Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

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1 The Journal of Fourier Analysis and Applications Volume, Issue 4, 25 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets Antoine Ayache and Yimin Xiao Communicated by Christian Houdré ABSTRACT. Let B H ={B H t, t R N } be an N, d-fractional Brownian sheet with index H = H,...,H N, N. The uniform and local asymptotic properties of B H are proved by using wavelet methods. The Hausdorff and packing dimensions of the range B H [, ] N, the graph GrB H [, ] N and the level set are determined.. Introduction For a given vector H = H,...,H N <H < for =,...,N, a one-dimensional fractional Brownian sheet B H ={B H t, t RN } with Hurst index H is a real-valued, centered Gaussian random field with covariance function given by E [ B H sbh t] = N s 2H + t 2H s t 2H, s,t R N.. 2 It follows from. that B H is an anisotropic Gaussian random field and B H t = a.s. for every t R N with at least one zero coordinate. Moreover, B H has the following invariance properties: i B H is self-similar in the sense that for all constants c>, {B } { H ct, t RN d N = H c } B H t, t RN..2 Math Subect Classifications. 6G5, 6G7, 42C4, 28A8. Keywords and Phrases. Fractional Brownian sheet, wavelet representation, modulus of continuity, Hausdorff dimension, packing dimension, range, graph, level set. Acknowledgements and Notes. Second author was partially supported by NSF grant DMS Birkhäuser Boston. All rights reserved ISSN DOI:.7/s

2 48 Antoine Ayache and Yimin Xiao ii Let W H ={W H t, t R N } be the Gaussian random field defined by { N W H t 2H B H t = t,...,t N if t = for all otherwise. Then, W H d = B H..3 In the above, = d means equality in the finite dimensional distributions. Fractional Brownian sheet has the following stochastic integral representation B H t = κ H t tn gt, swds,.4 where W ={Ws,s R N } is the standard real-valued Brownian sheet and gt, s = N [ t H /2 ] H /2 s + s + with s + = max{s, }, and where κ H is the normalizing constant given by [ κ 2 = N 2 g,s] 2 ds H so that E [ B H t2] = N t 2H for all t R N. Here =,,..., R N. Let B H,...,BH d be d independent copies of BH. Then the N, d-fractional Brownian sheet with Hurst index H = H,...,H N is the Gaussian random field B H ={B H t : t R N } with values in R d defined by B H t = B H t,...,bh d t, t R N..5 Note that, if N =, then B H is a fractional Brownian motion in R d with Hurst index H, ; ifn > and H = = H N = /2, then B H is the N, d-brownian sheet. Hence, B H can be regarded as a natural generalization of one parameter fractional Brownian motion in R d to N, d Gaussian random fields, as well as a generalization of the Brownian sheet. Another well known generalization is the multiparameter fractional Brownian motion X ={Xt, t R N }, which is a centered Gaussian random field with covariance function E[X i sx t] = 2 δ i s 2H + t 2H s t 2H, s, t R N,.6 where <H < is a constant and δ i =, if i = and, if i =, and where denotes the Euclidean norm in R N. Fractional Brownian sheets arise naturally in many areas, including in stochastic partial differential equations cf. Øksendal and Zhang [26], Hu, Øksendal and Zhang [6] and in studies of most visited sites of symmetric Markov processes cf. Eisenbaum and Khoshnevisan [4]. Recently, many authors have studied various properties of the fractional Brownian sheets. For example, Dunker [] has studied the small ball probability of an N, - fractional Brownian sheet. For the special class of fractional Brownian sheets such that

3 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 49 there is a unique minimum among H,...,H N, Mason and Shi [25] have obtained the exact rate for the small ball probability and have computed the Hausdorff dimension of some exceptional sets related to the oscillation of their sample paths. Belinski and Linde [6] give a different proof of the small ball probability result of Mason and Shi [25] and have also obtained a sharp estimate for the small ball probability in the case of N = 2 and H = H 2. More generally, by using different kinds of s-numbers Kühn and Linde [2] have determined the rate, up to a logarithmic factor, for the small ball probability and the optimality of series representations for the fractional Brownian sheet B H with an arbitrary index H = H,...,H N, N. Ayache and Taqqu [4] have derived optimal wavelet series expansions for the fractional Brownian sheet B H ; see also Dzhaparidze and van Zanten [2] for other optimal infinite series expansions. Xiao and Zhang [36] have proved a sufficient condition for the oint continuity of the local times of an N, d-fractional Brownian sheet B H. Kamont [8] and Ayache [3] have studied the box-dimension and the Hausdorff dimension of the graph set of an N, -fractional Brownian sheet B H using wavelet methods. The main obective of this article is to further investigate the asymptotic and fractal properties of the N, d-fractional Brownian sheet B H. We are particularly interested in describing the anisotropic nature of B H in terms of the Hurst index H = H,...,H N, N. We should mention that several authors have been interested in applying anisotropic Gaussian random fields to stochastic modelling; see, for example, Bonami and Estrade [9] for bone structure modelling, and Benson et al. [7] for modelling aquifer structure in hydrology. We hope that the results and techniques in this article will be helpful for studying more general anisotropic Gaussian random fields. The rest of this article is organized as follows. In Section 2, we study the uniform and local modulus of continuity and the law of the iterated logarithm of an N, -fractional Brownian sheet. Many authors have studied the asymptotic behavior of the sample functions of Gaussian random fields see, e.g., Albin [2] and Kôno [2] and the reference therein. Our approach is different from those in the aforementioned references and relies on the wavelet expansion of B H in terms of a Lemarié-Meyer wavelet basis for L 2 R. We remark that, even though the methods of Albin [2] and Kôno [2] based on metric entropy may be modified to prove our Theorems and 2 below, our wavelet-based approach has certain advantages. In particular, it allows us to prove that, with probability, the sample function B H t does not satisfy any local Hölder conditions with respect to the function σ +ε t + h, t, for any ε >, where σ 2 t + h, t = E [ B H t + h B H t 2] ; see Theorem 3. This implies that B H t is nowhere differentiable on R N. Theorem 3 is more difficult to establish directly due to the complex dependence structure of B H. It is worthwhile to mention that another way of proving the non-differentiability of B H is by investigating the regularity of the local times of B H. This approach relies on solving Problem 4.2 in Xiao and Zhang [36] and requires totally different techniques. We will deal with it elsewhere. In Section 3, we determine the Hausdorff and packing dimensions of the range B H [, ] N, the graph GrB H [, ] N and the level set L x ={t, N : B H t = x}. Our method for determining the Hausdorff dimension of GrB H [, ] N is different from that in Ayache [3], and is more reminiscent to the arguments in Xiao [3]. Our results suggest that, due to the anisotropy of B H in t, the sample paths of an N, d-fractional Brownian sheet B H are more irregular than those of the Brownian sheet or an N, d- fractional Brownian motion. Finally, we introduce some notation. Throughout this article, the underlying parameter space is R N,orR N + = [, N. A typical parameter, t R N is written as

