DISCRETE ROUGH PATHS AND LIMIT THEOREMS

Transcription

1 DISCRETE ROUGH PATHS AND LIMIT THEOREMS YANGHUI LIU AND SAMY TINDEL Abtract. In thi article, we conider it theorem for ome weighted type random um (or dicrete rough integral). We introduce a general tranfer principle from it theorem for unweighted um to it theorem for weighted um via rough path technique. A a by-product, we provide a natural explanation of the variou new aymptotic behavior in contrat with the claical unweighted random um cae. We apply our principle to derive ome weighted type Breuer-Major theorem, which generalize previou reult to random um that do not have to be in a finite um of chao. In thi context, a Breuer-Major type criterion in notion of Hermite rank i obtained. We alo conider ome application to realized power variation and to Itô formula in law. In the end, we tudy the aymptotic behavior of weighted quadratic variation for ome multi-dimenional Gauian procee. Content 1. Introduction 2 2. Dicrete rough path Definition and algebraic propertie Dicrete rough integral 9 3. Limit theorem General it theorem Single it theorem I Single it theorem II Double it theorem 2 4. Breuer-Major theorem Weighted Breuer-Major theorem I Weighted Breuer-Major theorem II Realized power variation and parameter etimation Stratonovich integral 4 5. Multi-dimenional Gauian procee Preinarie on Gauian rough path Unweighted it theorem Weighted it theorem 53 Reference 54 Key word and phrae. Dicrete rough path, Dicrete rough integral, Weighted random um, Limit theorem, Breuer-Major theorem. S. Tindel i upported by the NSF grant DMS

2 2 Y. LIU AND S. TINDEL 1. Introduction Let D n : = t < t 1 < < t n = 1 be a partition on [, 1]. Take a 1-increment proce h n t defined for, t D n uch that t and a weight proce y t defined for t n N D n. We conider a dicrete integral a a Riemann um of the form: J t (y; h n ) := y tk h n t k t k+1. (1.1) t k <t Recall that a claical it theorem for uch a proce i a tatement of the type: 1 a n J (1; h n ) = hn a n ω, a n. (1.2) Here a n i an increaing equence uch that n a n =, ω i a non-zero continuou proce and the it i uually undertood a a finite dimenional ditribution it. A typical example of (1.2) i the convergence of a renormalized random walk to Brownian motion (Donker theorem, ee [25]), but a wide range of more complex ituation can occur. Indeed, it i well-known that the rate of growth of a n and the nature of the it proce ω are determined by both the marginal tail of h n and it dependence tructure; ee e.g. [1, 11, 38, 39]. The it proce ω i necearily elf-imilar; ee [26]. In thi paper we are intereted in the following related problem: Problem 1. Given that h n converge to ome 1-increment proce, ay, the increment of a Wiener proce, what i the aymptotic behavior of the dicrete integral J (y; h n ) for a general weight y, and when would (or would not) the aymptotic behavior of J (y; h n ) be imilar to that of h n? Thi problem ha drawn a lot of attention in recent article due to it eential role in topic uch a normal approximation (e.g. [31, 32, 33]), time-dicretization baed numerical approximation (e.g. [16, 24, 28]), parameter etimation (e.g. [2, 9, 27, 3]), and the ocalled Itô formula in law (e.g. [3, 5, 17, 18, 2, 21, 22, 35, 36]). Let u, however, point out everal itation in the exiting reult: (1) Each proce h n i uually a functional of a Gauian proce x with tationary increment, living in a fixed finite um of chao; (2) The underlying Gauian proce x i one-dimenional; (3) Only the pecial cae y t = f(x t ), t i conidered for the weight function. (4) To the bet of our knowledge, there i no theoretical explanation for the variou unexpected aymptotic behavior of the dicrete integral oberved in e.g. [5, 18, 35, 37] o far. (5) Satifactory general criteria of convergence for equence of dicrete weighted integral are till rare. Thi i in harp contrat with the imple Breuer-Major type condition in the unweighted cae. The aim of the current paper i thu to give an account on it theorem for dicrete integral thank to rough path technique combined with Gauian analyi. In our etting, we will conider a general 1-increment proce h n and a general weight proce (y, y,..., y (l 1) ) with y = which i controlled by the increment of ome rough path x = (1, x 1,..., x l 1 ). Here l i ome contant in N. Notice that we will define the notion of controlled proce later in the paper, ee Definition 2.3 below, but we can oberve that thi cla of path include function of the form y = f(x) or olution of differential equation driven by x. Let u label the following hypothei:

