer to [4, 14]. The following two lemmas are from [8]. Lemma 2.1. Let T = {Tt , t ≥ 0} be an inverse α-stable subordinator with 0 < α < 1. Then for all 0 < a ≤ b < ∞ and all integers n ≥ 1,

1−α

1−α b−a b−a n!(b − a)nα n!(b − a)nα n ≤ E[|Tb − Ta | ] ≤ . (10) b Γ((n − 1)α + 2)Γ(α) b Γ(nα + 1) In the case a = 0, follow equality holds, E[|Tb |n ] =

bαn Γ(n + 1) . Γ(αn + 1)

(11)

Lemma 2.2. Let W H = {W H (t), t ∈ R} be a fractional Brownian motion of index H in R and Tt be an inverse α-stable subordinator 0 < α < 1. Assume that Tt is independent of W H . Then for all 0 < a < b < ∞ and all positive integers n,  C1 (n)(b − a)nα ≤ E |Z H (b) − Z H (a)|n/H ≤ C2 (n)(b − a)nα , (12) where n! C1 (n) = √ 2n/(2H) Γ π



n 1 + 2H 2

and n! C2 (n) = √ 2n/(2H) Γ π





b−a b

n 1 + 2H 2



Moreover, when a = 0 we have the equality H

E(|Z (b)|

n/H

Γ 1 ) = √ 2n/2H π



1−α

b−a b

1 Γ((n − 1)α + 2)Γ(α)

1−α

1 . Γ(nα + 1)

 + 12 Γ(n + 1) αn b . Γ(αn + 1)

n 2H

Lemma 2.3. For any η > 0, x∗ > 0, there exist C > 0 and h0 > 0 such that

 2 H H Hα ≤ (CT /h) exp −γx 1+2H−2Hα /(1 + η) sup |Z (t + s) − Z (t)| > xh P sup 0≤t≤T −h 0≤s≤h

(13)

(14)

(15)

(16)

for every x ≥ x∗ and T ≥ h > h0 , where γ is a constant depending only on H and α. Proof. From Lemma 3.4 in [8], we know that there exists a finite constant γ > 0, depending only on H, α, such that for all t ≥ 0, h > 0 and x > 0,    2 (17) P |Z H (t + h) − Z H (t)| > xhHα ≤ exp −γx 1+2H−2Hα . 

Then (16) follows from Lemma 2.4 in [7].

Lemma 2.4. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion wtih 2H(1 − α) < 1. Then for every function a(t), with limt→∞ a(t) = ∞, every θ ∈ R,

θ 1 H E exp a(t)Z (t) = Λ2 (θ), (18) lim t→∞ a(t)2/(1−2H+2Hα) tHα where 1 − 2H + 2Hα Λ2 (θ) = 2



H αα(1−α)

2H(1−α)/(1−2H+2Hα)

· θ2/(1−2H+2Hα) .

(19)

The proof of Lemma 2.4 relies on explicit calculation of the moments of Z H (t) and the following theorem in Valiron ([16], p.44).

4

WEIGANG WANG, ZHENLONG CHEN

∞ Lemma 2.5. Let f (z) = p=0 cp z p be an entire function such that cp = 0 for infinitely many p’s. For any r > 0, let M (r) = sup|z|=r |f (z)|. Then a necessary and sufficient condition for log M (r) =B rρ is that, for all values of ε and all sufficiently large integers p, we have 1 ρ/p pc ≤ B + ε, ρe p and there exists a sequence of integers pn , such that pn+1 = 1, lim n→∞ pn for which 1 n lim pn cρ/p = B. pn n→∞ ρe lim

(20)

r→∞

(21)

(22)

(23)

Proof of Lemma 2.4. From the moment generating function of a Gaussian random variable, for all θ ∈ R, we have

2

θ 2 θ H 2H a (t)T1 . (24) E exp Hα a(t)W (Tt ) = E exp t 2 We consider the Taylor series ∞    ET12Hn n M1 (r) := E exp rT12H = r . n! n=0