4 4 Antoine Ayache and Yimin Xiao t = t,...,t N, or occasionally, as c,ift = =t N = c. For any s, t R N such that s <t =,...,N, we define the closed interval or rectangle [s, t] = N [s,t ]. We use, and to denote the ordinary scalar product and the Euclidean norm in R m, respectively, no matter the value of the integer m. We will use c to denote an unspecified positive and finite constant which may not be the same in each occurrence. More specific constants in section i are numbered as c i,,c i,2,..., and so on. 2. Modulus of Continuity and Asymptotic Properties In this section, we investigate the uniform and local modulus of continuity, nowhere differentiability and the laws of the iterated logarithm of the fractional Brownian sheet B H ={B H t, t R N } with index H = H,H 2,...,H N,<H <. Our approach is based on the wavelet expansion of B H. For simplicity, we will suppose, in this section, that B H is real-valued i.e., d =, and we will use A, A,A 2,... to denote positive random variables. Let us first state the main results. Theorem. There exist a random variable A = A ω > of finite moments of any order and an event of probability such that for any ω, sup s,t [,] N B H s, ω B H t, ω N s t H log 3 + s t A ω. 2. Remark. Up to a constant, the inequality in Theorem is sharp. When H = =H N = 2 it agrees with the corresponding result for the Brownian sheet due to Orey and Pruitt [28], Theorem remains valid when [, ] N is replaced by any compact rectangle of R N. The event will be specified in the proof of Theorem. As we mentioned in the introduction, Theorem can also be proven using the method of proving Theorem in Kôno [2]. The proof we give below is based on the wavelet representation of B H. Next we give an upper bound for the asymptotic behavior of the fractional Brownian sheet B H t as t. Recall that, by the law of the iterated logarithm see, for example, Orey [27], the ordinary fractional Brownian motion {B H t, t R} with Hurst index H, satisfies with probability, B H t A + t H log log3 + t, t R, where A is a positive random variable. This result can be extended to the fractional Brownian sheet as follows.

5 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 4 Theorem 2. Let B H ={B H t, t R N } be a fractional Brownian sheet with index H = H,...,H N and let be the event of probability in Theorem. Then, there is a random variable A 2 > of finite moments of any order such that for all ω, B H t, ω N + t H log log3 + t A 2ω. 2.2 sup t R N Note that 2.2 is more concerned with the global property of B H t and it does not catch the local asymptotic behavior of B H t near t =. To study the asymptotic properties of B H at a fixed point t R N, we consider σ 2 s, t = E[ B H t B H s 2 ], s, t R N. 2.3 It follows from the proof of Lemma 8 below that for any closed interval I =[a,b], there is a finite constant c 2, > such that for all s, t I, σt,s c 2, N t s H. 2.4 Hence, one can apply the metric entropy method to prove a law of the iterated logarithm for B H. The following is a consequence of the proof of Theorem 2 in Kôno [2]. Proposition. There exists a positive and finite constant c 2,2 such that for every t I, with probability B H t + h B H t lim sup c h σt,t + h log log h 2, The next result is a partial inverse of 2.5 and implies that, for every ε>, the sample function B H t does not satisfy σ +ε -Hölder condition at any point, where σt,s is defined in 2.3. Theorem 3. Let B H ={B H t, t R N } be a real-valued fractional Brownian sheet with index H = H,...,H N. Then, there is an event 2 of probability such that for all ɛ>, t, ] N and all ω 2, lim sup h B H t + h, ω B H t, ω σt + h, t +ɛ =. 2.6 In order to prove Theorems, 2, and 3, we will use the wavelet representation of the fractional Brownian sheet introduced in Ayache [3]. To this end, we need to introduce the following notation. {2 J/2 ψ2 J x K,J,K Z Z} will be a Lemarié-Meyer wavelet basis for L 2 R see, for instance, Lemarié and Meyer [23], or Meyer [24]. Recall that such orthonormal bases satisfy the following properties: a ψ and its Fourier transform ψ belong to the Schwartz class SR, namely the space of all infinitely differentiable functions u which verify for all integers n and m, lim tm d nut =. t dt

6 42 Antoine Ayache and Yimin Xiao Recall that the Fourier transform of a function f L R is defined as fξ= R e iξ x fxdx for all ξ R. b ψ is even, compactly supported and vanishes in a neighborhood of the origin. More precisely, the support of ψ is contained in the domain {ξ : ξ 8π 3 }. 2π 3 For any α,, the functions ψ α and ψ α will denote, respectively the fractional primitive of order α + 2 and the fractional derivative of order α + 2 of the mother wavelet ψ, which are defined for all x R as ψ α x= e ixξ ψξ 2π R ξ α+/2 dξ and ψ α x= e ixξ ξ α+/2 ψξdξ π R In view of properties a and b of the Lemarié-Meyer wavelets, these definitions make sense and ψ α and ψ α are real-valued and belong to the Schwartz class SR. Moreover, we have for every ξ R, ψ α ξ = ψξ ξ α+/2 and ψ α ξ = ξ α+/2 ψξ. 2.8 For any H = H,...,H N, N and for all =,..., N Z N, k = k,...,k N Z N and t = t,...,t N R N,weset H,k t = N l= where for any α,, x R and any J, K Z 2, ψ H l l,k l t l, 2.9 ψ α J,K x = ψα 2 J x K ψ α K. 2. We are now in a position to give the wavelet representation of fractional Brownian sheet. Proposition 2 Ayache [3]. There is a sequence {ɛ,k,,k Z N Z N } of independent N, Gaussian random variables such that for any H = H,...,H N, N, the fractional Brownian sheet B H ={B H t, t R N } can be represented up to a multiplicative constant that only depends on H as: B H t, ω = 2,H ɛ,k ω H,k t, t RN, 2., k Z N Z N where,h = N l= l H l and where the series in 2. is, for each t, convergent in the L 2 -norm being the underlying probability space. In fact, the series 2. is also convergent in a much stronger sense. More precisely, we have the following. Proposition 3. For almost all ω, the series 2. is uniformly convergent in t, on any compact subset of R N. Proposition 3 can be obtained by using the same method as that of Ayache and Taqqu [4]. But since we are not interested in determining the rate of convergence of the series 2., this proposition can also be proved more simply as follows.

7 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 43 Proof. For simplicity we will only prove that, with probability, the series 2. converges uniformly in t [, ] N. For any n N and t R N, let Bn H t = 2,H ɛ,k H,k t, 2.2, k I n where I n = {, k Z N Z N : l nand k l nfor all l N }. Since the functions H,k are continuous and the random variables ɛ,k are symmetric and independent, the Itô-Nisio Theorem see Theorem 2.. in Kwapień and Woyczyński [22] implies that, for proving the uniform convergence of series 2. on [, ] N, it is sufficient to show that the sequence {Bn H t, t [, ]N } n N is weakly relatively compact in C[, ] N, the space of continuous functions on [, ] N equipped with the usual topology of uniform convergence. Observe that 2., 2.3, and 2.4 imply that there is a constant c 2,3 > such that for all n and all t, t [, ] N, B H E[ n t Bn H t ] 2 c t 2,3 t 2H, 2.3 where H = min{h,...,h N }. Since 2.3 and Theorem 2.3 in Billingsley [8] entail the weak relative compactness of the sequence {B H n t, t [, ]N } n N in C[, ] N, this finishes the proof of Proposition 3. The proof of Theorem mainly relies on the following two technical lemmas. The proof of Lemma is similar to that of Lemma 4 of Ayache [3], or Lemma 2 of Ayache and Taqqu [4]. Hence, it is omitted. Lemma. Let {ɛ m,m Z N } be a sequence of N, Gaussian random variables. Then, there is a random variable A 3 >, of finite moments of any order such that for almost all ω and for all m = m,...,m N Z N, N ɛ m ω A 3 ω log 3 + m i. 2.4 We will also make use of the following elementary inequality: There is a constant c>such that for all m,...,m N Z N, N N log 3 + m i c log3 + mi. 2.5 i= Lemma 2. For any α,, define the functions S α x, y = ψ α 2 J x K ψ α 2 J y K log3 + J + K 2.6 and J,K Z 2 2 Jα T α x = J,K Z 2 2 Jα i= i= ψ α J,K x log3 + J + K. 2.7 Then there is a constant c 2,4 > such that for all x, y [, ], one has S α x, y c 2,4 x y α log 3 + x y 2.8