3 DISCRETE ROUGH PATH 3 Hypothei 1.1. Take i =, 1,..., l 1. For any partition < 1 < < m 1 of [, 1] uch that m n and +1 1, we have m m up n (x i ; h n ) =, (1.3) J +1 where J (x i ; h n ) i defined by (1.1) and the it i undertood a a it in probability. We will be able to prove the following it theorem (ee Theorem 3.9 for a more precie tatement): Theorem 1.2. Conider an underlying rough path x = (1, x 1, x 2,..., x l 1 ), where l i ome contant in N (depending on the regularity of x). Let (y, y,..., y (l 1) ) be a proce on [, 1] whoe increment are controlled by x, and aume that h n atifie ome proper regularity hypothei. Suppoe that the following aumption are fulfilled: (i) We have the convergence (x, h n ) f.d.d. (x, W ) a n. Here and in the following W i a tandard Brownian motion independent of x, and f.d.d. tand for the finite dimenional ditribution it. (ii) Hypothei 1.1 hold true for i = 1,..., l 1. Then we have the following convergence in ditribution for the proce y: (x, J (y; h n )) f.d.d. (x, v), (1.4) where the integral v t = t y udw u ha to be undertood a a conditional Wiener integral. A alluded to above, Theorem 1.2 can be een a a general principle which allow to tranfer it theorem (1.3) taken on monomial of the rough path to the correponding it theorem involving controlled procee a weight. Therefore, potential application of thi reult are numerou (ee the aforementioned parameter etimation problem, Itô formula in law, or numerical cheme for rough differential equation), and will be detailed throughout the paper. A ha already been oberved in [18, 31, 33], the aymptotic behavior of (1.1) can be completely different from (1.4). One of the firt occurrence of thi kind of reult i provided by [31], where for a one-dimenional fractional Brownian motion x with Hurt parameter ν (, 1) the following it theorem i obtained: conider the increment 4 hn t = t k <t [(nν δx tk t k+1 ) 2 1], where δx tk t k+1 = x tk+1 x tk and (t k ),...,n tand for the uniform partition of [, 1]. Let f be a continuou function with proper regularity. Then, a n, we have: n 2ν 1 J 1 (f(x); h n ) L f (x )d. (1.5) Our approach allow to generalize Theorem 1.2 to handle it uch a (1.5), weighted by controlled procee. In addition, our reult provide an explanation of the appearance of f in the right-hand ide of (1.5), baed on the tructural undertanding of the dicrete integral from the rough path theory. Indeed, our next theorem how that the it 1 1 f (x 4 )d i the reult of a peed match between different level (1, x 1,..., x l 1 ) of the rough path x and