(25)

By Jensen’s inequality, for any constant γ ≥ 1 and nonnegative random variable X,  γ/γ γ/(γ+1)  ≤ E(X γ ) ≤ EX γ+1 . EX γ

(26)

Here γ denotes the largest integer ≤ γ. It follows from (26) with X = T1 , and γ = 2Hn, Lemma 2.1 and Stirling’s formula that 2Hn

 n T1 ∼ C1 n−1/2 (2H)2H(1−α) α−2Hα e1+2Hα−2H nn[2H(1−α)−1] , (27) E n! where C1 is the constant depending on H and α only. In the above xn ∼ yn means that limn→∞ xn /yn = 1. From (27), we know that M1 (r) is an analytic function on R if and only if H(1 − α) < 1/2. In the latter case, we choose ρ1 = [1 − 2H(1 − α)]−1 and we have that

ρ /n

2H(1−α)ρ1 ET12Hn 1 2H 1 1 n = . (28) lim n→∞ ρ1 e n! ρ1 αα/(1−α) Hence, Lemma 2.5 implies that log M1 (r) 1 = lim r→∞ r ρ1 ρ1 From (24), (25), (29) we get that 1



2H(1−α)ρ1

2H

.

αα/(1−α)

θ

H

(29)

E exp Hα a(t)W (Tt ) t

2 ρ 1 2 θ θ 2 1 a (t) · log M1 = lim 2 n→∞ [a2 (t) θ ]ρ1 2 2 2

2H(1−α)ρ1 2 ρ1 2H θ 1 = α/(1−α) ρ1 α 2 = Λ2 (θ). lim

t→∞

a(t)2/(1−2H+2Hα)

(30)



LARGE DEVIATIONS FOR SUBORDINATED F-BM AND APPLICATIONS

5

Lemma 2.6. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion. Then for any β( 12 + H − αH) < 1, every function a(t), with limt→∞ a(t) = ∞, every θ ∈ R,

1 a(t) H β lim = Λ1 (θ), log E exp θ αβH |Z (t)| (31) 1 t→∞ a(t)[1−β( 2 +H−αH)]−1 t where

⎧ β( 1 +H−αH) βH(1−α) −αβH 2 1 ⎪ 1−β( 1 +H−αH) 1−β( 1 +H−αH) 1−β( 1 +H−αH) ⎪ 2 2 2 ⎪ [1 − β( + H − αH)]α β H ⎨ 2 Λ1 (θ) = −1 1 ⎪ · θ[1−β( 2 +H−αH)] ⎪ ⎪ ⎩ 0

if θ > 0,

(32)

if θ ≤ 0.

Proof. Similar to the proof of Lemma 2.4, from Lemma 2.2 , the independence between Tt and W H (t), we derive that ∞   β  E|Z H (t)/tαH |nβ n r = M2 (r) := E exp r Z H (t)/tαH n! n=0

=

=

∞  E|W H (1)|nβ ETtbβH n r n! · tαβHn n=0    ∞ √1 2nβ/2 Γ nβ + 1 ET1nβH  2 2 π

n!

n=0

(33) rn .

It follows from (27), (33) that 1 1 1 1 E|Z H (t)/tαH |nβ ∼ C2 n− 2 [α−αβH β β( 2 +H−αH) H βH(1−α) e1−β( 2 +H−αH) ]n n[β( 2 +H−αH)−1]n , (34) n!

where C2 is the constant depending only on H and α. From (34), we know that M2 (r) is an analytic function on R if and only if β( 12 + H − αH) < 1. In the latter case, when θ > 0, we choose ρ2 = [1 − β( 12 + H − αH)]−1 then 1 n n→∞ ρ2 e lim



E|Z n (t)/tαH |nβ n!