8 44 Antoine Ayache and Yimin Xiao and T α x c 2, Proof. First we prove 2.8. Since the function ψ α belongs to the Schwartz class SR, its derivative of any order n, satisfies nψ x c dx 2,5 3 + x 2, x R, 2.2 where c 2,5 > is a constant that only depends on n. For any x, y [, ], satisfying x = y, there is a unique integer J such that 2 J < x y 2 J. 2.2 We decompose S α x, y into the following 3 parts: S α, x, y= 2 Jα ψ α 2 J x K ψ α 2 J y K log3 + J + K, 2.22 J = K= J S α,2 x, y= 2 Jα ψ α 2 J x K ψ α 2 J y K log3 + J + K, 2.23 J = K= S α,3 x, y= 2 Jα ψ α 2 J x K ψ α 2 J y K log3 + J + K 2.24 J =J + K= and derive upper bounds for S α, x, y, S α,2 x, y and S α,3 x, y separately. Without loss of generality, we will assume x<y. It follows from the Mean-Value Theorem that for any integers <J J and K Z, there is ν 2 J x,2 J y, such that ψ α 2 J x K ψ α 2 J y K = 2 J x y dψα ν K dx By using 2.2 and 2.2 we derive dψα dx ν K c2,5 3 + ν K 2 c 2, J x K ν 2 J x 2 c 2, J x K To estimate S α, x, y, we note that for all integers J, 2.26 gives dψα dx ν K c2,5 2 + K 2 J x 2 c2,5 + K So 2.22, 2.25, and 2.27 entail that S α, x, y c 2,6 x y, 2.28 where the constant c 2,6 = c K= 2 J α log3+ J + K 2,5 J =. + K 2

9 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 45 Next we proceed to derive an upper bound for S α,2 x, y. Combining 2.23, 2.25, 2.26, and 2.5 yields J S α,2 x, y c 2,5 x y = c 2,5 x y J = K= J c 2,5 x y J = K= J J = K= J 2 J α log3 + J + K J x K 2 2 J α log 3 + J + K + 2 J x J x 2 J x K 2 2 J α log 3 + J + 2 J + K + K 2 c 2,7 x y 2 log J α 3 + J + 2 J. J = 2.29 In the above, z denotes the integer part of z and <c 2,7 < is a constant. Note that for any J N, log 3 + J + 2 J log 2 J J J It follows from 2.29, 2.3, and 2.2 that S α,2 x, y c 2,7 x y 2 J α J + 3 J J = c 2,8 x y 2 J + α J c 2,9 x y α 3 + log x y. Now, let us give an upper bound for S α,3 x, y. For every z [, ], let θ J z = 2 Jα ψ α 2 J z K log3 + J + K J =J + K= Then for every x, y [, ], S α,3 x, y θ J x + θ J y Thus, it is sufficient to bound θ J z, uniformly in z [, ]. Note that for any fixed J N and any z [, ], 2.2 and 2.5 imply that ψ α 2 J z K log3 + J + K log3 + J + K c 2, J z K 2 K= = c 2,5 K= log 3 + J + K + 2 J z J z 2 J z K 2 log 3 + J + 2 J + K K= 2.34 c 2,5 K= c 2, + J, 2 + K 2

10 46 Antoine Ayache and Yimin Xiao where c 2, is a finite constant. It follows from 2.34, 2.32 and some simple calculations that for every z [, ], θ J z c 2, J =J + 2 Jα + J c 2, 2 αx 2 + xdx c 2, 2 J α + J. It is clear that 2.33, 2.35, and 2.2 entail that, J 2.35 S α,3 x, y c 2,2 x y α log 3 + x y, 2.36 where c 2,2 > is a constant independent of x and y. Combining 2.28, 2.3, and 2.36 yields 2.8. To prove 2.9, we observe that, by 2., T α x = S α x, for every x [, ]. Therefore, 2.9 follows from 2.8 immediately. We are now in a position to prove Theorem. Proof of Theorem. Observe that for any s, t [, ] N, one has that B H t B H s N B H s,...,s i,t i,t i+,...,t N B H s,...,s i,s i,t i+,...,t N 2.37 i= with the convention that, when i =, B H s,...,s i,t i,t i+,...,t N = B H t and when i = N, B H s,...,s i,s i,t i+,...,t N = B H s. Another convention that will be used in the sequel is that, when i =, i l= ψ H l l,k l s l =and when i = N, Nl=i+ ψ H l l,k l t l =. Let be the event of probability on which Proposition 3 and 2.4 hold. For every fixed integer i N, it follows from Propositions 2 and 3, Lemmas and 2, 2. and 2.5 that for every ω and s, t [, ]N, B H s,...,s i,t i,t i+,...,t N B H s,...,s i,s i,t i+,...,t N i N A 4 T Hl s l T Hl t l S Hi s i,t i 2.38 l= A 5 s i t i H i l=i+ log s i t i, where the random variables A 4 > and A 5 > are of finite moments of any order. Finally, Theorem follows from inequalities 2.37 and Now we prove Theorem 2. Proof of Theorem 2. For any α,, let T α x be defined by 2.7. We first show that there is a constant c 2,3 >, depending on α only, such that for all x R, T α x c 2,3 + x α log log3 + x. 2.39

11 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 47 It follows from 2.9 that the inequality 2.39 is satisfied when x. It remains to show that it is also true when x >. Our approach is similar to the proof of Lemma 2. For any x >, we choose an integer J such that 2 J x < 2 J and then write T α x as the sum of the following 3 parts: and T α, x = T α,2 x = T α,3 x = J J = K= J = J K= J = K= 2 Jα ψ α J,K x log3 + J + K, Jα ψ α J,K x log3 + J + K Jα ψ α J,K x log3 + J + K First, let us derive an upper bound for T α, x. For any integers <J J and K Z, 2., the Mean-Value Theorem and 2.2 imply that ψ J,K α x c 2,5 2 J x 3 + ν K 2 c 2,5 2J x 2 + K Putting 2.44 into 2.4 yields T α, x c 2,4 x 2.45 for some finite constant c 2,4 >. To derive an upper bound for T α,2 x, we note that T α,2 x R α,2 x + R α,2, 2.46 where R α,2 x = J J = K= Applying 2.2 with n = and 2.5, we have J R 2,α x c 2,5 2 Jα ψ α 2 J x K log3 + J + K J = K= 2 Jα log3 + J + K J x K 2 J c 2,5 2 log Jα 4 + J + 2 J x. J = Hence, with some elementary computation, we have J R α,2 x c 2,6 2 log αt 5 + t + 2 t x dx c 2,7 2 J α log 5 + J + 2 J x c 2,8 + x α log log3 + x,

