Derivatives of local times for some Gaussian fields II
aa r X i v : . [ m a t h . P R ] O c t Derivatives of local times for some Gaussian fields II
Minhao Hong and Fangjun Xu ∗ October 23, 2020
Abstract
Given a (2 , d )-Gaussian field Z = (cid:8) Z ( t, s ) = X H t − e X H s , s, t ≥ (cid:9) , where X H and e X H are independent d -dimensional centered Gaussian processes satisfyingcertain properties, we will give the necessary condition for existence of derivatives of the localtime of Z . Keywords:
Gaussian fields, Derivatives of local time, Necessary condition.
Subject Classification: Primary 60F25; Secondary 60G15, 60G22.
Local times for Gaussian processes or fields are important in the probability theory. Recently,their derivatives received much attention, see, for example, [5, 3] and references therein. In [3], weconsider derivatives of local time for a (2 , d )-Gaussian field Z = (cid:8) Z ( t, s ) = X H t − e X H s , s, t ≥ (cid:9) , where X H and e X H are independent processes from a class of d -dimensional centered Gaussianprocesses satisfying certain local nondeterminism property. A sufficient condition for existence ofderivatives of the local time of Z was given. Then, under the condition, derivatives of the localtime are shown to be H¨older continuous in both time and space variables. Moreover, under somemild assumption, the sufficient condition is necessary for existence of derivatives of the local timeat the origin. However, when the location is not the origin, the necessity of the condition is stillopen. To the best of our knowledge, for local times of Gaussian processes or fields, in most cases,one just gave sufficient conditions for existence of local times and their derivatives. Only in thecase where the location is the origin, there are a few papers showing the necessity of the sufficientcondition, see, [6, 8] for local times, [5] for the first derivative of local time and [2, 3] for both localtimes and their derivatives. Moreover, the Gaussian processes in [3] can be Bifractional Brownianmotions and Subfractional Brownian motions, which are more general than the fractional Brownianmotions in [6, 8, 2, 5].In this paper, when the location is not the origin, we will show that, if the Gaussian processesin [3] satisfying some additional properties, then the sufficient condition H H H + H (2 | k | + d ) < ∗ M. Hong is partially supported by National Natural Science Foundation of China (Grant No.11871219). F. Xuis partially supported by National Natural Science Foundation of China (Grant No.11871219, No.11871220). H H H + H (2 | k | + d ) <
1. Tomake our results as general as possible, we only consider the necessary condition for existence ofderivatives of local times here and would pose less restrictions on the Gaussian processes. For H ∈ (0 , X H = { X Ht : t ≥ } be a d -dimensional centered Gaussian stochastic process whosecomponents X H,ℓ (1 ≤ ℓ ≤ d ) are independent and identically distributed, and satisfy the followingproperty: (P1) Bounds on the second moment of increments : for any
T >
0, there exist two positiveconstants κ T, ,H ≤ κ T, ,H depending only on T and H , such that for any 0 ≤ s < t < T , κ T, ,H ( t − s ) H ≤ E h ( X H,ℓt − X H,ℓs ) i ≤ κ T, ,H ( t − s ) H . Let G d be the class of all such d -dimensional centered Gaussian processes and G d , the class of d -dimensional centered Gaussian processes in G d possessing the additional property: (P2) Bounds on the covariance of increments on disjoint intervals : there exists a nonnegativedecreasing function β ( γ ) : (1 , ∞ ) → R with lim γ →∞ β ( γ ) = 0, such that, for any 0 < s < t < T with t − ss ≤ γ , (cid:12)(cid:12)(cid:12) E (cid:2) ( X H,ℓt − X H,ℓs ) X H,ℓs (cid:3)(cid:12)(cid:12)(cid:12) ≤ β ( γ ) h E (cid:0) X H,ℓt − X H,ℓs (cid:1) i h E (cid:0) X H,ℓs (cid:1) i . According to results in [1, 4, 7], we can easily see that the following d -dimensional Gaussianprocesses are in G d , .(i) Bifractional Brownian motion (bi-fBm) . The covariance function for components of this processis given by E ( X H,ℓt X H,ℓs ) = 2 − K (cid:2) ( t H + s H ) K − | t − s | H K (cid:3) , where H ∈ (0 ,