ρ2 /n =

1 −αβHρ2 β( 1 +H−αH)ρ2 βH(1−α)ρ2 α β 2 H . ρ2

(35)

Hence from Lemma 2.5 we can get lim

r→∞

1 log M2 (r) 1 = α−αβHρ2 β β( 2 +H−αH)ρ2 . ρ 2 r ρ2

(36)

It fellow from (33),(36) that lim



1 1

−1

a(t) log E exp θ αβH |Z H (t)|β t



a(t)[1−β( 2 +H−αH)] 1 ρ2 = lim ρ log M2 (θa(t)) · θ t→∞ (θa(t)) 2 1 1 = α−αβHρ2 β β( 2 +H−αH)ρ2 H βH(1−α)ρ2 · θρ2 ρ2

t→∞

(37)

β( 1 +H−αH)

βH(1−α) −αβH 2 −1 1 1 1 1 1 = [1 − β( + H − αH)]α 1−β( 2 +H−αH) β 1−β( 2 +H−αH) H 1−β( 2 +H−αH) · θ[1−β( 2 +H−αH)] . 2

When θ < 0, for the one hand,



1 1

a(t)[1−β( 2 +H−αH)]

−1

a(t) log E exp θ αβH |Z H (t)|β t

≤ 0.

(38)

6

WEIGANG WANG, ZHENLONG CHEN

On the other hand, by the concavity of log(x),



a(t) H β log E exp θ |Z (t)| 1 t→∞ a(t)[1−β( 2 +H−αH)]−1 tαβH

a(t)1−ρ2 ≥ lim E θ αβH |Z H (t)|β t→∞ t   H = θE|Z (1)|β lim a(t)1−ρ2 = 0. 1

lim

(39)

t→∞

Hence lim

t→∞



1 1

a(t)[1−β( 2 +H−αH)]

−1

a(t) log E exp θ αβH |Z H (t)|β t

= 0.

(40) 

Combining (37) and (40) finishes the proof of Lemma 2.6. 3. Large deviations results

Theorem 3.1. Let Z H = {Z H (t), t ≥ 0} be a real-valued subordinated fractional Brownian motion wtih 2H(1 − α) < 1. Then for every function a(t), with limt→∞ a(t) = ∞, and every Borel set D ⊆ R,   Z H (t) 1 lim sup log P ∈ D ≤ − inf Λ∗2 (x) (41) 2/(1−2H+2Hα) ¯ a(t)2/(1−2H+2Hα)−1 tHα x∈D t→∞ a(t) and lim inf t→∞



1 a(t)2/(1−2H+2Hα)

log P

Z H (t) a(t)2/(1−2H+2Hα)−1 tHα

 ∈D

≥ − inf o Λ∗2 (x), x∈D

(42)

where 2H(1−α) 2Hα 2 1 Λ∗2 (x) = ( + H − αH)α 1+2H−2Hα H − 1+2H−2Hα x 1+2H−2Hα . 2

(43)

Proof. Note that the function Λ2 (θ) =

1 − 2H + 2Hα 2



H

2H(1−α)/(1−2H+2Hα)

αα(1−α)

· θ2/(1−2H+2Hα)

in Theorem 2.4 on R. It follows from the G¨artner-Ellis theorem ([6], Theorem 2.3.6)  is differentiable Z H (t) 2/(1−2H+2Hα) satisfies a large deviation principle with the that the pair a(t)2/(1−2H+2Hα)−1 tHα , a(t) good rate function Λ∗2 (x) = sup(θx − Λ1 (θ)) θ∈R

2H(1−α) 2Hα 2 1 = ( + H − αH)α 1+2H−2Hα H − 1+2H−2Hα x 1+2H−2Hα , 2

which is the Frenchel-Legendre transform of Λ1 . (44) fellows from proof of Theorem 3.1.