12 48 Antoine Ayache and Yimin Xiao where the last inequality follows from 2.4. Similarly, 2.2 and 2.5 imply that R α,2 = J J = K= 2 Jα ψ α K log3 + J + K J c 2,9 2 Jα log3 + J J = c 2,2 + x α loglog3 + x. 2.5 Combining 2.46, 2.47, and 2.5 we obtain that for any x >, T α,2 x c 2,2 + x α log log3 + x. 2.5 Now, let us derive an upper bound for T α,3 x. It follows from 2.43 and 2. that for all x R, where for any x R, R α,3 x = J = K= T α,3 x R α,3 x + R α,3, 2.52 Using 2.2 again one obtains that for all x R, R α,3 x c 2,5 = c 2,5 c 2,22 2 Jα ψ α 2 J x K log3 + J + K J = K= J = K= J = 2 Jα log3 + J + K J x K 2 2 Jα log 3 + J + 2 J x + K J x 2 J x K 2 2 Jα log 3 + J + 2 J x Note that for any x R, log 3 + J + 2 J x log 3 + J + 2 J + log3 + x It follows from 2.52, 2.53, 2.54, and 2.55 that for any x R, T α,3 x c 2,23 log3 + x It is clear that 2.56, 2.5, and 2.45 entail Finally, 2., 2.9, 2.39, 2.5, and Lemma imply that there is a random variable A 6 > of finite moments of any order such that for all ω and t = t,...,t N R N, B H t, ω A 6 N T H t c 2,24 A 6 N + t H log log3 + t.

13 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 49 This finishes the proof of Theorem 2 with A 2 = c 2,24 A 6. Remark 2. Observe that while proving Theorem 2, we have obtained the following result which will be used in the proof of Proposition 4. Namely, there is a random variable A 2 > of finite moments of any order such that for all n, t = t,...,t N R N and ω, one has B n H t, ω ɛ,k ω,k t,k Z N Z N 2,H A 2 ω N + t H log log3 + t The rest of this section is devoted to the proof of Theorem 3. First, we need to fix some more notation. For any λ = λ,λ 2,...,λ N R N and for each integer n N, we denote by λ n the vector of R N defined as λ n = λ,...,λ n,λ n+,...,λ N 2.58 with the convention that λ = λ 2,λ 3,...,λ N and λ N = λ,λ 2,...,λ N. For convenience, we may sometimes write λ as λ n, λ n [see, e.g., 2.69]. For any H = H,...,H N, N, =,..., N Z N, k = k,...,k N Z N and t = t,...,t N R N,weset Ĥ n ĵ n, k n tn = n l= ψ H l l,k l t l N l=n+ ψ H l l,k l t l 2.59 with the convention that Ĥ t ĵ, k = N l=2 ψ H l l,k l t l and Ĥ N t ĵ N, k N = N N l= ψ H l l,k l t l. Recall that the function ψ H l l,k l t l is defined in 2.. Let us now introduce a wavelet transform that allows us to construct a sequence of independent and identically distributed fractional Brownian sheets on R N starting from a fractional Brownian sheet on R N. Proposition 4. Let B H ={B H t, t R N } be an N, -fractional Brownian sheet with index H = H,...,H N. For every n {,...,N}, n,k n Z Z and we define t n = t,...,t n,t n+,...,t N R N, C n,k n tn = 2 n +H n R B H tψ H n 2 n t n k n dtn, 2.6 where ψ H n is the wavelet introduced in 2.7. Then C n,k n n,k n Z Z is a sequence of independent and identically distributed fractional Brownian sheets on R N with index Ĥ n. Proof. Note that Property b of Lemarié-Meyer wavelets and 2.8 imply that ψ H n =. Therefore one has that, ψ H n 2 n t n k n dtn =. 2.6 R

14 42 Antoine Ayache and Yimin Xiao For every p, q Z N Z N and for every t n R N, define I t n ; p, q; n,k n = 2 n p,q H tψ H n 2 n t n k n dtn We claim that R I t n ; p, q; n,k n = δpn,q n ; n,k n Ĥ n p n, q tn n, 2.63 where δp n,q n ; n,k n =, if p n,q n = n,k n and δp n,q n ; n,k n = otherwise. To verify 2.63, note that 2.9, 2., 2.59, and 2.6 imply that 2 n R H p,q tψ H n 2 n t n k n dtn = 2 n Ĥ n p n, q n tn R ψ H n 2 p n t n q n ψ H n 2 n t n k n dtn Since the functions ψ H n and ψ H n are real-valued, it follows from Parseval s formula, 2.8 and the orthonormality of the Lemarié-Meyer wavelets 2 J/2 ψ2 J x K, J Z, K Z, that 2 n ψ H n 2 p n t n q n ψ H n 2 n t n k n dtn R = 2 p n e i2 pn q n 2 n k n ξ n ψh n 2 p n ξ ψ H n n 2 n ξ n dξn R = 2 H n+/2p n n p n e i2 pn q n 2 n k n ξ n ψ 2 p n ξ n ψ 2 nξ n dξn R = 2 H n+/2p n n + n ψ 2 p n t n q n ψ 2 n t n k n dtn R = 2 H n+/2p n n δp n,q n ; n,k n Combining 2.64 and 2.65 yields Next it follows from Proposition 3, 2.6, 2.62, and 2.63, that almost surely, for every t n R N, C n,k n tn = 2 p,h ɛ p,q I tn ; p, q; n,k n p,q Z N Z N = 2 pn,ĥn ɛ n, p n,k n, q n Ĥn 2.66 p n, q tn n. p n, q n Z N Z N Observe that we are allowed to interchange the order of integration and summation in deriving the first equality in 2.66 because the function t n ψ H n2 nt n k n belongs to SR, the partial-sum processes Bm H t [which are defined in 2.2] converge to BH t uniformly on all compact sets and because of Also, observe that Propositions 2 and 3 entail that, with probability, the series 2.66 is uniformly convergent in t n,on any compact of R N and that the Gaussian field {C n,k n t n, t n R N } is a fractional Brownian sheet on R N, with index Ĥ n. Finally, observe that the Gaussian random sequences {ɛ n, p n,k n, q n, p n, q n Z N Z N } n,k n Z Z are independent and have the same distribution, so are the fractional Brownian sheets {C n,k n t n, t n R N }, n,k n Z Z.