1) and K ∈ (0 , K = 1 gives the fractional Brownian motion (fBm) withHurst parameter H = H .(ii) Subfractional Brownian motion (sub-fBm) . The covariance function for components of thisprocess is given by E ( X H,ℓt X H,ℓs ) = t H + s H − (cid:2) ( t + s ) H + | t − s | H (cid:3) , where H ∈ (0 , X H and e X H be independent Gaussian processes in G d with parameters H , H ∈ (0 , Z = (cid:8) Z ( t, s ) = X H t − e X H s , s, t ≥ (cid:9) (1.1)is a (2 , d )-Gaussian field.For any ε > k = ( k , · · · , k d ) with all k i being nonnegative integers, let p ( k ) ε ( x ) = ∂ k ∂x k · · · ∂x k d d p ε ( x ) = ι | k | (2 π ) d Z R d (cid:16) d Y i =1 y k i i (cid:17) e ιy · x e − ε | y | dy, where p ε ( x ) = πε ) d e − | x | ε and | k | = d P i =1 k i . 2or any T > x ∈ R d , if L ( k ) ε ( T, x ) := Z T Z T p ( k ) ε ( X H t − e X H s + x ) ds dt (1.2)converges to some random variable in L when ε ↓
0, we denote the limit by L ( k ) ( T, x ) and call itthe k -th derivative of local time for the (2 , d )-Gaussian field Z . If it exists, L ( k ) ( T, x ) admits thefollowing L -representation L ( k ) ( T, x ) = Z T Z T δ ( k ) ( X H t − e X H s + x ) ds dt. (1.3)The following are main results of this paper. Theorem 1.1
Assume that X H = { X H t : t ≥ } and e X H = { e X H t : t ≥ } are two independentGaussian processes in G d , with parameters H , H ∈ (0 , , respectively. For any x = 0 , if H H H + H (2 | k | + d ) ≥ , then there exist positive constants c and c such that lim inf ε ↓ E [ | L ( k ) ε ( T, x ) | ] h d, | k | H ,H ( ε ) ≥ c e − c | x | , where h d, | k | H ,H ( ε ) = ε H H H H − d −| k | if H H H + H (2 | k | + d ) > ε − ) if H H H + H (2 | k | + d ) = 1 . (1.4) Theorem 1.2
Assume that X H = { X H t : t ≥ } and e X H = { e X H t : t ≥ } are two independentGaussian processes in G d with parameters H , H ∈ (0 , , respectively. We further assume that(i) | k | is even or (ii) E [ X H , t X H , s ] ≥ and E [ e X H , t e X H , s ] ≥ for any < s, t < T . Then, if H H H + H (2 | k | + d ) ≥ , there exists a positive constant c such that lim inf ε ↓ E [ | L ( k ) ε ( T, | ] h d, | k | H ,H ( ε ) ≥ c . Remark 1.3
For independent Gaussian processes X H and e X H in G dL as defined in [3], if theyare also in G d , , then, for x = 0 , L ( k ) ( T, x ) exists in L if and only if H H H + H (2 | k | + d ) < . Remark 1.4
For independent Gaussian processes X H and e X H in G dL as defined in [3], ifthey also satisfy the assumptions in Theorem 1.2, then L ( k ) ( T, exists in L if and only if H H H + H (2 | k | + d ) < . Moreover, the definition of d -dimensional bi-fBm implies that its componentshave nonnegative covariance function. The non-negativity of covariance function for componentsof sub-fBm follows from [1]. Remark 1.5