4 4 Y. LIU AND S. TINDEL the fact that f(x), f (x),..., f (l 1) (x) are the correponding weight procee. Specifically, we hall get the following it theorem (ee Theorem 3.11 for a multidimenional verion). Theorem 1.3. Let y, x and h n be procee defined a in Theorem 1.2, and recall that J (y; h n ) i the increment defined by (1.1). Suppoe that x and h n verify the following aumption: (i) There i ome τ {1,..., l 1} uch that J t (x τ ; h n ) (t )ϱ in probability for all, t D n uch that < t, where ϱ i ome contant, and J t (x i ; h n ) in probability for all i < τ. (ii) Hypothei 1.1 hold true for i = τ + 1,..., l 1. Then the following convergence hold true in probability: ( ) J (y; n hn ) = y (τ) t dt ϱ. A more complicated ituation of aymptotic behavior i oberved in [5, 32, 35]. Thi uually correpond to a tranition in term of roughne for the underlying rough path x. For example, in the critical cae when ν = 1 in (1.5), and for the ame f and 4 hn a in (1.5), one obtain the convergence: n 1/2 J 1 (f(x); h n ) d σ f(x )dw f (x )d, (1.6) where σ i ome contant and recall that W i a tandard Brownian motion independent of x. An explanation of the above aymptotic behavior according to the technique of rough path i that the two level 1 and x 2 give contribution of the ame order in the it theorem. Thi i then reflected into the fact that the component f(x) and f (x) (repectively, th and 2nd derivative of f(x) a a controlled proce) give contribution of the ame order. Our generalization of (1.6) i thu the following double it theorem. Theorem 1.4. Let y, x and h n be procee defined a in Theorem 1.2. Furthermore, we aume that x and h n fulfill the following condition: (i) We have the convergence (x, h n ) f.d.d. (x, W ) a n. (ii) There exit a contant ϱ and ome τ : < τ < l uch that for any partition < 1 < < m 1 on [, 1] atifying m n, i+1 i 1 m, and =, m = t, we have the convergence in probability: J +1 m n j (x τ ; h n ) = (t )ϱ. (1.7) (iii) Hypothei 1.1 hold true for i {1,..., l 1} \ {τ}. Then we obtain the following it for the increment J (y; h n ): ( ( ) ) (x, J (y; h n )) f.d.d. x, y t dw t + y (τ) t dt ϱ. A mentioned previouly, Theorem 1.2, 1.3 and 1.4 are abtract tranfer principle from monomial of a rough path to a controlled proce for it theorem of the form (1.2). For

5 DISCRETE ROUGH PATH 5 ake of illutration, let u mention an important application of thi tranfer principle we will encounter in the article, namely a weighted type Breuer-Major theorem. e t 2 dt q Recall that the Hermite polynomial of order q i defined a H q (t) = ( 1) q e t2 2 dq 2, and we denote by γ the tandard normal ditribution. We conider the following Breuer-Major type criterion: Hypothei 1.5. Take l N. Let f L 2 (γ) be a function uch that we have the expanion f = q=d a qh q for a given d 1 and a d. We uppoe that the coefficient a q atify: a 2 qq!q 2(l 1) <. (1.8) q=d Following i our weighted type Breuer-Major theorem: Theorem 1.6. Let x be a one-dimenional fbm with Hurt parameter ν < 1. Suppoe that 2 Hypothei 1.5 hold true for f L 2 (γ) with ome l N and d 1. Let (y, y,..., y l 1 ) be a proce controlled by x. We define a equence {h n ; n 1} of increment by h n t := n 1/2 t k <t f(nν δx tk t k+1 ) for, t in the partition D n, uch that < t. (i) When d > 1 1, and l i the mallet integer uch that l >, we have the convergence: 2ν 2ν ) (x, J (y; h n )) (x, f.d.d. σ ν,d y t dw t, a n, where σ ν,d i a contant which can be computed explicitly and where we recall that J (y; h n ) i defined by (1.1). (ii) When d = 1 2ν (x, J (y; h n )) f.d.d. (iii) When d < 1 2ν and l = d + 1, the following convergence hold true: ( (x, σ ν,d y t dw t + 1 ) dad ) y u (d) du, a n. 2 and l = d + 1, we have the convergence in probability: n ( 1 2 νd) J t (y; h n ) ( 1 ) dad t 2 y (d) u du, a n. The proof of Theorem 1.6 i baed on our general Theorem Let u oberve that Theorem 1.6 improve on the reference on weighted Breuer-Major theorem quoted above in the following way: (i) The function f i not aumed to be in a finite um of chao. In fact a convenient ufficient condition for (1.8) to be fulfilled i that the function f i an element of C 2l 2. (ii) Multidimenional verion of Theorem 1.6 (baed on [1]) are eaily conceived, where f(n ν δx tk t k+1 ) in the definition of h n i replaced by f(n ν δx 1 t k t k+1,..., n ν δx d t k t k+1 ), for a d- dimenional Gauian proce (x 1,..., x d ). (iii) The weight y in Theorem 1.6 i obviouly a controlled proce intead of a mere function of x. It i worth noting again that the cla of controlled procee include olution of differential ytem driven by x.