2 1+2H−2Hα

(44)

> 1. That finishes the 

When a(t) = tHα , Theorem 3.1 gives Theorem 3.1 in [8]. The condition 2H(1 − α) < 1 is to H ensure that EeθZ (t) < ∞ and Theorem 3.1 in [8] should also have this condition. An interesting question is to establish get the large deviation for subordinated fractional Brownian motion when H EeθZ (t) = ∞? Theorem 1.1 answers this question. Proof of Theorem 1.1. Similar to the proof of Theorem 3.1. From Lemma 2.6, by the G¨artner-Ellis theorem, the pair

|Z H (t)|β 1/[1−β( 12 +H−αH)] , a(t) 1 1 a(t)β( 2 +H−αH)/[1−β( 2 +H−αH)] tαβH

LARGE DEVIATIONS FOR SUBORDINATED F-BM AND APPLICATIONS

7

satisfies a large deviation principle with rate function Λ∗1 (x) = sup(θx − Λ2 (θ)) θ∈R ⎧ 2(1−α)H 2 2αH ⎨( 1 + H − αH)α 1+2H−2αH H − 1+2H−2αH x β(1+2H−2αH) , = 2 ⎩∞,

if x > 0,

(45)

if x ≤ 0. 

4. Tail Probability of Supremum We study the asymptotic properties of

H P sup Z (s) > x 0≤s≤t

as x → ∞.

(46)

The main result of this section is Theorem 1.2. Recall from [1] that a random variable T has asymptotically Weibullian tail distribution if P(T > t) = Ctr exp(−βtα )(1 + o(1))

(47)

as t → ∞, where α, β, C > 0, γ ∈ R. Write T ∈ W(α, β, γ, C) if T satisfies (47). Lemma 4.1. Let {Tt , t ≥ 0} be the inverse of α-stable subordinator process. Then for every t > 0, we have

1 −1 , (1 − α)(αt−1 )α/(1−α) , , Aα tα/2(1−α) , Tt ∈ W (48) 1−α 2(1 − α)  −1/2 . where Aα = 2π(1 − α)αα/(2(1−α)) Proof. Let {Xt , t ≥ 0} be the α-stable subordinator specified by (8). From Lemma 1 in [9], we know that



α

t t P (Tt > x) = P > x = P X1 < 1/α X1 x 

α/2(1−α)

−α/(1−α)  t t (49) ∼ Aα exp −(1 − α)αα/(1−α) x1/α x1/α   = Aα tα/2(1−α) x−1/2(1−α) exp −(1 − α)(αt−1 )α/(1−α) x1/(1−α) as x → ∞, where Aα is defined as (48). That is

1 −1 −1 α/(1−α) α/2(1−α) , (1 − α)(αt ) , Aα t Tt ∈ W , . 1−α 2(1 − α)  The following lemmas are Lemma 2.1 and Lemma 4.2 in [1]. Lemma 4.2. Let X ∈ W(α1 , β1 , γ1 , C1 ), Y ∈ W(α2 , β2 , γ2 , C2 ) be independent non-negative random variables. Then the product XY ∈ W(α, β, γ, C) with α 1 α2 α= , α1 + α2 

α1 /(α1 +α2 )  α /(α +α ) α2 α1 2 1 2 α2 /(α1 +α2 ) α1 /(α1 +α2 ) β2 + , β = β1 α2 α1 α1 α2 + 2α1 γ2 + 2α2 γ1 , 2(α1 + α2 ) √ 1 (α1 β1 )(α2 −2γ1 +2γ2 )/2(α1 +α2 ) (α2 β2 )(α1 −2γ2 +2γ1 )/2(α1 +α2 ) . C = 2πC1 C2 √ α1 + α2 γ=

8

WEIGANG WANG, ZHENLONG CHEN

Lemma 4.3. Let B H (t) be an fBm with H ∈ (0, 1), we have the following statements (i) If H ∈ (0, 1/2), then

1 1 1 H −(H+1)/(2H) − 3, √ 2 . sup B (t) ∈ W 2, , 2 H H π t∈[0,1] (ii) If H = 1/2, then

(iii) If H ∈ (1/2, 1), then

(50)



2 1 . sup B H (t) ∈ W 2, , −1, √ 2 2π t∈[0,1]

(51)



1 1 . sup B (t) ∈ W 2, , −1, √ 2 2π t∈[0,1]