For each N ∈ N , define the ( N, d ) -Gaussian field Z N = n N X j =1 X j,H j t j : t j ≥ , j = 1 , . . . , N o , here X j,H j t j are independent d -dimensional Gaussian processes in G d . Replace Z in L ( k ) ε ( T, x ) and L ( k ) ( T, x ) by Z N and denote the new terms by L ( k ) N,ε ( T, x ) and L ( k ) N ( T, x ) , respectively. Usingthe methodologies developed here, we could obtain similar results for the existence of L ( k ) N ( T, x ) in L . That is,(i) Assume that X j,H j t j ( j = 1 , . . . , N ) are independent d -dimensional Gaussian processes in G d , , x = 0 and | k | + d ≥ N P j =1 1 H j . Then there exist positive constants c and c such that lim inf ε ↓ E [ | L ( k ) N,ε ( T, x ) | ] h d, | k | H ,H ,...,H N ( ε ) ≥ c e − c | x | , where h d, | k | H ,H ,...,H N ( ε ) = ε N P j =1 12 Hj − d −| k | if | k | + d > N P j =1 1 H j ln(1 + ε − ) if | k | + d = N P j =1 1 H j . (ii) Assume that X j,H j t j ( j = 1 , . . . , N ) are independent d -dimensional Gaussian processes in G d , | k | is even or all X j,H j have nonnegative covariance functions, and | k | + d ≥ N P j =1 1 H j . Thenthere exists a positive constant c such that lim inf ε ↓ E [ | L ( k ) N,ε ( T, | ] h d, | k | H ,H ,...,HN ( ε ) ≥ c . The case N = 1 is easy. It follows from simplified proofs of Theorems 1.1 and 1.2. Moreover, ourmethodologies also work for derivatives of local times of L´evy processes or fields. After some preliminaries in Section 2, Sections 3 and 4 are devoted to the proofs of Theorems1.1 and 1.2, respectively. Throughout this paper, if not mentioned otherwise, the letter c , with orwithout a subscript, denotes a generic positive finite constant whose exact value may change fromline to line. For any x, y ∈ R d , we use x · y to denote the usual inner product and | x | = ( d P i =1 | x i | ) / .Moreover, we use ι to denote √− In this section, we give three lemmas. The first two will be used in the proof of Theorem 1.1 andthe last one in the proof of Theorem 1.2.
Lemma 2.1
Assume that k ∈ N ∪ { } and ε > . Then, for any a, b, c, x ∈ R with a > , c > and ∆ = c + ε − ( b − ε ) a +2 ε > , we have ( − k π Z R exp n −
12 ( y a + 2 y y b + y c ) − ε y − y ) + y ) + ιy x o y k ( y − y ) k dy = k X ℓ =0 k + ℓ X m =0: even k − m X n =0: even c k,ℓ,m,n ( a + 2 ε ) − m +12 ( ε − ba + 2 ε ) k + ℓ − m ∆ − (2 k − m ) − − n x k − m − n e − x , (2.1)4 here c k,ℓ,m,n = ( − ℓ − m + n (cid:18) kℓ (cid:19)(cid:18) k + ℓm (cid:19)(cid:18) k − mn (cid:19) ( m − n − and we use the convention = 1 for the case x = 0 ∈ R .Proof. Let L be the left hand side of the equality (2.1). It is easy to show that L = ( − k π Z R exp n − y ( a + 2 ε ) − y y ( b − ε ) − y ( c + ε ) + ιy x o y k ( y − y ) k dy = k X ℓ =0 ( − k + ℓ π (cid:18) kℓ (cid:19) Z R exp n − y ( a + 2 ε ) − y y ( b − ε ) − y ( c + ε ) + ιy x o y k + ℓ y k − ℓ dy = k X ℓ =0 k + ℓ X m =0: even ( − k + ℓ √ π (cid:18) kℓ (cid:19)(cid:18) k + ℓm (cid:19) ( m − a + 2 ε ) − k − ℓ − − m ( ε − b ) k + ℓ − m Z R y k − m e − y ∆+ ιy x dy = k X ℓ =0 k + ℓ X m =0: even 2 k − m X n =0: even c k,ℓ,m,n ( a + 2 ε ) − m +12 ( ε − ba + 2 ε ) k + ℓ − m ∆ − (2 k − m ) − − n x k − m − n e − x . Lemma 2.2
Assume that k ∈ N ∪ { } and ε > . Then, for any a , b , c , a , b , c , x ∈ R with a , a , c , c > and ∆ ′ = c + c + a + 2 b + ε − ( b − b − a − ε ) a + a +2 ε > , we have ( − k π Z R exp (cid:26) − (cid:0) [ y a + 2 y y b + y c ] + [( y − y ) a + 2( y − y ) y b + y c ] (cid:1)(cid:27) × exp n − ε y − y ) + y ) + ιy x o y k ( y − y ) k dy = k X ℓ =0 k + ℓ X m =0: even k − m X n =0: even c k,ℓ,m,n ( a + a + 2 ε ) − m +12 ( ε + b + a − b a + a + 2 ε ) k + ℓ − m (∆ ′ ) − (2 k − m ) − − n x k − m − n e − x ′ , where c k,ℓ,m,n = ( − ℓ − m + n (cid:18) kℓ (cid:19)(cid:18) k + ℓm (cid:19)(cid:18) k − mn (cid:19) ( m − n − and we use the convention = 1 for the case x = 0 ∈ R .Proof. Note that the integral in the above statement can be written as Z R exp n − y ( a + a + 2 ε ) − y y ( b − a − b − ε ) − y ( c + c + a + 2 b + ε ) + ιy x o y k ( y − y ) k dy. Then the desired result follows from Lemma 2.1.