15 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 42 Let us show that the increments of the Gaussian field {C n,k n t n, t n [, ] N } can be controlled uniformly in the indices n and k n. Lemma 3. Let be the event of probability in Theorem. For every n N and x,y [, ] N, let τ n xn, ŷ n = x l y l H l log 3 + x l y l l =n Then there is a random variable A 7 > of finite moments of any order, such that for every n N, k n {,,...,2 n}, x n [, ] N, ŷ n [, ] N and ω, C n,k n xn,ω C n,k n ŷn,ω A 7 ω 2 nh n τ n xn, ŷ n Proof. Following the same lines as in the proofs of Theorems and 2, one can show that there is a random variable A 8 >, of finite moments of any order, such that for every t n R, x n [, ] N, ŷ n [, ] N and ω, one has B H t n, x n,ω B H t n, ŷ n,ω A8 ω τ n xn, ŷ n + tn H n log log3 + t n It follows from 2.6 and 2.69 that for all n N, k n {,,...,2 n}, x n [, ] N and ŷ n [, ] N, Cn,k n xn,ω C n,k n ŷn,ω 2 n+h n B H t n, x n,ω B H t n, ŷ n,ω ψ H n 2 n t n k n dtn 2.7 R A 8 ω 2 n+h n τ n xn, ŷ n R + t n H n log log3 + t n ψ H n 2 n t n k n dtn, where the last integral is finite since ψ H n belongs to SR. Moreover, the integral I n,k n = 2 n + t n H n log log3 + t n ψ H n 2 n t n k n dtn R can be bounded independently of n N and k n {,,...,2 n}. Indeed, by setting u = 2 nt n k n, we derive that I n,k n 2 + u H n log log4 + u ψ H n u du <. 2.7 R Thus, 2.69 follows from 2.7 and 2.7. In the following, without loss of generality, we will suppose that H = min{h,...,h N } Lemma 4. Let be the event with probability in Theorem. If there exist some ɛ>, s = s, ŝ, ], ] N and ω such that B H s + h, ω B H s, ω lim sup h σs + h, s +ɛ <, 2.73

16 422 Antoine Ayache and Yimin Xiao then there is a constant c 2,25 > only depending on s, ɛ and ω such that for every N and k {,,...,2 }, C,k ŝ,ω c 2,25 2 ɛh + 2 s k +ɛh Proof. When 2.73 holds, we can find two constants c 2,26 > and η> [depending on s, ɛ and ω] such that for all h R N satisfying h η, B H s + h, ω B H s, ω c 2,26 σs + h, s +ɛ In particular, for h = h,,..., with h η, wehave B H s + h, ŝ,ω B H s, ŝ,ω c 2,26 h H +ɛ Observe that Theorem 2 implies that by increasing the value of c 2,26, 2.76 holds for all h R. Hence, it follows from 2.6, 2.6, and 2.76 that for every n N and k n {,,...,2 n}, C,k ŝ,ω 2 +H B H t, ŝ B H s, ŝ ψ H 2 t k dt R c 2,26 2 +H t s H +ɛ ψ H 2 t k dt. R Setting u = 2 t k in this last integral, one obtains that C,k ŝ,ω c2,26 2 H 2 u + 2 k s H +ɛ ψ H u du R c 2,27 2 ɛh u H +ɛ ψ H u du R + c 2,27 2 ɛh 2 s k H +ɛ ψ H u du c 2,25 2 ɛh + 2 s k +ɛh, for some finite constant c 2,25 >. This proves The proof of Theorem 3 mainly relies on the following technical lemma, which is, to a certain extent, inspired by Lemma 4. in Ayache, Jaffard, and Taqqu [5]. Lemma 5. There is an event 3 with probability satisfying the following property: For all arbitrarily small η>, ɛ>and for all ω 3, there exist a real number ν> and an integer, such that for all integers,, k {,,...,3 } N with 3 k [η, ] N and for all t [, ], there exists k {,,...,2 } such that t 2 k 2 H ɛ R and C,k 3 k,ω ν Proof. Let us fix two constants ɛ, η,. Since the distribution of the fractional Brownian sheet {C,k x, x [η, ] N } is independent of the indices and k see

17 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 423 Proposition 4, there is a constant ν η >, depending on η only, such that for all x [η, ] N, N and k {,,...,2 }, Var C,k x νη Thus, for all N, k {,,...,2 } and x [η, ] N, P C,k x ν η 2 π. 2.8 For any N,weset { D η, = 2 k : k {,,...,3 } N and 3 k [η, ] N } 2.8 and 2 L ɛ, = 2 H ɛ, 2.82 where x is the integer part of x. Observe that each k {,,...,2 } can be written as k = 2 H ɛ q + r, 2.83 where q {,,...,L ɛ, } and r {,,..., 2 H ɛ }. Also observe that for all t [, ] and N, there is q {,,...,L ɛ, } and r {,,..., 2 H ɛ } such that 2 H ɛ t q + r 2. Hence, we have for all r {,,..., 2 H ɛ }, 2 H ɛ t q + r 2 ɛh and 2.77 follows from this last inequality. Now we consider the event E η,ɛ, defined by E η,ɛ, = L ɛ, d D η, q= 2 H ɛ r= { C, 2 H ɛ q+r d ν η } It follows from the independence of the random variables C, 2 H ɛ q+r d, see Proposition 4, 2.8, 2.8, and 2.82 that P E η,ɛ, 3 + N 2 2 H ɛ Lɛ, + π exp c 2,28 + log 2 2 H ɛ. π Since = PE η,ɛ, <, the Borel-Cantelli Lemma implies that P =, Eη,ɛ, c J = =J

18 424 Antoine Ayache and Yimin Xiao where E c η,ɛ, denotes the complementary of the event E η,ɛ,. Finally, we take 3 = m N n N J = and it has the desired properties of Lemma 5. =J E c m+, n+, The following lemma is a consequence of Lemmas 3 and 5. Lemma 6. Let 2 = 3. Then, for all ɛ>small enough, t, t, ], ] N and all ω 2, there exist ν>and an integer, satisfying the following property: For every integer,, there exists k {,,...,2 } such that t 2 k 2 H ɛ and C,k t,ω ν/ Proof. Let us fix t = t, t, ], ] N, ɛ> small enough and ω 2. Observe that there is an η> such that t, t, ] [η, ] N, and for every N, there is a k {,,...,3 } N with 3 k [η, ] N, such that t 3 k N Lemma 5 implies that for some real ν>, integer, and every integer, there is k {,,...,2 } satisfying t 2 k 2 H ɛ and C,k 3 k, ω ν On the other hand, applying Lemma 3 with n =, together with 2.72 and 2.87, one obtains that for any <θ<h, there is a constant c 2,29 > such that for all N, C,k t,ω C,k 3 k, ω 2 H θ c 2,29 2 θ Combining 2.89 and 2.9, we have that for all,, C,k t,ω C,k 3 k, ω C,k t,ω C,k 3 k, ω ν c 2, H θ 2 θ. 2.9 By choosing θ> small enough so that lim 23 H θ 2 θ =, we see that 2.9 implies the existence of an integer,, such that 2.86 holds for all,.