(52)

H

Proof of Theorem 1.2. We only prove the theorem when H ∈ (0, 1/2), the other cases can be proved similarly. From Lemma 4.1 and (47), we can get that

1 −1 (53) , (1 − α)αα(1−α) , , Aα . T1H ∈ W H(1 − α) 2H(1 − α) Hence by Lemma 4.2, Lemma 4.3 and (53), we know that ˜ 1 , β˜1 , γ˜1 , c˜1 ), T1H sup B H (s) ∈ W(α

(54)

s∈[0,1]

where α ˜ 1 , β˜1 , γ˜1 , c˜1 are defined as (7). From the independence between {W H (t), t ≥ 0} and {Tt , t ≥ 0}, the self-similarity of fBm, the continuous of Tt and (54) we have





sup W H (s) > x P sup Z H (s) > x = P sup W H (Ts ) > x = P 0≤s≤t

0≤s≤t



TtH

=P

H

∼ c˜1

0≤s≤1

x tαH 1

γ˜1



exp −β˜1



αH

=P t

sup B (s) > x

0≤s≤1

= P T1H sup B H (s) >

0≤s≤Tt



x



tαH  x α˜ 1

tαH

T1H

H



sup B (s) > x

0≤s≤1

(55)

, as x → ∞. 

By Theorem 1.2, we derive the following corollaries. Corollary 4.4. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion. For every function a(t) with limt→∞ a(t) = ∞, we have the following statements hold (i)If H ∈ (0, 1/2), then

  1 γ ˜ α ˜ H as t → ∞. (56) sup Z (s) > x ∼ c˜1 (a(t)x) 1 exp −β˜1 (a(t)x) 1 P αH t a(t) 0≤s≤t (ii)If H = 1/2, then

  1 γ ˜ α ˜ H sup Z (s) > x ∼ c˜2 (a(t)x) 2 exp −β˜2 (a(t)x) 2 P αH t a(t) 0≤s≤t (iii)If H > 1/2, then

  1 γ ˜ α ˜ H sup Z (s) > x ∼ c˜3 (a(t)x) 3 exp −β˜3 (a(t)x) 3 P αH t a(t) 0≤s≤t where α ˜ i , β˜i , γ˜i , c˜i , i = 1, 2, 3 are defined as Theorem 1.2.

as t → ∞.

(57)

as t → ∞.

(58)

LARGE DEVIATIONS FOR SUBORDINATED F-BM AND APPLICATIONS

Proof. For every a(t) with limt→∞ a(t) = ∞, and every x > 0, from (54), (55) we have



1 H T H sup Z (s) > x = P T1 sup B (s) > a(t)x P αH t a(t) 0≤s≤t 0≤s≤1  γ ˜ α ˜ ∼ c˜1 (a(t)x) 1 exp −β˜1 (a(t)x) 1 , as t → ∞.

9

(59) 

Corollary 4.5. For every H ∈ (0, 1), there exist two constants A1 , A2 , such that for every t ≥ 0, x > 0,



x2/(1+2H−2Hα) H (60) P sup |Z (s)| > x ≤ A1 exp −A2 2Hα/(1+2H−2Hα) . t 0≤s≤t Proof. For (54), there exist two constants A1 > 0, A2 = β˜1 /2 such that for every x > 0,     H H P T1 sup B (s) > x ≤ A1 exp −A2 xα˜ 1 .

(61)

s∈[0,1]



By (55) we can complete the proof.