Lemma 2.3
Assume that k ∈ N ∪ { } and ε > . Then, for any a, b, c ∈ R with a, c > and ( a + ε )( c + ε ) − b > , we have ( − k π Z R exp n −
12 ( y a + 2 y y b + y c ) − ε y + y ) o y k y k dy = k X ℓ =0 , even c k,ℓ b k − ℓ (( a + ε )( c + ε ) − b ) k − ℓ +12 , where c k,ℓ = ( ℓ − (cid:0) kℓ (cid:1) (2 k − ℓ − .Proof. This follows from similar arguments as in the proof of Lemma 2.1.5
Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1.
Proof.
We divide the proof into several steps.
Step 1.
Recall the definition of L ( k ) ε ( T, x ) in (1.2). Using Fourier transform, L ( k ) ε ( T, x ) = ι | k | (2 π ) d Z T Z T Z R d e ιz · ( X H u − e X H v + x ) e − ε | z | d Y i =1 z k i i dz du dv. Hence E [ | L ( k ) ε ( T, x ) | ] = ( − | k | (2 π ) d Z [0 ,T ] Z R d e − (cid:2) E ( z · X H t + z · X H t ) + E ( z · e X H s + z · e X H s ) (cid:3) × e − ε ( | z | + | z | )+ ι ( z + z ) · x d Y i =1 z k i ,i d Y i =1 z k i ,i dz dz dt ds, where z = ( z , , · · · , z ,d ) and z = ( z , , · · · , z ,d ).For i = 1 , · · · , d , we first introduce the following notations I i ( H, t , t , z , z ) = e − E [ z ,i · ( X H,it − X H,it )+ z ,i · X H,it ] e I i ( H, t , t , z , z ) = e − E [ z ,i · ( e X H,it − e X H,it )+ z ,i · e X H,it ] K i ( ε, z , z ) = e − ε ( z ,i + z ,i )+ ι ( z ,i + z ,i ) x i z k i ,i z k i ,i . Then we define F ( t , t , s , s , x i ) = ( − k i π Z R I i ( H , t , t , z , z + z ) e I i ( H , s , s , z , z + z ) K i ( ε, z , z ) dz ,i dz ,i F ( t , t , s , s , x i ) = ( − k i π Z R I i ( H , t , t , z , z + z ) e I i ( H , s , s , z , z + z ) K i ( ε, z , z ) dz ,i dz ,i . Now we can obtain that E h | L ( k ) ε ( T, x ) | i = 2(2 π ) d "Z D d Y i =1 F ( t , t , s , s , x i ) dt ds + Z D d Y i =1 F ( t , t , s , s , x i ) dt ds =: 2(2 π ) d ( I ( ε ) + I ( ε )) , (3.1)where D = { < s < t < T, < s < t < T } . Step 2.