6 6 Y. LIU AND S. TINDEL (iv) A mentioned above, the ingle and double iting phenomenon in Theorem 1.6 can be explained in term of peed match on different level of the rough path above x. (v) The olution to Problem 1 above i expreed eaily in term of the Hurt parameter ν of x and the Hermite rank d of f. Throughout the paper we will give an account on other application of our general Theorem , uch a realized power variation, convergence of trapezoidal Riemann um and quadratic variation of multidimenional Gauian procee. A the reader might ee, the improvement (i)-(v) mentioned above will be a contant of our rough path method. Let u briefly explain the general methodology we have followed for our proof, eparating the general principle from the application. (a) The proof of Theorem i motly baed on rough path type expanion for the weight proce y and a more claical coare graining argument (alo called big block/mall block in the literature). By handling the remainder term thank to rough path technique, the convergence of J (y; h n ) i reduced to thoe of J (y; ζ 1 ), J (y ; ζ 2 ),..., J (y (l 1) ; ζ l ), where each ζ i i a dicrete proce of the form ζj i = J +1 (x i ; h n ). The convergence of thee quantitie are further reduced to thoe of J (1; ζ 1 ), J (1; ζ 2 ),..., J (1; ζ l ), uch a thoe in Hypothei 1.1 and relation (1.7). The random procee J (1; ζ 1 ), J (1; ζ 2 ),..., J (1; ζ l ) will be the elementary brick for our iting procedure. (b) Our application, uch a the weighted type Breuer-Major Theorem 1.6, heavily rely on the criteria developed in Theorem Thi ingredient i combined with ome Malliavin calculu technique in order to handle the building brick J (1; ζ i ). More pecifically, in cae of the weighted Breuer-Major theorem 1.6, we hall invoke integration by part on the Wiener pace. Thi tep i imilar to what i done in [32]. However, due to our rough path reduction of the problem, we only have to conider integration by part to compute moment of the elementary brick x i t k H q (n ν δx tk t k+1 ) (a oppoed to g(x tk )f(n ν δx tk t k+1 ) for a general nonlinear function g). Thi reduction to computation in finite chao i one of the crucial tep which allow to derive the Breuer-Major type criteria (1.8) for a general function f. The paper i organized a follow. In Section 2 we introduce the concept of dicrete rough path and dicrete rough integral and recall ome baic reult of the rough path theory. In Section 3, we prove our general it theorem including Theorem 1.2, Theorem 1.3 and Theorem 1.4. In Section 4, we apply them to the one-dimenional fractional Brownian motion, which allow u to derive a weighted type Breuer-Major theorem. We alo conider application of the weighted type Breuer-Major theorem to parameter etimation and Itô formula in law. In Section 5, we conider the it theorem of a weighted quadratic variation in the multi-dimenional Gauian etting. Notation: For implicity, we conider uniform partition, that i, we denote t k = k for each n k, n N. Take, t [, 1]. We denote by S k (, t) the implex {(t 1,..., t k ) [, 1] k ; t 1 t k < t}, and for implicity we will write S k for S k (, 1). In contrat, whenever we deal with a dicrete interval [, t) D n, we et S k (, t) = {(t 1,..., t k ) Dn; k t 1 < < t k < t}, and imilarly, when = and t = 1 we imply write S k. Throughout the paper we work on a probability pace (Ω, F, P ). If X i a random variable, we denote by X Lp the L p -norm of X. The letter K tand for a contant which can change

