Operator Fractional Brownian Sheet and Martingale Differences
aa r X i v : . [ m a t h . P R ] J a n OPERATOR FRACTIONAL BROWNIANSHEET AND MARTINGALE DIFFERENCES
Hongshuai Dai , Guangjun Shen § and Liangwen Xia July 29, 2018
Abstract
In this paper, inspired by the fractional Brownian sheet of Riemann-Liouville type,we introduce the operator fractional Brownian sheet of Riemman-Liouville type, andstudy some properties of it. We also present an approximation in law to it based onthe martingale differences.
Keywords:
Fractional Brownian sheet; Operator fractional Brownian sheet of Riemann-Liouville type; Martingale differences; Weak convergence
MSC(2010):
1. Introduction
Self-similar processes, first studied rigorously by Lamperti [18] under the name “semi-stable”, are stochastic processes that are invariant in distribution under suitable scalingof time and space. There has been an extensive literature on self-similar processes. Werefer to Vervaat [25] for general properties, to Samorodnitsky and Taqqu [24][Chaps.7 and8] for studies on Gaussian and stable self-similar processes and random fields.The fractional Brownian motion (fBm) as a well-known self-similar process has beenstudied extensively. Many results about weak approximation to fBms have been estab-lished recently. See [12, 19] and the references therein. We point out that the fBm doesnot represent a casual time-invariant system as there is no well-defined impulse responsefunction. Hence, based on the Riemann-Liouville fractional integral, Barnes and Allan[4] introduced the fractional Riemann-Liouville (RL) Brownian motion (RL-fBm). RL-fBms share with fBms many properties which include self-similarity, regularity of samplepaths, etc.- with one notable exception that its increment process is nonstationary. Formore information on RL-fBms, refer to Lim [20] and the references therein. On the otherhand, there are two typical multiparameter extensions of fBms, one of which is the frac-tional Brownian sheet introduced by Kamont [16]. Fractional Brownian sheets have beenstudied extensively as a representative of anisotropic Gaussian random fields. For moreinformation, refer to [2, 3] and [26, 27]. Inspired by the study of RL-fBms and fractional [1] School of Statistics, Shandong University of Finance and Economics, Jinan, 250014 China[2] Department of Mathematics, Anhui Normal University, Wuhu, 241000 China § Corresponding author. Brownian sheets, Dai [8] introduced the multifractional Riemann-Liouville Brownian sheetand studied the weak limit theorem for it.The definition of self-similarity has been extended to allow scaling by linear operatorson multidimensional space R d , and the corresponding processes are called operator self-similar processes. We refer to [17], [18], [22] and the references therein. We note thatDidier and Pipiras [14, 15] introduced the operator fractional Brownian motions (ofBm inshort) as an extension of fBms and studied their properties. Similar to fBms, weak limittheorems for ofBms have also attracted a lot of interest. Recently, Dai and his coauthors[9]-[11] presented some weak limit theorems for some kinds of ofBms.In contrast to the extensive study on the multiparameter extension of fBms, there islittle work studying the multiparameter extension of ofBms. Inspired by the study ofthe fractional Brownian sheet and the operator fractional Brownian motion of Riemann-Liouville type introduced by Dai [10], we will introduce a new random field, which wecall the operator fractional Brownian sheet of Riemann-Liouville type, and present anapproximation to it.Most of the estimates of this paper contain unspecified constants. An unspecified posi-tive and finite constant will be denoted by C , which may not be the same in each occur-rence. Sometimes we shall emphasize the dependence of these constants upon parameters.At the end of this section, we point out that all processes considered here are assumedto be proper. We say that a process { X ( t ); t ∈ R d + } is proper if for each t ∈ R d + thedistribution of X ( t ) is full; that is, the distribution is not contained in a proper hyperplane.The rest of this paper is organized as follows. In Section 2, we introduce the opera-tor fractional Brownian sheet of Riemann-Liouville type and state some properties. Wepresent an approximation in law to it in Section 3. A final note is presented at the end ofthis paper.