This corollary recovers Theorem 3.5 in [8] and Theorem 4.5 in [11] in the case of a = 0 and b = t. Form Corollary 4.4, we can get the next corollary. Corollary 4.6. For every a(t) with limt→∞ a(t) = ∞ and x ≥ 0, we have

1 1 H ˜ α˜ , sup |Z (s)| > x = −βx log P αH lim ˜ t→∞ a(t)α t a(t) 0≤s≤t

(62)

where α ˜=α ˜ 1 and β˜ = β˜1 are defined as (7). Theorem 4.7. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion. Then for every function a(t), with limt→∞ a(t) = ∞, and every Borel set D ⊆ R,

1 1 H sup log P Z (s) ∈ D ≤ − inf I(x) (63) lim sup α ˜ ¯ tαH a(t) 0≤s≤t x∈D t→∞ a(t) and lim inf t→∞

1 log P a(t)α˜



1 sup Z H (s) ∈ D tαH a(t) 0≤s≤t

≥ − inf o I(x), x∈D

(64)

˜ α˜ , α where I(x) = β|x| ˜=α ˜ 1 and β˜ = β˜1 are defined as (7). Proof. We follow the proof of Theorem 2.2.3 in [6] to prove (63) and (64) respectively. (i) Since the distribution of {Z H (s), s ≥ 0} is symmetric, (62) implies that for every x > 0,

1 1 H ˜ α˜ sup log P Z (s) > x = −βx (65) lim ˜ t→∞ a(t)α tαH a(t) 0≤s≤t and



1 1 H ˜ α˜ . lim sup Z (s) < −x = −β|x| log P αH (66) ˜ t→∞ a(t)α t a(t) 0≤s≤t ¯ (63) is obvious. For 0 ∈ ¯ let a = inf D, ¯ b = sup D, ¯ we might For every Borel set D ⊆ R, if 0 ∈ D, / D, let 0 < a ≤ b as well, and we can get the same result at the case of a ≤ b < 0 similarly. For (65),

1 1 H lim sup sup log P Z (s) ∈ D α ˜ tαH a(t) 0≤s≤t t→∞ a(t)

1 1 H (67) sup Z (s) ≥ a = −I(a) log P αH ≤ lim sup α ˜ t a(t) 0≤s≤t t→∞ a(t) = − inf I(x). ¯ x∈D

10

WEIGANG WANG, ZHENLONG CHEN

(ii) For every x ∈ Do . When x > 0, ∃δ > 0, such that (x, x + δ) ⊂ D. Let

1 H sup Z (t) > x . A(t, x) = P αH t a(t) 0≤s≤t We have that

1 H sup Z (s) ∈ D tαH a(t) 0≤s≤t

1 1 H sup Z (s) ∈ (x, x + δ) log P αH ≥ a(t)α˜ t a(t) 0≤s≤t 1 log [A(t, x) − A(t, x + δ)] = a(t)α˜ 

1 1 A(t, x + δ) . = log A(t, x) − log 1 − a(t)α˜ a(t)α˜ A(t, x)

1 log P a(t)α˜



(68)

For (59), we know that 

1 A(t, x + δ) 1 A(t, x + δ) lim = lim − log 1 − ˜ t→∞ a(t)α t→∞ A(t, x) a(t)α˜ A(t, x)   (x + δ)γ˜ 1 α ˜ α ˜ α ˜ ˜ = lim − (x + δ) } exp{− βa(t) − x t→∞ a(t)α˜ xγ˜ = 0. Hence by (65), (68) and (69) we get that

1 1 1 H lim inf sup Z (s) ∈ D ≥ lim inf log P αH log A(t, x) = −I(x). ˜ ˜ t→∞ a(t)α t→∞ a(t)α t a(t) 0≤s≤t When x = 0, for small enough δ > 0, (δ/2, δ) ⊂ D, we have

1 1 H sup log P Z (s) ∈ D a(t)α˜ tαH a(t) 0≤s≤t

1 1 H sup log P Z (s) ∈ (δ/2, δ) ≥ a(t)α˜ tαH a(t) 0≤s≤t 1 log [A(t, δ/2) − A(t, δ)] = a(t)α˜ 

1 1 A(t, δ) . = log A(t, δ/2) − log 1 − a(t)α˜ a(t)α˜ A(t, δ/2) Similarly to (69),



1 A(t, δ) =0 log 1 − ˜ t→∞ a(t)α A(t, δ/2) lim

and lim inf t→∞

1 log P a(t)α˜



1 sup Z H (s) ∈ D tαH a(t) 0≤s≤t

Let δ → 0, we have 1 lim inf log P ˜ t→∞ a(t)α



≥ lim inf t→∞

When x < 0, using (66), we have (70) similarly. This completes the proof of Theorem 4.7.