We estimate lim inf ε ↓ I ( ε ) h d, | k | H ,H ( ε ) . Note that I ( ε ) = Z D d Y i =1 F ( t , t , s , s , x i ) dt ds. Making the change of variables y = z and y = z + z gives F ( t , t , s , s , x i ) = ( − k i π Z R I i ( H , t , t , y , y ) e I i ( H , s , s , y , y ) K i ( ε, y , y − y ) dy ,i dy ,i .
6n order to calculate the above integral, we set a = E [( X H , t − X H , t ) ] + E [( e X H , s − e X H , s ) ] b = E [( X H , t − X H , t ) X H , t ] + E [( e X H , s − e X H , s ) e X H , s ] c = E [( X H , t ) ] + E [( e X H , s ) ]∆ = c + ε − ( b − ε ) a + 2 ε . For any ( t , t , s , s ) ∈ D , it is easy to see that a, b, c > ac + aε + 2 cε + ε − b + 2 bεa + 2 ε ≥ cε + ε a + 2 ε > , where we use | b | ≤ √ ac ≤ a + c in the first inequality.By Lemma 2.1, F ( t , t , s , s , x i ) equals k i X ℓ =0 k i + ℓ X m =0:even 2 k i − m X n =0:even c k i ,ℓ,m,n ( a + 2 ε ) − m +12 ( ε − ba + 2 ε ) k i + ℓ − m ∆ − (2 k i − m ) − − n x k i − m − ni e − x i , where c k i ,ℓ,m,n = ( − ℓ − m + n (cid:0) k i ℓ (cid:1)(cid:0) k i + ℓm (cid:1)(cid:0) k i − mn (cid:1) ( m − n − γ > t , t , s , s ) ∈ D , using the Cauchy-Schwartz inequality and properties (P1) and (P2) , we can show that (cid:12)(cid:12) E [( X H , t − X H , t ) X H , t ] (cid:12)(cid:12) ≤ c (cid:2) ( γ H + γ − H ) a + β ( γ ) a (cid:3) , where γ H a comes from the case γ < t − t t < γ , γ − H a from the case t − t t ≥ γ , and β ( γ ) a fromthe case t − t t ≤ γ . Similarly, (cid:12)(cid:12) E [( e X H , s − e X H , s ) e X H , s ] (cid:12)(cid:12) ≤ c (cid:2) ( γ H + γ − H ) a + β ( γ ) a (cid:3) . Hence (cid:12)(cid:12)(cid:12) ε − ba + 2 ε (cid:12)(cid:12)(cid:12) ≤
12 + c ( γ H + γ − H + γ H + γ − H ) a + β ( γ ) a a + 2 ε ≤ c (cid:16) γ H + γ H + β ( γ )( a + 2 ε ) (cid:17) . Let D γ = D ∩ n < t − t t < T ∧ γ , T < t < T , < s − s s < T ∧ γ , T < s < T o . (3.2)For any ( t , t , s , s ) ∈ D γ , using the property (P2) , we can show that c ( T H + T H ) ≤ ∆ ≤ c ( T H + T H ) (3.3)provided that γ is very large and ε is very small.Note that for any α >
0, the function h ( w ) = w α e − w ∈ (0 , α α e − α ] when w ∈ (0 , + ∞ ). Choosing γ large enough giveslim inf ε ↓ I ( ε ) h d, | k | H ,H ( ε ) ≥ lim inf ε ↓ (1 − c β ( γ )) d h d, | k | H ,H ( ε ) Z D ( a + 2 ε ) −| k |− d ∆ − d e − | x | dt ds, < c β ( γ ) < ε ↓ I ( ε ) h d, | k | H ,H ( ε ) ≥ c lim inf ε ↓ (1 − c β ( γ )) d h d, | k | H ,H ( ε ) Z D γ ( a + 2 ε ) −| k |− d ( T H + T H ) − d e − | x | c T H T H dt ds ≥ c e − | x | c T H T H lim inf ε ↓ (1 − c β ( γ )) d h d, | k | H ,H ( ε ) Z D γ (( t − t ) H + ( s − s ) H + 2 ε ) −| k |− d dt ds ≥ c e − | x | c T H T H , (3.4)where in the last inequality we use Lemma A.3. in [3] and the property (P1) . Step 3.