2. Operator Fractional Brownian Sheet
In this section, we first introduce the operator fractional Brownian sheet of Riemann-Liouville type and then study some properties of it. For any x ∈ R d , x T denotes thetranspose of x . Let (Ω , F , P ) be a probability space and {F t,s ; ( t, s ) T ∈ R } be a familyof sub- σ -fields of F such that F t,s ⊆ F t ′ ,s ′ for any ( t, s ) T < ( t ′ , s ′ ) T with the usual partialorder. Moreover, for any stochastic process Y = { Y ( t, s ); ( t, s ) T ∈ R } , we denote by∆ ( t,s ) Y ( t ′ , s ′ ) the increment of Y over the rectangle ( t, t ′ ] × ( s, s ′ ], that is,∆ ( t, s ) Y ( t ′ , s ′ ) = Y ( t ′ , s ′ ) − Y ( t, s ′ ) − Y ( t ′ , s ) + Y ( t, s ) . Let σ ( A ) be the collection of all eigenvalues of a linear operator A on R d . Let λ A = min { Re λ : λ ∈ σ ( A ) } and Λ A = max { Re λ : λ ∈ σ ( A ) } . Moreover, given any linear operator A on R d and t >
0, we define the power operator t A = ∞ X k =0 (log t ) k A k k ! . Next, we recall the operator fractional Brownian motion of Riemann-Liouville type intro-duced by Dai [10]. Let D be a linear operator on R d with 0 < λ D , Λ D <
1. We define the2
FBMS and Martingale Differences operator fractional Brownian motion of Riemann-Liouville type ˜ X = { ˜ X ( t ); t ∈ R + } withexponent D by ˜ X ( t ) = Z t ( t − u ) D − I/ dW ( u ) , (2.1)where W ( u ) = { W ( u ) , ..., W d ( u ) } T is a standard d -dimensional Brownian motion and I is the d × d identity matrix.Based on (2.1), we can define the operator fractional Brownian sheet of Riemann-Liouville type X = (cid:8) X ( t, s ); ( t, s ) T ∈ R (cid:9) as follows. Definition 2.1
Let ˜ B be the standard Brownian sheet. The operator fractional Browniansheet of Riemann-Liouville type X = (cid:8) X ( t, s ); ( t, s ) T ∈ R (cid:9) is defined by X ( t, s ) = Z t Z s ( t − u ) D − I ( s − v ) D − I B ( du, dv ) , (2.2)where B ( du, dv ) = (cid:0) B ( du, dv ) , ..., B d ( du, dv ) (cid:1) T with B i being independent copies of ˜ B ,and D is a linear operator on R d with 0 < λ D , Λ D < Remark 2.1
Let x + = max { x, } . From (2.2) and Mason and Xiao [21], we get that X isan R d -valued Gaussian random field with mean zero vector and for any ( t , t ) T , ( s , s ) T ∈ R E (cid:2) X ( t , t ) X T ( s , s ) (cid:3) = Z ∞ Z ∞ (cid:2) ( t − u ) + ( t − v ) + ] D − I · (cid:2) ( s − v ) + ( s − u ) + (cid:3) D ∗ − I dudv, (2.3)where D ∗ is the adjoint operator of D .It is obvious that the equation (2.2) is well defined. Next, we study some propertiesof the random field X . We first introduce the following notation. Let k x k denote theusual Euclidean norm of x ∈ R d . Similar to Dai, Shen and Kong [13], End ( R d ) denotesthe set of linear operators on R d (endomorphisms). Furthermore, we will not distinguishan operator D ∈ End ( R d ) from its associated matrix relative to the standard basis of R d .For any A ∈ End ( R d ), let k A k = max k x k =1 k Ax k be the operator norm of A . Next, werecall the definition of operator self-similar processes. Recall that an R d -valued stochasticprocess ˜ Y = { ˜ Y ( t ); t ∈ R } is said to be operator self-similar (o.s.s.) if it is continuous inlaw at each t ∈ R , and there exists D ∈ End ( R d ) such that (cid:8) ˜ Y ( ct ) (cid:9) D = (cid:8) c D ˜ Y ( t ) (cid:9) for all c > , where D = denotes the equality of all finite-dimensional distributions. Theorem 2.1