(70)

(71)

(72)

1 log A(t, δ/2) = −I(δ/2). a(t)α˜

1 sup Z H (s) ∈ D tαH a(t) 0≤s≤t

(69)

(73)

≥ −I(0) = 0.

(74)



LARGE DEVIATIONS FOR SUBORDINATED F-BM AND APPLICATIONS

11

5. Application When H = 1/2, α = 1/2. [5] investigated that {st (·) : 1 ≤ t < ∞} is relatively compact in C[0, 1], where W (T (xt)) st (x) = 5/4 −3/4 1/4 , 0 ≤ x ≤ 1. 2 3 t (log log t)3/4 So we can get the law of iterated logarithm of W (T (t)). In this section we will investigate the law of iterated logarithm of Z H (t). Theorem 5.1. Let Z H = {Z H (t), t ≥ 0} be real-valued subordinated fractional Brownian motion. Then we have the following law of iterated logarithm lim sup t→∞

Z H (t) 1 tHα (log log t) 2 +H−Hα

1 1 ≤ ( + H − αH)−( 2 +H−αH) α−Hα H H(1−α) , a.s. 2

(75)

Proof. We prove the theorem in two cases 2H(1 − α) < 1 and 2H(1 − α) ≥ 1 separately. 1 (i) In the case of 2H(1 − α) < 1. Let a(t) = (c log log t) 2 −H+Hα , where c = ( 12 + H − 2Hα

2H(1−α)

αH)−1 α− 1+2H−2Hα H 1+2H−2Hα . From Theorem 3.1, we have that for any x > 0,   2 Z H (t) 1 lim log P > x = −c−1 x 1+2H−2Hα . 2/(1−2H+2Hα) Hα [2/(1−2H+2Hα)]−1 t→∞ a(t) t a(t)

That is 1 log P lim t→∞ log log t



Z H (t) 1 tHα (c log log t) 2 +H−Hα

 >x

2

= −x 1+2H−2Hα .

(76)

(77)

1

Let b(t) = tHα (c log log t) 2 +H−Hα . For any θ > 1, ε > 0, let tk = θk , when k is large enough, from (77) we have that H

ε d Z (tk ) P > 1 + ε ≤ (k log θ)−(1+ 2 ) , (78) b(tk )  H ∞ 2 where d = 1+2H−2Hα > 1. It follows from k=1 P Zb(t(tk )k ) > 1 + ε < ∞ and the Borel-Cantell lemma, we derive lim sup k→∞

Z H (tk ) ≤ 1 + ε, a.s. b(tk )

(79)

Z H (tk ) ≤ 1, a.s. b(tk )

(80)

Since ε > 0 is arbitrary, we have lim sup k→∞

Next, we prove that this is true for t → ∞. Note that, for every t > 0, there exist k such that θk ≤ t < θk+1 , hence Z H (t) ≤ Z H (tk ) + |Z H (t) − Z H (tk )| ≤ Z H (tk ) +

max

tk ≤t
|Z H (t) − Z H (tk )|.

For η = 1, x∗ = 1, from Lemma 2.3, there exist C > 0, h0 > 0 such that

 2 sup |Z H (t + s) − Z H (t)| > xhHα ≤ (CT /h) exp −(γ/2)x 1+2H−2Hα P sup 0≤t≤T −h 0≤s≤h

(81)

(82)

for every x ≥ x∗ and T ≥ h > h0 . Hence for large enough k, we have P

maxtk ≤tε b(tk ) 



εb(tk ) ≤P sup sup |Z (t + s) − Z (t)| > (tk+1 − tk )Hα (tk+1 − tk )Hα 0≤t≤tk 0≤s≤tk+1 −tk