We estimate lim inf ε ↓ I ( ε ) h d, | k | H ,H ( ε ) . Note that I ( ε ) = Z D d Y i =1 F ( t , t , s , s , x i ) dt ds. Making the change of variables y = z and y = z + z gives F ( t , t , s , s , x i ) = ( − k i π Z R I i ( H , t , t , y , y ) e I i ( H , s , s , y − y , y ) K i ( ε, y , y − y ) dy ,i dy ,i . In order to calculate the above integral, we set a = E [( X H , t − X H , t ) ] , a = E [( e X H , s − e X H , s ) ] ,b = E [( X H , t − X H , t ) X H , t ] , b = E [( e X H , s − e X H , s ) e X H , s ] e = E [( X H , t ) ] , e = E [( e X H , s ) ]∆ ′ = e + e + a + 2 b + ε − ( b − b − a − ε ) a + a + 2 ε . It is easy to see that, for any ( t , t , s , s ) ∈ D , a , a , e , e > ′ ≥ a e + a a + a e + 2 a b + 2 a b + 2 b b + ε ( e + e ) + ε a + a + 2 ε ≥ ε ( e + e ) + ε a + a + 2 ε > , where we use | b | ≤ √ a e ≤ a + e and | b | ≤ √ a e ≤ a + e in the first two inequalities.By Lemma 2.1, F ( t , t , s , s , x i ) equals k i X ℓ =0 k i + ℓ X m =0:even 2 k i − m X n =0:even c k i ,ℓ,m,n ( a + a + 2 ε ) − m +12 ( ε + b + a − b a + a + 2 ε ) k i + ℓ − m (∆ ′ ) − (2 k i − m ) − − n x k i − m − n e − x ′ , where c k i ,ℓ,m,n = ( − ℓ − m + n (cid:0) k i ℓ (cid:1)(cid:0) k i + ℓm (cid:1)(cid:0) k i − mn (cid:1) ( m − n − t , t , s , s ) ∈ D , using the Cauchy-Schwartz inequality and properties (P1) and (P2) , we can show that (cid:12)(cid:12)(cid:12) ε + b + a − b a + a + 2 ε (cid:12)(cid:12)(cid:12) ≤ | b − b | a + a + 2 ε ≤ c (cid:16) γ H + γ H + β ( γ )( a + a + 2 ε ) (cid:17) . ε ↓ I ( ε ) h d, | k | H ,H ( ε ) ≥ lim inf ε ↓ h d, | k | H ,H ( ε ) Z D ( a + a + 2 ε ) −| k |− d ∆ − d e − | x | dt ds − c β ( γ ) lim sup ε ↓ h d, | k | H ,H ( ε ) Z D ( a + a + 2 ε ) −| k |− d dt ds ≥ − c β ( γ ) lim sup ε ↓ h d, | k | H ,H ( ε ) Z D (( t − t ) H + ( s − s ) H + 2 ε ) −| k |− d dt ds ≥ − c β ( γ ) . Letting γ ↑ + ∞ gives lim inf ε ↓ I ( ε ) h d, | k | H ,H ( ε ) ≥ . (3.5) Step 3.
Combining (3.1), (3.4) and (3.5) giveslim inf ε ↓ E [ | L ( k ) ε ( T, x ) | ] h d, | k | H ,H ( ε ) ≥ lim inf ε ↓ I ( ε ) h d, | k | H ,H ( ε ) + lim inf ε ↓ I ( ε ) h d, | k | H ,H ( ε ) ≥ c e − | x | c T H T H . This completes the proof.
In this section, we give the proof of Theorem 1.2.
Proof.