The random field X = { X ( t, s ); ( t, s ) T ∈ R } is an operator self-similarGaussian random field with exponent D . Moreover, X has a version with continuoussample paths a.s.. Proof:
We first check the operator self-similarity. For every c >
0, we have X ( ct, cs ) = Z ∞ Z ∞ ( ct − u ) D − I + ( cs − v ) D − I + dB ( u, v ) D = c D − I Z ct Z cs ( t − uc ) D − I ( s − vc ) D − I dB ( u, v ) D = c D X ( t, s ) , since B ( cu, cv ) D = c I B ( u, v ) and z D y D = ( zy ) D for any z > , y > . (2.4)Next, we check the sample continuity. Choose any t = ( t , t ) T , s = ( s , s ) T ∈ R .Without loss of generality, we assume that s < t with the usual partial order, and k t − s k ≤
1. By some calculations, we have∆ s X ( t ) = Z ∞ Z ∞ (cid:16)(cid:0) t − u (cid:1) D − I + − (cid:0) s − u (cid:1) D − I + (cid:17)(cid:16)(cid:0) t − v (cid:1) D − I + − (cid:0) s − v (cid:1) D − I + (cid:17) B ( du, dv ) . (2.5)Hence, (cid:13)(cid:13) ∆ s X ( t ) (cid:13)(cid:13) = d X i =1 (cid:16) Z ∞ Z ∞ d X j =1 F i,j ( t, s, u, v ) B j ( du, dv ) (cid:17) , (2.6)where F ( t, s, u, v ) = (cid:16) ( t − u ) D − I + − ( s − u ) D − I + (cid:17)(cid:16) ( t − v ) D − I + − ( s − v ) D − I + (cid:17) = (cid:0) F i,j ( t, s, u, v ) (cid:1) d × d . Noting that R ∞ R ∞ P dj =1 F i,j ( t, s, u, v ) B j ( du, dv ) is a Gaussian random variable, we getfrom (2.6) that for any even k ∈ NE h(cid:13)(cid:13) ∆ s X ( t ) (cid:13)(cid:13) i k ≤ C h Z ∞ Z ∞ (cid:13)(cid:13) F ( t, s, u, v ) (cid:13)(cid:13) dudv i k . (2.7)On the other hand, we have (cid:13)(cid:13) F ( t, s, u, v ) (cid:13)(cid:13) ≤ C k F ( t, s, u, v ) k × k F ( t, s, u, v ) k , (2.8)where F ( t, s, u, v ) = (cid:0) t − u (cid:1) D − I + − (cid:0) s − u (cid:1) D − I + , and F ( t, s, u, v ) = (cid:0) t − v (cid:1) D − I + − (cid:0) s − v (cid:1) D − I + . FBMS and Martingale Differences Now, we look at Z ∞ (cid:13)(cid:13) F ( t, s, u, v ) (cid:13)(cid:13) dudv. By using the same method as in Dai, Hu and Lee [11], we have Z ∞ k ( t − u ) D − I + − ( s − u ) D − I + k du ≤ C ( t − s ) λ D − δ . (2.9)Similarly, Z ∞ (cid:13)(cid:13) F ( t, s, u, v ) (cid:13)(cid:13) dv ≤ C ( t − s ) λ D − δ . (2.10)From Maejima and Mason [21], and (2.7)-(2.10), we can get that for any δ > λ D − δ > E h(cid:13)(cid:13) ∆ s X ( t ) (cid:13)(cid:13) k i ≤ C h ( t − s ) λ D − δ × ( t − s ) λ D − δ i k ≤ C (cid:13)(cid:13) t − s (cid:13)(cid:13) ( λ D − δ ) k . (2.11)The sample continuity follows from Garsia [6] and (2.11). (cid:3)
3. Limit Theorem
One aim of this paper is to present an approximation in law to the operator fractionalBrownian sheet of Riemann-Liouville type X via the martingale differences. In order toreach it, we first recall some facts about the martingale differences. Similar to Wang, Yanand Yu [26], we use the definitions and notations introduced in the basic work of Cairoliand Walsh [7] on stochastic calculus in the plane. For any n = ( n , n ) T ∈ N × M with N = { , · · · , n } and M = { , · · · , m } , let ˜ F n := F n ,m W F n ,n , the σ − fieldsgenerated by F n ,m and F n ,n . Now, we recall the definition of the strong martingale. Definition 3.1