19 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 425 We are now in a position to prove Theorem 3. Proof of Theorem 3. Let 2 be the event in Lemma 6. Suppose, ad absurdum, that there is ɛ>small enough, t = t, t, ], ] N and ω 2 such that B H t + h, ω B H t, ω lim sup h σt + h, t +ɛ <. Then, it follows from Lemma 4 that there is a constant c 2,25 > such that for every N and k {,,...,2 }, C,k t,ω c 2,25 2 ɛh + 2 t k +ɛh On the other hand, Lemma 6 implies the existence of k {,,...,2 } satisfying 2.85, 2.86, and Hence, for every integer,, <ν/2 c 2,25 2 ɛh + 2 s k +ɛh c 2,25 2 ɛh + 2 ɛh + +ɛh Since ɛ>can be taken arbitrarily small, one may suppose that + ɛh <. Then one obtains that lim 2 ɛh + 2 ɛh + +ɛh =, which contradicts This finishes the proof of Theorem Hausdorff Dimension of the Range, Graph, and Level Sets In this section, we study the Hausdorff and packing dimensions of the range B H [, ] N = { B H t : t [, ] N }, the graph GrB H [, ] N = { t, B H t : t [, ] N } and the level set L x = {t, N : B H t = x}, where x R d is fixed. We refer to Falconer [5] for the definitions and properties of Hausdorff and packing dimensions. In order to state our theorems conveniently, we further assume <H... H N <. 3. Theorem 4. Let B H ={B H t, t R N + } be an N, d-fractional Brownian sheet with Hurst index H = H,...,H N satisfying 3.. Then with probability, dim H B H [, ] N N = min d; 3.2 H and dim H GrB H [, ] N k =min N H if = k H k H + N k + H k d if H k H +N k + H k d, k N; N H d, N H k H d< k H, 3.3

20 426 Antoine Ayache and Yimin Xiao where H :=. Remark 3. When d =, 3.3 coincides with the result of Ayache [3]. The second equality in 3.3 is verified by the following lemma, whose proof is elementary and is omitted. Denote k H k N κ := min + N k + H k d, k N; H. Lemma 7. Assume 3. holds. We have i If d N H, then κ = N H. ii If l H d< l H for some l N, then κ = l and κ N l + d,n l + d + ]. H H l H + N l + H l d 3.4 As usual, the proof of Theorem 4 is divided into proving the upper and lower bounds separately. In the following, we first show that the upper bounds in 3.2 and 3.3 follow from Theorem and a covering argument. Proof of the upper bounds in Theorem 4. For the proof of the upper bound in 3.2, we note that clearly dim H B H [, ] N d a.s., so we only need to prove the following inequality: dim H B H [, ] N N H a.s. 3.5 For any constants <γ <γ <H N, it follows from Theorem that there is a random variable A of finite moments of all orders such that for all ω, sup s,t [,] N B H s, ω B H t, ω N s t γ A ω. 3.6 Let ω be fixed and then suppressed. For any integer n 2, we divide [, ]N into m n sub-rectangles {R n,i } with sides parallel to the axes and side-lengths n /H =,...,N, respectively. Then N m n c 3, n H. 3.7 and B H [, ] N can be covered by B H R n,i i m n and, by 3.6, we see that the diameter of the image B H R n,i satisfies diamb H R n,i c 3,2 n +δ, 3.8

21 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 427 where δ = max{h γ /H, N}. We choose γ γ,h for each such that δ N γ > N H. Hence, for γ = N γ, it follows from 3.7 and 3.8 that m n i= [ diamb H R n,i ] γ N c 3,3 n H n δγ as n. This proves that dim H B H [, ] N γ a.s. By letting γ H along rational numbers, we derive 3.5. Now we turn to the proof of the upper bound in 3.3. We will show that there are several different ways to cover GrB H [, ] N, each of which leads to an upper bound for dim H GrB H [, ] N. For each fixed integer n 2, we have GrB H [, ] N m n R n,i B H R n,i. 3.9 i= It follows from 3.8 and 3.9 that GrB H [, ] N can be covered by m n cubes in R N+d with side-lengths c 3,4 n +δ and the same argument as the above yields dim H GrB H [, ] N N H a.s. 3. We fix an integer k N. Observe that each R n,i B H R n,i can be covered by l n,k cubes in R N+d of sides n H k, where by 3.6 N=k H l n,k c 3,5 n k H n H +δd k. Hence, GrB H [, ] N can be covered by m n l n,k cubes in R N+d with sides n H k. Denote k H k η k = + N k + γ k d. H Recall from the above that we can choose the constants γ k and γ N such that δ> γ k H k. Some simple calculations show that m n l n,k n H k ηk c3,6 n δ γ k H k d as n. This implies that dim H GrB H [, ] N η k almost surely. Therefore for every k =,...,N, dim H GrB H [, ] N k H k H + N k + H k d. 3.

22 428 Antoine Ayache and Yimin Xiao Combining 3. and 3. yields the upper bound in 3.3. For proving the lower bounds in Theorem 4, we need several lemmas. Lemma 8. For any ε>, there exist positive and finite constants c 3,7 and c 3,8 such that for all s, t [ε, ] N, N c 3,7 s t 2H E[ B H s B H t 2 ] c 3,8 N s t 2H. 3.2 Remark 4. The upper bound in 3.2 holds for all s, t [, ] N and c 3,8 is independent of ε. However, the constant c 3,7 depends on ε. Proof of Lemma 8. In order to prove the upper bound in 3.2, we use. and write E [ B H s BH t 2] as N s 2H N + t 2H N 2 s 2H + t 2H s t 2H s 2H N N 2 N + t 2H N s 2H + t 2H s t 2H t 2H N N 2 N + t N s N 2HN s 2H + t 2H s t 2H 2 := T + T 2 + T 3. s 2H N N 3.3 It is clear that T 2 c t N s N 2H N and T 3 t N s N 2H N for all s, t [ε, ] N. Therefore the upper bound in 3.2 follows from 3.3 and induction on N. It seems to be quite involved to apply a similar elementary method to prove the lower bound in 3.2; see Xiao and Zhang [36, pp ] for an application of this argument to a similar, but easier problem. Instead, we will proceed by making use of the stochastic integral representation.4. We believe that our argument below will be useful in further studying other problems such as the sharp Hölder conditions for the local times and the exact Hausdorff measure of the image and graph sets of fractional Brownian sheet B H. Let Y ={Yt,t R N + } be the Gaussian random field defined by Yt = = t t tn tn gt,rwdr N t r H 2 Wdr, t R N Then by the independence of the Brownian sheet on different quadrants of R N,wehave E [ B H s BH t 2] κ 2 E[ Ys Yt 2] H for all s, t [ε, ] N. For every t [ε, ] N, we decompose the rectangle [,t] into the following disoint union: N [,t]=[,ε] N Rt ε, t, 3.5

23 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 429 where Rt ={r [, ] N : r i ε, if i =, ε < r t } and ε, t can be written as a union of 2 N N sub-rectangles of [,t]. It follows from 3.4 and 3.5 that for every t [ε, ] N, Yt = [,ε] N gt,rwdr + := Xε,t + N Rt gt,rwdr + gt,rwdr ε,t N Y t + Zε, t. 3.6 Since the processes Xε,t, Y t N and Zε, t are defined by the stochastic integrals over disoint sets, they are independent. Only the Y t s will be useful for proving the lower bound in 3.2. Now let s, t [ε, ] N and N be fixed. Without loss of generality, we assume s t. Then Y E[ t Y s ] 2 2 = gt,r gs, r dr + g 2 t,rdr, 3.7 Rs Rs,t where Rs,t ={r [, ] N : r i ε, if i =, s <r t }. By 3.7 and some elementary calculations we derive Y E[ t Y s ] 2 g 2 t,rdr = Rs,t [,ε] N k = c 3,9 t s 2H, t k r k 2H k t s t r 2H dr where c 3,9 is a positive constant depending on ε and H k k N only. It follows from 3.6, 3.7, and 3.8 that for all s, t [ε, ] N, 3.8 [ Ys ] 2 E Yt N c 3,9 [ Y E s Y t ] 2 N s t 2H. 3.9 This proves 3.2 with c 3,7 = c 3,9 κ 2 H. Lemma 9 below is proved in Xiao and Zhang [36, p. 22] which will be used to derive a lower bound for dim H B H [, ] N. Lemma is needed for determining a lower bound for dim H GrB H [, ] N. Both lemmas will be useful in the proof of Theorem 5. Lemma 9. Let <α<and ε>be given constants. Then for any constants δ>2α, M>and p>, there exists a positive and finite constant c 3,, depending on ε, δ, p and M only, such that for all <A M, ε ds ε A + s t 2α p dt c 3, A p δ