7 DISCRETE ROUGH PATH 7 from line to line. The letter G denote a generic a.. finite random variable. We denote by a the integer part of a. 2. Dicrete rough path In thi ection, we introduce the concept of dicrete rough path and dicrete rough integral, and recall ome baic reult of the rough path theory. Then we derive our main etimate on dicrete rough integral Definition and algebraic propertie. Thi ubection i devoted to introduce the main rough path notation which will be ued in the equel. Let V be a finite dimenional vector pace. We denote by C k (V) the et of function g : S k V uch that g t1 t k = whenever t i = t i+1 for i k 1. Such a function will be called a (k 1)-increment. We define the operator δ a follow: k+1 δ : C k (V) C k+1 (V), (δg) t1 t k+1 = ( 1) i g t1 ˆt i t k+1, where ˆt i mean that thi particular argument i omitted. For example, for f C 1 (V) and g C 2 (V) we have i=1 δf t = f t f and δg ut = g t g u g ut. (2.1) A fundamental property of δ, which i eaily verified, i that δδ =, where δδ i conidered a an operator from C k (V) to C k+2 (V). Let u now introduce the notion of rough path which will be ued throughout the paper. Definition 2.1. Conider ν (, 1), l N uch that l 1 ν and p > 1. Let x = (x1,..., x l ) be a continuou path on S 2 and with value in l k=1 (Rd ) k. For p > et and define a ν-hölder emi-norm a follow: x k uv 1/k x k L [,t], p,ν := up p (u,v) S 2 ([,t]) v u, (2.2) ν x p,ν := x 1 p,ν + + x l p,ν. (2.3) We call x a (L p, ν, l)-rough path (or imply a rough path) if the following propertie hold true: (1) the emi-norm x k [,t],p,ν in (2.2) are finite. In thi cae we ay that x k, k = 1,..., l are repectively in C ν (S 2, (R m ) k ). For convenience, we denote x k p,ν := x k [,1], p,ν. (2) For all k {2,..., l}, x k atifie the identity δx k ut = k 1 x k j u x j ut. (2.4) j=1

8 8 Y. LIU AND S. TINDEL Remark 2.2. Our definition of rough path differ lightly from the uual one in everal apect: (i) We don t impoe l = 1, o that the order of our rough path might be lower than in ν the tandard theory. In the equel we will introduce another parameter α (, 1) uch that νl + α > 1. (ii) We conider a rough path with value in L p, and meaure it regularity by looking at increment of the form x k t Lp for (, t) S 2. In thi paper, we are motly concerned with dicrete um. Recall that we are conidering dicrete implexe related to partition of [, 1], which are denoted by S 2. We now introduce a general notion of dicrete controlled proce. Definition 2.3. Fix α > and let l be the mallet integer uch that νl + α > 1. Let V be ome finite dimenional vector pace. Let y, y, y,..., y (l 1) be continuou procee on [, 1] uch that y = y () =. For convenience, we will alo write: y () = y, y (1) = y, y (2) = y,.... Suppoe that y take value in V, and y (k) take value in L((R d ) k, V) for all k = 1,..., l 1. For (, t) S 2 and k =, 1,..., l 2 we denote. We call (y (),..., y (l 1) ) a dicrete V-valued rough path in L p con- if r (k) t Lp K(t ) (l k)ν for all k =, 1,..., l 1. The dicrete path and r (l 1) t trolled by (x, α) = δy (l 1) t r (k) t = δy (k) t y (k+1) x 1 t y (l 1) x l k 1 t, (2.5) (y (),..., y (l 1) ) i controlled by (x, α) almot urely if r (k) t G y (t ) (l k)ν, k =,..., l 1 for ome finite random variable G y. Remark 2.4. In ome of our computation below we will rephrae (2.5) for k = a the following identity for (, t) S 2 : y t = where we take x 1 by convention. l 1 y (i) x i t + r () t, (2.6) We firt label a imple algebraic property relating the remainder r (k). i= Lemma 2.5. Let y = (y (),..., y (l 1) ) be a dicrete rough path in L p controlled by (x, α) for all p > 1. Then the following identity hold true for all (, u, t) S 3: δr () ut = In particular, we have the following etimate for p > 1: l 1 r ux (i) i ut. (2.7) i=1 δr () ut Lp K(t ) νl. (2.8) Proof. By the definition of r () in (2.5) and the expreion (2.1) of δg for g C 2 (V), ome elementary computation yield: l 1 ut = δr () i=1 l 1 y (i) x i t + i=1 l 1 y u (i) x i ut + i=1 y (i) x i u = l 1 i=1 l 1 δy u (i) x i ut i=2 y (i) δx i ut, (2.9)