An integrable process Y = { Y ( n ) , n ∈ N × M } is called a strong mar-tingale if:(i) Y is adapted;(ii) Y vanishes on the axes;(iii) E (cid:2) ∆ n Y ( m ) | ˜ F n (cid:3) = 0 for any n ≤ m ∈ N × M with the usual partial order.Let (cid:8) ξ ( n ) = ( ξ ( n ) i,j , F ( n ) i,j ) (cid:9) n ∈ N be a sequence such that for all E h ξ ( n ) i +1 ,j +1 |F ( n ) i,j i = 0 , where F ( n ) i,j = F ( n ) i,n W F ( n ) n,j with F ( n ) k,l being the σ − fields generated by all ξ ( n ) r,s , r ≤ k, s ≤ l .Then we call (cid:8) ξ ( n ) = ( ξ ( n ) i,j , F ( n ) i,j ) (cid:9) n ∈ N a martingale differences sequence.5 It is well known that if the martingale differences sequence { ξ ( n ) } satisfies the followingcondition ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 (cid:0) ξ ( n ) i,j (cid:1) → t · s in the sense of L , then the sequence ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 ξ ( n ) i,j converges weakly to the Brownian sheet, as n goes to infinity (see for example, Morkvenas[23].) Recently, Wang, Yan and Yu [26] extended this work to the fractional Brown-ian sheet. If { ξ ( n ) } is a square integrable martingale differences sequence satisfying thefollowing two conditions: lim n →∞ n ( ξ ( n ) i,j ) = 1 , a.s. (3.1)for any 1 ≤ i, j ≤ n , and max ≤ i,j ≤ n | ξ ( n ) i,j | ≤ Cn , a.s. (3.2)for some C ≥
1, then, based on { ξ ( n ) } , the authors of [26] constructed a sequence toconverge weakly to the fractional Brownian sheet. Inspired by these results, we want tostudy the weak limit theorem for the operator fractional Brownian sheet of Riemann-Liouville type X introduced in Definition 2.1. Similar to Wang, Yan and Yu [26], weassume that < λ D , Λ D < η ( n ) i,j = ( ξ ( n ) i,j, , ..., ξ ( n ) i,j,d ) T , (3.3)and B n ( t, s ) = ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 η ( n ) i,j , (3.4)where ξ ( n ) i,j,k , k = 1 , · · · , d, are independent copies of ξ ( n ) i,j .From the above arguments, we obtain that (cid:8) ( η ( n ) i,j , F ( n ) i,j ) (cid:9) n ∈ N is still a sequence of squareintegrable martingale differences on the probability space (Ω , F , P ).For any n ≥ t, s ) T ∈ [0 , , define X n ( t, s ) = Z t Z s ( t − u ) D − I + ( s − v ) D − I + B n ( du, dv )= n ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 η ( n ) i,j Z ini − n Z jnj − n ( t − u ) D − I + ( s − v ) D − I + dudv. (3.5)Then, we have the following approximation. As a prelude to giving the result, let D ([0 , ) = D ([0 , , R d ) . FBMS and Martingale Differences Theorem 3.1
Let < λ D , Λ D < . The sequence of processes { X n ( t, s ); ( t, s ) T ∈ [0 , } given by (3.5) converges weakly, as n → ∞ in D ([0 , ) , to the operator fractional Brow-nian sheet of Riemann-Liouville type { X ( t, s ); ( t, s ) T ∈ [0 , } given by (2.2) . The proof of Theorem 3.1 is based on a series of technical results.