By Lemma 2.3, E [ | L ( k ) ε ( T, | ] = ( − | k | (2 π ) d Z [0 ,T ] Z R d e − (cid:2) E ( z · X H t + z · X H t ) + E ( z · e X H s + z · e X H s ) (cid:3) × e − ε ( | z | + | z | ) d Y i =1 z k i ,i d Y i =1 z k i ,i dz dz dt ds = 1(2 π ) d Z [0 ,T ] d Y i =1 (cid:16) k i X ℓ =0 , even c k i ,ℓ b k i − ℓ (( a + ε )( c + ε ) − b ) ki − ℓ +12 (cid:17) dt ds, where a = E [( X H , t ) ] + E [( e X H , s ) ], b = E [ X H , t X H , t ] + E [ e X H , s e X H , s ] and c = E [( e X H , s ) ] + E [( X H , t ) ]. According to the assumption (i) | k | is even or (ii) E [ X H , t X H , s ] ≥ E [ e X H , t e X H , s ] ≥ < s, t < T , E [ | L ( k ) ε ( T, | ] ≥ π ) d d Y i =1 c k i , Z [0 ,T ] b | k | (cid:16) ( a + ε )( c + ε ) − b (cid:1) | k | + d dt ds. Recall the definition of D γ in (3.2). For any ( t , t , s , s ) ∈ D γ with γ large enough, by theproperty (P1) and the Cauchy Schwartz inequality, b = E [( X H , t ) ] + E [( e X H , s ) ] + E [( X H , t − X H , t ) X H , t ] + E [( e X H , s − e X H , s ) e X H , s ] ≥ c ( t H + s H − t H γ H − s H γ H ) ≥ c ( t H + s H ) . a = E [( X H , t − X H , t ) ] + E [( e X H , s − e X H , s ) ], a = E [( e X H , s ) ] + E [( X H , t ) ] and a = E [( X H , t − X H , t ) X H , t ] + E [( e X H , s − e X H , s ) e X H , s ]. Then0 < ( a + ε )( c + ε ) − b = ( a + 2 a + a + ε )( a + ε ) − ( a + a ) ≤ a + ε )( a + ε ) . Therefore, for γ large enough,lim inf ε ↓ E [ | L ( k ) ε ( T, | ] h d, | k | H ,H ( ε ) ≥ lim inf ε ↓ c h d, | k | H ,H ( ε ) Z [0 ,T ] b | k | (cid:16) ( a + ε )( c + ε ) − b (cid:1) | k | + d dt ds ≥ lim inf ε ↓ c h d, | k | H ,H ( ε ) Z D γ ( t H + s H ) | k | (( t − t ) H + ( s − s ) H + ε ) | k | + d ( t H + s H + ε ) | k | + d dt ds ≥ c , where in the last inequality we use Lemma A.3. in [3] and the property (P1) . References [1] T. Bojdeckia, L. Gorostizab and A. Talarczyk: Sub-fractional Brownian motion and its rela-tion to occupation times.
Statistics & Probability Letters , , 405–419, 2004.[2] J. Guo, Y. Hu and Y. Xiao: High-order derivative of intersection local time for two independentfractional Brownian motions. J. Theor. Probab. , , 1190–1201, 2019.[3] M. Hong and F. Xu: Derivatives of local times for some Gaussian fields. J. Math. Anal. Appl. , , 123716, 2020.[4] C. Houdr´e and J. Villa: An example of infinite dimensional quasi-helix. In Stochastic Models(Mexico City, 2002), Contemp. Math. , , Amer. Math. Soc., Providence, RI, 195–201, 2003.[5] A. Jaramillo, I. Nourdin and G. Peccati: Approximation of fractional local times zero energyand weak derivatives. arXiv: 1903.08683.[6] D. Nualart and S. Ortiz-Latorre: Intersection local time for two independent fractional Brow-nian motions. J. Theor. Probab. , , 759–757, 2007.[7] J. Song, F. Xu and Q. Yu: Limit theorems for functionals of two independent Gaussianprocesses. Stochastic Process. Appl. , , 4791–4836, 2019.[8] D. Wu and Y. Xiao: Regularity of intersection local times of fractional Brownian motions. J.Theor. Probab. , (4), 972–1001, 2010. Minhao Hong
School of Statistics, East China Normal University, Shanghai 200262, China [email protected]
Fangjun Xu
Key Laboratory of Advanced Theory and Application in Statistics and Data Science - MOE, Schoolof Statistics, East China Normal University, Shanghai, 200062, ChinaNYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North,Shanghai, 200062, China [email protected], [email protected]@gmail.com, [email protected]