24 43 Antoine Ayache and Yimin Xiao Lemma. Let α, β and η be positive constants. For A> and B>, let J := JA,B = dt A + t α β B + t η. 3.2 Then there exist finite constants c 3, and c 3,2, depending on α, β, η only, such that the following hold for all reals A, B > satisfying A /α c 3, B: i If αβ >, then ii if αβ =, then J c 3,2 A β α B η ; 3.22 iii Proof. J c 3,2 B η log + BA /α ; 3.23 if <αβ< and αβ + η =, then J c 3,2 B αβ+η By a change of variable, we have J = B η B dt A + B α t α β + t η To prove i and ii, we note that, if B<, then we can split the integral in 3.25 so that J = B η dt A + B α t α β + t η + B η B dt A + B α t α β + t η If B, then J is bounded by the first term in Hence, in the following, it is sufficient to consider the case <B<. When αβ >, by using 3.26 and changing the variables again, we get J B η A β α B η dt A + B α t α β + BA /α c A β α B + η c A β α B + c η B αβ+η c 3,2 A β α B, η A β α +ηα B η B ds + s α β + BA /α dt A + B α t α β t η A β α +ηα ds s αβ+η BA /α ds + s α β s η 3.27 where in deriving the third and the last inequalities, we have used the fact that αβ > and BA /α c 3,.

25 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 43 When αβ =, similar to 3.27, we have J B η BA /α 2 BA /α B η ds + s α β + A ηα ds + s + A ηα c 3,2 B η log + BA /α. BA /α BA /α ds s +η ds + s α β s η 3.28 Hence, 3.23 holds. Finally, we consider the case <αβ<. If we further have αβ + η<, then it follows from 3.2 that J dt < and 3.24 holds. So it only remains to t consider the case <αβ<and αβ αβ+η + η>. For simplicity, we assume c 3,2 = and split the integral in 3.25 as J= B A /α B η dt B A + B α t α β + + t η ds A /α A β α B η + s α β + c A β α B + BA /α η A β α B η Since αβ + η>, we have J c A β α B + η A β α B η c A β α B + c η B αβ+η c 3,2 B αβ+η, B A /α dt A + B α t α β + t η ds + s α β + B A /α s η ds s αβ + A /α B A /α η BA /α c BA α αβ ds s αβ+η where the last inequality follows from the assumption that <αβ<and BA /α c. 3, Thus, 3.24 holds and the proof of Lemma is finished. Proof of the lower bounds in Theorem 4. First we prove the lower bound in 3.2. Note that for any ε,, dim H B H [, ] N dim H B H [ε, ] N. Hence, by Frostman s theorem see e.g., Kahane [7, Chapter ], it is sufficient to show that for all <γ < min{d, N H }, E γ = [ε,] N [ε,] N E B H s B H t γ ds dt <. 3.3 Since <γ <d,wehave< E γ <, where is a standard d-dimensional normal vector. Combining this fact with Lemma 8, we have E γ c 3,3 ds dt ds N N γ/2 dt N s t 2H ε ε ε ε

26 432 Antoine Ayache and Yimin Xiao To prove the above integral is finite, we observe that for any <γ <min{d, N H }, there exists an integer k N such that N =k+ H <γ N =k H, 3.33 where N =N+ H :=. In the following, we will only consider the case of k =, the remaining cases are simpler because they require less steps of integration using Lemma 9. Now assuming 3.33, we choose positive constants δ 2,...,δ N such that δ > 2H for each 2 N and δ δ N < γ 2 < 2H + δ δ N Applying Lemma 9 to 3.32 with A = N s t 2H and p = γ/2, we obtain from 3.32 that E γ c 3,4 +c 3,4 ds dt ds N ε ε ε ε N s t 2H γ/2 /δn dt N By repeatedly using Lemma 9 to the integral in 3.35 for N 2 steps, we derive that E γ c 3,5 + c 3,5 ds s t 2H γ/2 /δ2 + +/δ N dt ε ε Since the δ s satisfy 3.34, we have 2H [ γ/2 /δ2 + +/δ N ] <. Thus, the integral in the right-hand side of 3.36 is finite. This proves 3.3. Now we prove the lower bound in 3.3. Since dim H GrB H [, ] N dim H B H [, ] N always holds, we only need to consider the case when k H d< k H for some k N Here and in the sequel, H :=. Let <ε<and <γ < k H k H + N k + H k d be fixed, but arbitrary, constants. By Lemma 7, we may and will assume γ N k + d,n k + d +. In order to prove dim H GrB H [ε, ] N γ a.s., again by Frostman s theorem, it is sufficient to show [ ] G γ = s t 2 + B H s B H t 2 γ/2 ds dt < [ε,] N [ε,] N E

27 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 433 Since γ>d, we note that for a standard normal vector in R d and any number a R, [ ] E a c γ/2 3,6 a γ d, see e.g., Kahane [7, p. 279]. Consequently, we derive that G γ c 3,6 [ε,] N [ε,] N σs,t d ds dt, 3.39 s t γ d where σ 2 s, t = E [ B H s BH t2]. This was obtained by Ayache [3] for d =. By Lemma 8 and a change of variables, we have G γ c 3,7 dt N N t H d N γ d dt. 3.4 t In order to show the integral in 3.4 is finite, we will integrate [dt ],...,[dt k ] iteratively. Furthermore, we will assume k > in If k =, we can use 3.24 to obtain 3.44 directly. We integrate [dt ] first. Since H d >, we can use 3.22 of Lemma with A = N =2 t H and B = N =2 t to get G γ c 3,8 dt N N=2 t H d /H N=2 γ d dt t We can repeat this procedure for integrating dt 2,...,dt k. Note that, if d = k H, then we need to use 3.23 to integrate [dt k ] and obtain G γ c 3,9 dt N N=k γ d log t + N=k dt k < t Note that the last integral is finite since γ d < N k +. d> k H, then 3.22 gives On the other hand, if G γ c 3,2 dt N N=k t H d k /H dt k γ d N=k t We integrate [dt k ] in 3.43 and by using 3.24, we see that G γ c 3,2 dt N N=k+ γ d+hk d k t H dt k++ <, 3.44 since γ d + H k d k H <N k. Combining 3.42 and 3.44 yields This completes the proof of Theorem 4.