Lemma 3.1
Let { X n ( t, s ) } be the family of processes defined by (3.5) . Then for any s = ( s , s ) T < t = ( t , t ) T < u = ( u , u ) T ∈ [0 , , E h(cid:13)(cid:13) ∆ s X n ( t ) (cid:13)(cid:13) (cid:13)(cid:13) ∆ t X n ( u ) (cid:13)(cid:13) i ≤ C ( u − s ) H ( u − s ) H , (3.6) where H = λ D − δ with < δ < λ D − .Proof: From (3.5), we have∆ s X n ( t )= Z t s Z t s (cid:16) ( t − u ) D − I + − ( s − u ) D − I + (cid:17)(cid:16) ( t − v ) D − I + − ( s − v ) D − I + (cid:17) B n ( du, dv )= ⌊ nt ⌋ X i =1 ⌊ nt ⌋ X j =1 n Z ini − n Z jnj − n (cid:16) ( ⌊ nt ⌋ n − u ) D − I + − ( ⌊ ns ⌋ n − u ) D − I + (cid:17) × (cid:16) ( ⌊ nt ⌋ n − v ) D − I + − ( ⌊ ns ⌋ n − v ) D − I + (cid:17) dudvη ( n ) i,j . It follows from (3.2) and the Cauchy-Schwarz inequality that E h(cid:13)(cid:13) ∆ s X n ( t ) (cid:13)(cid:13) i = E (cid:20)(cid:13)(cid:13)(cid:13) P ⌊ nt ⌋ i =1 P ⌊ nt ⌋ j =1 n R ini − n R jnj − n (cid:16) ( ⌊ nt ⌋ n − u ) D − I + − ( ⌊ ns ⌋ n − u ) D − I + (cid:17) × (cid:16) ( ⌊ nt ⌋ n − v ) D − I + − ( ⌊ ns ⌋ n − v ) D − I + (cid:17) dudvη ( n ) i,j (cid:13)(cid:13)(cid:13) (cid:21) ≤ C P ⌊ nt ⌋ i =1 (cid:16) R ini − n k ( ⌊ nt ⌋ n − u ) D − I + − ( ⌊ ns ⌋ n − u ) D − I + k du (cid:17) × P ⌊ nt ⌋ j =1 (cid:16) R jnj − n k ( ⌊ nt ⌋ n − v ) D − I + − ( ⌊ ns ⌋ n − v ) D − I + k dv (cid:17) ≤ C (cid:18) R t k ( ⌊ nt ⌋ n − u ) D − I + − ( ⌊ ns ⌋ n − u ) D − I + k du (cid:19) (cid:18) R t k ( ⌊ nt ⌋ n − v ) D − I + − ( ⌊ ns ⌋ n − v ) D − I + k dv (cid:19) ≤ C (cid:18) R k ( ⌊ nt ⌋ n − u ) D − I + − ( ⌊ ns ⌋ n − u ) D − I + k du (cid:19) (cid:18) R k ( ⌊ nt ⌋ n − v ) D − I + − ( ⌊ ns ⌋ n − v ) D − I + k dv (cid:19) . (3.7)From Dai, Hu and Lee [11], we obtain that Z k ( ⌊ nt ⌋ n − u ) D − I + − ( ⌊ ns ⌋ n − u ) D − I + k du ≤ C ( ⌊ nt ⌋ n − ⌊ ns ⌋ n ) H , where H = λ D − δ . Then (3.7) can be bounded by C (cid:16) ⌊ nt ⌋ − ⌊ ns ⌋ n (cid:17) H (cid:16) ⌊ nt ⌋ − ⌊ ns ⌋ n (cid:17) H . (3.8)Hence, for any s < t < u ∈ [0 , , we have E h(cid:13)(cid:13) ∆ s X ( t ) (cid:13)(cid:13) (cid:13)(cid:13) ∆ t X ( u ) (cid:13)(cid:13) i ≤ C (cid:2) E (cid:2) k ∆ t X ( u ) k (cid:3) (cid:2) E (cid:2) k ∆ s X ( t ) k (cid:3) ≤ C (cid:16) ⌊ nt ⌋ − ⌊ ns ⌋ n (cid:17) H (cid:16) ⌊ nt ⌋ − ⌊ ns ⌋ n (cid:17) H × (cid:16) ⌊ nu ⌋ − ⌊ nt ⌋ n (cid:17) H (cid:16) ⌊ nu ⌋ − ⌊ nt ⌋ n (cid:17) H . (3.9)Hence, if u − s ≥ n , then (cid:12)(cid:12)(cid:12) ⌊ nu ⌋ − ⌊ ns ⌋ n (cid:12)(cid:12)(cid:12) H ≤ C | ( u − s ) | H . (3.10)Conversely, if u − s < n , then either u and t or t and s belong to a same subinterval[ mn , m +1 n ) for some integer m . Hence (3.10) still holds. The other term follows a similardiscussion. The proof is now completed. (cid:3) Since X n ( t, s ), n ∈ N , are null on the axes, by using the criterion given by Bickel andWichura [5], and Lemma 3.1, we can get the following lemma. Lemma 3.2
The sequence { X n ( t, s ); ( t, s ) T ∈ [0 , } is tight in D ([0 , ) . Now, in order to prove Theorem 3.1, it suffices to show the following lemma which statesthat the law of all possible weak limits is the law of the operator fractional Brownian sheetof Riemann-Liouville type X . Lemma 3.3