  Cθ γc2 k 2 ≤ . exp − log log θ θ−1 2(θ − 1)(2Hα)/(1+2H−2Hα) H

H

 (83)

12

WEIGANG WANG, ZHENLONG CHEN

Note θ > 1, we get

∞  maxtk ≤t ε < ∞. P b(tk )

k=1

By the Borel-Cantelli lemma and the arbitrary of ε we have that maxtk ≤t
(84)

lim

From (80), (81) and (84), we have lim sup t→∞

maxtk ≤t
(85)

and lim sup t→∞

Z H (t) (log log t)

1 2 +H−Hα

tHα

1 1 ≤ ( + H − αH)−( 2 +H−αH) α−Hα H H(1−α) , a.s. 2

(86)

(ii) In the case of 2H(1 − α) > 1. We can choose β > 0, such that β( 12 + H − αH) < 1. Let 2αH

1

2(1−α)H

a(t) = (c log log)1−β( 2 +H−αH) , where c = ( 12 + H − αH)−1 α− 1+2H−2αH H 1+2H−2αH . Similar to the proof of part(i), from Theorem 1.1, we can have that

lim sup t→∞

|Z H (t)|β 1 (log log t)β( 2 +H−Hα) tHαβ

1 1 ≤ ( + H − αH)−β( 2 +H−αH) α−Hαβ H H(1−α)β , a.s. 2

(87)

So we can get (75).  Acknowledgements. This project is supported by National Natural Science Foundation of China (Grant No.11371321). References [1] M. Arendarczyk and K. Debicki, Asymptotics of supremum distribution of a Gaussian process over a Weibullian time, Bernoulli, 17 (2011) 194-210. [2] W. Barreto-Souza and L.R.G.Fpntes, Long-rang Trap Models on Z and Quasistable Processes, J. Theor. Probab., 28 (2015) 1500-1519. [3] P.Becker-kern, M.M. Meerschaert and H.P.Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab., 32 (2004) 730-756. [4] J. Bertion, L´ evy Processes, Cambridge University Press, 1996. [5] E. Cs´ aki, A. F¨ oldes and P. R´ ev´ esz, Strassen Theorems for a class of iterated processes, Tran. Amer. Math. Soc., 349 (1997) 1153-1167. [6] A. Dembo and O. Zeitouni, Large Deviation Techiniques and Applications, second ed., Springer, 1998. [7] E. Cs´ aki and M. Cs¨ org˝ o, Inequalities for increments of stochastic processes and moduli of continuty, Ann. Prob., 20 (1992) 1031-1062. [8] J. Gajda and M. Magdziarz, Large deviations for subordinated Brownian motion and application, Stat. Prob. Lett., 88 (2014) 149-156. [9] J. Hawkes, A lower Lipschitz condition for the stable subordinator, Z. Wahrscheinlichkeitstheorie verw. Geb., 17 (1971) 23-32. [10] M. Magdziarz and R. L. Schilling, Asymptotic properties of Brownian motion delayed by inverse subordinators, P. Am. Math. Soc., 143 (2015) 4485-4501. [11] M.M. Meerscharet, E. Nane and Y.M. Xiao, Large deviations for local time fractional Brownian motion and applications, J. Math. Anal. Appl., 346 (2008) 432-445. [12] M.M. Meerscharet, E. Nane and Y.M. Xiao, Correlated continuous time random walks, Stat. Prob. Lett., 79 (2009) 1194-1202. [13] M.M. Meerscharet and H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab., 41 (2004) 632-638. [14] K. Sato, L´ evy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. [15] C. Stone, The set of zereo of a semi stable process, Illinois J. Math., 7 (1963) 631-637. [16] G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949.

LARGE DEVIATIONS FOR SUBORDINATED F-BM AND APPLICATIONS

13

(Weigang Wang) Department of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, P. R. China E-mail address: wwgys [email protected] (Zhenlong Chen) Department of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, P. R. China E-mail address: [email protected]