28 434 Antoine Ayache and Yimin Xiao The following result is on the Hausdorff dimension of the level set L x = {t, N : B H t = x}. Theorem 5. Let B H ={B H t, t R N + } be an N, d-fractional Brownian sheet with Hurst index H = H,...,H N satisfying 3.. If N H <dthen for every x R d, L x = a.s. If N H >d, then for any x R d and <ε<, with positive probability dim H Lx [ε, ] N k H k = min + N k H k d, k N H 3.45 k H k k k = + N k H k d, if d<. H H H Remark 5. When N H = d, we believe that for every x R d, L x = a.s. However, we have not been able to prove this statement. In the Brownian sheet case, this was proved by Orey and Pruitt [28, Theorem 3.4]. It also follows from a result of Khoshnevisan and Shi [9]. Proof. The second equality in 3.45 follows from Lemma 7. First we prove dim H Lx [ε, ] N k H k min + N k H k d, k N a.s H and L x = a.s. whenever the right-hand side of 3.46 is negative. It can be verified that the latter is equivalent to N H <d. For an integer n, divide the interval [ε, ] N into m n = n sub-rectangles R n,l of side lengths n /H =,,N. Let <δ<be fixed and let τ n,l be the lower-left vertex of R n,l. Then { } { P x B H R n,l P max B H s B H t } n δ ; x B H R n,l s,t R n,l { + P max B H s B H t } >n δ s,t R n,l { P B H τ n,l x 3.47 n δ} + e cn2δ c 3,22 n δd. N H In the above we have applied Lemma 2. in Talagrand [29] to get the second inequality. If N H <d, we choose δ>such that δd > N H. Let N n be the number of rectangles R n,l such that x B H R n,l. It follows from 3.47 that N H EN n c 3,22 n n δd as n Since the random variables N n are integer-valued, 3.48 and Fatou s lemma imply that a.s. N n = for infinitely many integers n. Therefore L x = a.s. Now we assume N H >dand define a covering {R n,l } of L x [ε, ] N by R n,l = R n,l, ifx B H R n,l and R n,l = otherwise. For every k N, R n,l can be

29 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 435 N=k+ H H covered by n k cubes of side length n Hk. Thus, we can cover the level set L x [ε, ] N H by a sequence of cubes of side length n k. Let δ, be an arbitrary constant and let η = k H k H + N k H k δd. It follows from 3.47 that E [ mn n l= c 3,23 n N=k+ Hk H n H ηl{x B k H R n,l } N H + N =k+ Hk H ηh ] k δd = c3, Fatou s lemma implies that the η-dimensional Hausdorff measure of L x [ε, ] N is finite a.s. and thus dim H L x [ε, ] N η almost surely. Letting δ along rational numbers, we obtain To prove the lower bound in 3.45, we assume k H d< k H for some k N. Let δ>be a small constant such that γ := k H k H + N k H k + δd > N k. 3.5 Note that, if we can prove that there is a constant c 3,24 >, independent of δ and γ, such that { P dim H Lx [ε, ] N } γ c 3,24, 3.5 then the lower bound in 3.45 will follow by letting δ. Our proof of 3.5 is based on the capacity argument due to Kahane see [7]. Similar methods have been used by Adler [], Testard [3], and Xiao [3]. Let M + γ be the space of all non-negative measures on RN with finite γ -energy. It is known cf. Adler [] that M + γ is a complete metric space under the metric µdtµds µ γ = R N R N t s γ We define a sequence of random positive measures µ n on the Borel sets C of [ε, ] N by µ n C = 2πn d/2 exp n B H t x 2 dt C 2 = exp ξ 2 C R d 2n + i ξ,b H t x dξ dt It follows from Kahane [7], or Testard [3] that, if there are constants c 3,25 >,c 3,26 > such that E µ n c 3,25, E µ n 2 c 3,

30 436 Antoine Ayache and Yimin Xiao and E µ n γ <+, 3.55 where µ n = µ n [ε, ] N, then there is a subsequence of {µ n }, say {µ nk }, such that µ nk µ in M + γ and µ is strictly positive with probability c2 /2c 3,25 3,26. In this case, it follows from 3.53 that the measure µ has its support in L x [ε, ] N almost surely. Hence, Frostman s theorem yields 3.5 with c 3,24 = c 2 /2c. 3,25 3,26 It remains to verify 3.54 and By Fubini s theorem we have E µ n = = = [ε,] N [ε,] N [ε,] N [ε,] N e i ξ,x exp ξ 2 R d 2n e i ξ,x exp R d 2π n + σ 2 t 2π + σ 2 t E exp i ξ,b H t dξ dt n + σ 2 t ξ 2 dξ dt 2 d/2 exp d/2 exp x 2 2σ 2 t dt x 2 2 n + σ 2 t dt := c 3, Denote by I 2d the identity matrix of order 2d, CovB H s, B H t the covariance matrix of B H s, B H t, Ɣ = n I 2d +CovB H s, B H t and ξ, η the transpose of the row vector ξ, η. Then E µ n 2 = = [ε,] N [ε,] N [ε,] N e i ξ+η,x exp [ε,] N R d R d [ε,]n 2π d exp detɣ [ε,]n 2π d 2 ξ, ηɣ ξ, η dξ dηds dt 2 x,xɣ x, x ds dt [ detcov B H s, B H t] ds dt. d/2 It can be proven see Xiao and Zhang [36, p. 24] that for all s, t [ε, ] N, 3.57 detcov B H s, BH t = N c 3,27 s 2H t 2H N N s t 2H. s 2H + t 2H s t 2H Combining 3.57, 3.58 and applying Lemma 9 repeatedly, we obtain E µ n 2 c 3,28 [ [ε,] N [ε,] N N s t 2H ] d/2 ds dt := c < ,26 In the above, we have omitted the proof of c 3,26 < since it is very similar to in the proof of Theorem 4. Therefore we have shown 3.54 holds.

31 Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets 437 Similar to 3.57, we have ds dt E µ n γ = [ε,] N [ε,] N s t γ c 3,29 [ε,] N c 3,3 dt N R d R d e i ξ+η,x exp [ε,] N N s t γ N s t 2H d/2 N t H d N γ dt, t 2 ξ,ηɣξ,η dξ dη ds dt 3.6 where the two inequalities follow from 3.58 and a change of variables. Note that the last integral in 3.6 is similar to 3.4. By using Lemma in the same way as in the proof of , we see that for any γ defined in 3.5, E µ n γ <+. This proves 3.55 and hence Theorem 5. By using the relationships among the Hausdorff dimension, packing dimension and the box dimension see Falconer [5], Theorems 4 and 5 and their proofs of the upper bounds, we derive the following analogous result on the packing dimensions of B H [, ] N, GrB H [, ] N and L x. Theorem 6. Let B H ={B H t, t R N + } be an N, d-fractional Brownian sheet with Hurst index H = H,...,H N satisfying 3.. Then with probability, dim P B H [, ] N N = min d; H, 3.6 dim P GrB H [, ] N k H k N =min +N k+ H k d, k N; H H If N H >d, then for any x R d and <ε<, dim P Lx [ε, ] N k H k = min + N k H k d, k N H 3.63 with positive probability. Remark 6. In light of Theorems 4, 5, and 6, it would be interesting to determine the exact Hausdorff and packing measure functions for B H [, ] N,GrB H [, ] N and the level set L x. In the special case of the Brownian sheet, the Hausdorff measure of the range and graph were evaluated by Ehm [3]. However, his method relies crucially on the independent increment property of the Brownian sheet and is not applicable to the fractional Brownian sheet B H in general. Moreover, no packing measure results have been proven even for random sets determined by the ordinary Brownian sheet. Related to these problems, we mention that the Hausdorff measure functions for the range and graph of an N, d-fractional Brownian motion X have been obtained by Talagrand [29] and Xiao [33, 34]; and the exact packing measure functions for X[, ] N have been studied by Xiao [32, 35]. Their methods are useful for studying the fractional Brownian sheet B H as well.