The family of random fields X n ( t, s ) defined by (3.5) converges, as n tendsto infinity, to the operator fractional Brownian sheet of Riemann-Liouville type X in thesense of finite-dimensional distributions. In order to prove Lemma 3.3, we need a technical result. Before we present this result,we first introduce the following notation.( t − u ) D − I + = (cid:0) ˜ K i,j ( t, u ) (cid:1) d × d and (cid:16) ⌊ nt ⌋ n − u (cid:17) D − I + = (cid:0) ˜ K ni,j ( t, u ) (cid:1) d × d . Lemma 3.4
For any ( t k , s k ) T , ( t l , s l ) T ∈ [0 , and q, m ∈ { , · · · , d } , we have that n P ni =1 P nj =1 R ini − n R jnj − n ˜ K nq,m ( t k , u ) ˜ K nm,q ( s k , v ) dudv R ini − n R jnj − n ˜ K nq,m ( t l , u ) ˜ K nm,q ( s l , v ) dudv ( ξ ( n ) i,j,q ) (3.11)8 FBMS and Martingale Differences converges to Z Z ˜ K q,m ( t k , u ) ˜ K m,q ( s k , v ) ˜ K q,m ( t l , u ) ˜ K m,q ( s l , v ) dudv, a.s. (3.12) as n tends to infinity.Proof: It is obvious that (3.11) is equivalent to n P ni =1 n R ini − n ˜ K nq,m ( t k , u ) du R ini − n ˜ K nq,m ( t l , u ) du · P nj =1 n R jnj − n ˜ K nm,q ( s k , v ) dv R jnj − n ˜ K nm,q ( s l , v ) dv ( ξ ( n ) i,j,q ) . (3.13)By using the same method as the proof of Lemma 8 in Dai, Hu and Lee[11], we can provethe lemma. (cid:3) Next, we prove Lemma 3.3.
Proof of Lemma 3.3.
Let a , ..., a Q ∈ R and ( t , s ) T , ..., ( t Q , s Q ) T ∈ [0 , . Next, weprove that the random vector Y n = Q X k =1 a k X n ( t k , s k )converges in distribution, as n tends to infinity, to the Gaussian random vector˜ X = Q X k =1 a k X ( t k , s k ) . By the well-known Cram´er-Wold device, see Whitt [28] for example, in order to prove theabove statement, we only need to show that as n → ∞ bY n D → b ˜ X, (3.14)where b = ( b , b , · · · , b d ) and D → denotes convergence in distribution.For conciseness of the paper, let( t − u ) D − I + ( s − v ) D − I + = K ( t, s, u, v ) = (cid:0) K ( t, s, u, v ) , · · · , K d ( t, s, u, v ) (cid:1) T , where K j ( t, s, u, v ) = (cid:0) K j, ( t, s, u, v ) , · · · , K j,d ( t, s, u, v ) (cid:1) . Then, we have bY n = d X q =1 Q X k =1 ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 n Z ini − n Z jnj − n a k b q K q ( ⌊ nt k ⌋ n , ⌊ ns k ⌋ n , u.v ) η ( n ) i,j dudv = d X m =1 d X q =1 Q X k =1 ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 n Z ini − n Z jnj − n a k b q K q,m ( ⌊ nt k ⌋ n , ⌊ ns k ⌋ n , u, v ) ξ ( n ) i,j,m dudv, and b ˜ X = d X m =1 d X q =1 Q X k =1 Z Z a k b q K q,m ( t k , s k , u, v ) B m ( du, dv ) . Since ξ ( n ) i,j,m , m = 1 , · · · , d, are independent, in order to prove (3.14), we only need to show d X q =1 Q X k =1 ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 n Z ini − n Z jnj − n a k b q K q,m ( ⌊ nt k ⌋ n , ⌊ ns k ⌋ n , u, v ) ξ ( n ) i,j,m dudv D → d X q =1 Q X k =1 Z Z a k b q K q,m ( t k , s k , u, v ) B m ( du, dv ) . (3.15)For convenience, we introduce the following notation. Y ( n ) i,j = d X q =1 Q X k =1 n Z ini − n Z jnj − n a k b q K q,m ( ⌊ nt k ⌋ n , ⌊ ns k ⌋ n , u, v ) ξ ( n ) i,j,m dudv. Then, (3.15) can be rewritten as ⌊ nt ⌋ X i =1 ⌊ ns ⌋ X j =1 Y ( n ) i,j D → d X q =1 Q X k =1 Z Z a k b q K q,m ( t k , s k , u, v ) B m ( du, dv ) . (3.16)Inspired by Wang, Yan and Yu [26], in order to prove (3.16), we first prove the followingLindeberg condition lim n →∞ n X i =1 n X j =1 E h ( Y ( n ) i,j ) {| Y ( n ) i,j | >ε } (cid:12)(cid:12)(cid:12) F ( n ) i − ,j − i = 0 (3.17)for all ε > (cid:16) n Z ini − n Z jnj − n K q,m ( ⌊ nt k ⌋ n , ⌊ st k ⌋ n , u, v ) ξ ( n ) i,j,m dudv (cid:17) ≤ n (cid:0) ξ ( n ) i,j,m (cid:1) (cid:18) Z ini − n Z jnj − n (cid:12)(cid:12) K q,m ( ⌊ nt k ⌋ n , ⌊ st k ⌋ n , u, v ) (cid:12)(cid:12) dudv (cid:19) ≤ Cn ( ξ ( n ) i,j,m (cid:1) Z ini − n Z jnj − n (cid:12)(cid:12) K q,m ( ⌊ nt k ⌋ n , ⌊ st k ⌋ n , u, v ) (cid:12)(cid:12) dudv. (3.18)It is easy to verify that there exists some δ > λ D − δ > Z ini − n (cid:13)(cid:13) ( t − u ) D − I + (cid:13)(cid:13) du ≤ C Z n − n (1 − u ) λ D − − δ + du, (3.19)since 0 < λ D − δ < t ∈ [0 , FBMS and Martingale Differences Noting the form of K , we get from (3.18) and (3.19) that (cid:16) n Z ini − n Z jnj − n K q,m ( ⌊ nt k ⌋ n , ⌊ st k ⌋ n , u, v ) ξ ( n ) i,j,m dudv (cid:17) ≤ Cn ( ξ ( n ) i,j,m ) δ n , (3.20)where δ n = Z n − n (1 − u ) λ D − − δ + du. It follows from (3.18) and (3.20) that (cid:16) Y ( n ) i,j (cid:17) ≤ C d X q =1 Q X k =1 n a k b q ( ξ ( n ) i,j,m ) δ n . (3.21)On the other hand, {| Y ( n ) i,j | > ε } = {| Y ( n ) i,j | > ε } . (3.22)Hence, from (3.21) and (3.22), n | Y ( n ) i,j | > ε o ⊆ n Cn ( ξ ( n ) i,j,m ) δ n > ǫ o . (3.23)Consequently, E (cid:20) ( Y ( n ) i,j ) {| Y ( n ) i,j | >ε } (cid:12)(cid:12)(cid:12) F ( n ) i − ,j − (cid:21) ≤ C E (cid:20) n ( ξ ( n ) i,j,m ) δ n { Cn ( ξ ( n ) i,j,m ) δ n >ε } (cid:12)(cid:12)(cid:12) F ( n ) i − ,j − (cid:21) ≤ Cδ n E (cid:20) { Cn ( ξ ( n ) i,j,m ) δ n >ε } (cid:12)(cid:12)(cid:12) F ( n ) i − ,j − (cid:21) (3.24)for all i, j = 1 , , ..., n . Hence, from (3.1) and (3.24), n X i =1 n X j =1 E (cid:20) ( Y ( n ) i,j ) {| Y ( n ) i,j | >ε } (cid:12)(cid:12)(cid:12) F ( n ) i − ,j − (cid:21) ≤ n X i =1 n X j =1 Cδ n E (cid:20) { Cn ( ξ ( n ) i,j,m ) δ n >ε } (cid:12)(cid:12)(cid:12) F ( n ) i − ,j − (cid:21) ≤ Cδ n n X i =1 n X j =1 E [1 { Cδ n >ε } ] → n → ∞ ) , because δ n → { Cδ n >ε } = 0 for large enough n .In order to prove (3.14), we also need to show that n X i =1 n X j =1 h Y ( n ) i,j i P → E h d X q =1 Q X k =1 Z Z a k b q K q,m ( t k , s k , u, v ) B m ( du, dv ) i , (3.25)11 where P → denotes convergence in probability. For convenience, we define˜ B m ( t, s, u, v ) = d X q =1 b q K q,m ( t, s, u, v ) . Note that the right-hand side of (3.25) is equivalent to Q X i,j =1 a i a j Z Z ˜ B m ( t i , s i , u, v ) ˜ B m ( t j , t j , u, v ) dudv. (3.26)Next, we look at the left-hand side of (3.25). In fact, we have Y ( n ) i,j = Q X k =1 a k Z ini − n Z jnj − n ˜ B m ( ⌊ nt k ⌋ n , ⌊ ns k ⌋ n , u, v ) ξ ( n ) i,j,m dudv. (3.27)Hence, (cid:16) Y ( n ) i,j (cid:17) = (cid:0) ξ ( n ) i,j,m (cid:1) Q X k,l =1 a k a l Z ini − n Z jnj − n ˜ B m ( ⌊ nt k ⌋ n , ⌊ ns k ⌋ n , u, v ) dudv · Z ini − n Z jnj − n ˜ B m ( ⌊ nt l ⌋ n , ⌊ ns l ⌋ n , u, v ) dudv. (3.28)Here, we point out that the entry K q,m ( t, s, u, v ) takes the form of d X i =1 ˜ K q,i ( t, u ) ˜ K q,i ( s, v ) ˜ K i,m ( t, u ) ˜ K i,m ( s, v ) . Hence, it follows from Lemma 3.4 and (3.26)-(3.28) that (3.25) holds.From the above arguments, we can easily get that the lemma holds. (cid:3)
Now, we prove Theorem 3.1.
Proof of Theorem 3.1:
Theorem 3.1 is a direct consequence of Lemmas 3.2 and 3.3,because tightness and the convergence of finite dimensional distributions imply weak con-vergence (see Bickel and Wichura [5]).
4. Final Note
In this work, based on the fractional Brownian motion of Riemann-Liouville type, weintroduce the operator fractional Brownian sheet of Riemann-Liouville type X and presentan approximation to it via martingale differences. In Definition 2.1, λ D and Λ D areassumed to be at (0 , X , λ D andΛ D can be at a larger interval than (0 , X to enjoy some nice properties. It follows from Mason and Xiao [22] thatfor an operator self-similar random filed { ˆ X ( t, s ) } with exponent ˆ D , if λ ˆ D >
0, thenˆ X (0 ,
0) = (0 , · · · , T a.s. Furthermore, if ˆ X (1 ,
0) is proper and E [ k ˆ X (1 , k ] < ∞ , thenΛ ˆ D ≤
1. In this paper, the operator fractional Brownian sheet of Riemann-Liouville type12
FBMS and Martingale Differences { X ( t, s ) } is assumed to be proper for ( t, s ) T = (1 , T , and X (0 ,
0) = (0 , · · · , T a.s.Hence, we assume 0 < λ D , Λ D < d = 1), a fractional Brownian sheet { W α,β ( t, s ) } with two parameters α, β ∈ (0 ,
1) can be defined as W α,β ( t, s ) = Z R f α ( t, u ) f β ( s, v ) ˜ B ( dv, du ) , (4.1)where f H ( t, u ) = ( t − u ) H − + − ( − u ) H − + . Hence in the one-dimensional case ( d = 1), X defined by (2.2) is a special kind of fractional Brownian sheets (with α = β ) of Riemann-Liouville type. In fact, inspired by (4.1), one could like to define the operator fractionalBrownian sheet of Riemann-Liouville type ˆ X = { ˆ X ( t, s ); ( t, s ) T ∈ R } byˆ X ( t, s ) = Z ∞ Z ∞ ( t − u ) D − I + ( s − v ) ˆ D − I + B ( du, dv ) , where ˆ D is a linear operator on R d with 0 < λ ˆ D , Λ ˆ D < X is well defined. However, in such case, we can not get Theorem 3.1according to our method. Acknowledgments
We thank the referee and the editor for their time and comments.Dai was supported by the National Natural Science Foundation of China (No.11361007),the Shandong Natural Science Foundation (No. ZR2014AM021) and the Fostering Projectof Dominant Discipline and Talent Team of Shandong Province Higher Education In-stitutions. Shen was supported by the National Natural Science Foundation of China(No.11271020), the Distinguished Young Scholars Foundation of Anhui Province (No.1608085J06), and the Top Talent Project of University Discipline (Speciality) (No. gxb-jZD03)
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