9 DISCRETE ROUGH PATH 9 where we have ued the fact that δx 1 ut =. Therefore, invoking (2.5) and (2.4) again we obtain δr () ut = = l 1 i=1 l 1 i=1 l 2 r ux (i) i ut + r (i) ux i ut. l 1 i=1 j=i+1 l 1 y (j) x j i u x i ut i=2 y (i) i 1 j=1 x i j u x j ut Thi conclude the identity (2.7). The inequality (2.8) follow by taking L p -norm on both ide of (2.7) and taking into account the aumption that x i t Lp K(t ) νi and r (i) t Lp K(t ) (l i)ν. An eential technical tool ued in the equel i the dicrete ewing lemma. It i recalled below, the reader being referred to [28] for a proof. Let u begin with the definition of dicrete 1-increment. Definition 2.6. Let π be a partition on [, 1]. We denote by C 2 (π, X ) the collection of increment R defined on S 2 with value in a Banach pace (X, ) uch that R tk t k+1 = for k =, 1,..., n 1. Similarly to the continuou cae (relation (2.1) and (2.2)), we define the operator δ and ome Hölder emi-norm on C 2 (π, X ) a follow: R uv δr ut = R t R u R ut, and R µ = up for µ >. (u,v) S 2 u v µ For R C 2 (π, X ) and µ >, we alo et δr µ = up (,u,t) S 3 δr ut t µ. (2.1) The pace of function R C 2 (π, X ) uch that δr µ < i denoted by C µ 2 (π, X ). The ewing lemma for element of C µ 2 (π, X ) can be tated a follow. Lemma 2.7. For a Banach pace X, an exponent µ > 1 and R C µ 2 (π, X ) a in Definition 2.6, the following relation hold true: R µ K µ δr µ, where K µ = 2 µ l µ Dicrete rough integral. In thi ubection, we derive upper-bound etimate for ome dicrete integral defined a Riemann type um. Namely, let f and g be function on S 2. For a generic partition D n = { = t < < t n = 1} of [, 1], we et ε(t) = t k for t (t k 1, t k ]. (2.11) We define the dicrete integral of f with repect to g a: J t (f; g) := f ε()tk g tk t k+1, (, t) S 2. (2.12) t k <t l=1

10 1 Y. LIU AND S. TINDEL Similarly, if f i a path on the grid = t < < t n = 1, then we define the dicrete integral of f with repect to g a: J t (f; g) := δf ε()tk g tk t k+1, (, t) S 2. (2.13) t k <t Remark 2.8. Notice that in (2.11), ε(t) i the upper endpoint of the partition when t (t k 1, t k ]. A a reult, the firt term of the Riemann um (2.13) i alway vanihing. In addition, we alo have J t k+1 t k (f; g) = for all (t k, t k+1 ) S 2. The next propoition give a baic etimate for dicrete integral. In the following, V and V tand for ome finite dimenional vector pace. Propoition 2.9. Let y = (y (),..., y (l 1) ) be a dicrete rough path on [, 1], controlled by (x, α) in L 2, and let h be a 1-increment defined on S 2 with value in V. Suppoe that h atifie J t (x i ; h) L2 K(t ) α+νi, (2.14) for i =, 1,..., l 1 and (, t) S 2, where we recall that l i an integer uch that α+νl > 1. Then we have the etimate which i valid for (, t) S 2. J t (r () ; h) L1 K(t ) νl+α, (2.15) Proof. In order to bound the increment R t := J t (r () ; h), we firt note that R tk t k+1 =, due to the fact that r () t k t k =. Let u now calculate δr: for (, u, t) S 3, it i readily checked that δr ut = J t (r () ; h) J u (r () ; h) Ju(r t () ; h) = (r () t k r () ut k )h tk t k+1. u t k <t Writing r () t k r () ut k = δr () ut k + r u () and invoking relation (2.7), we thu obtain δr ut = l 1 r uj (i) u(x t i ; h). (2.16) i= Now take the L 1 -norm on both ide of (2.16), take into account condition (2.14) and the hypothei νl + α > 1, and then apply Lemma 2.7. Thi eaily yield the deired etimate (2.15). 3. Limit theorem In thi ection, we firt prove a general it theorem for dicrete integral. Then we will handle two more pecific ituation which arie often in application.

11 DISCRETE ROUGH PATH General it theorem. Recall that the dicrete integral J t (y; h) i defined in (2.12). In thi ubection, we prove a general it theorem for J t (y; h). Theorem 3.1. Let V and V be two finite-dimenional vector pace. Let (y (),..., y (l 1) ) be a dicrete V-valued rough path on [, 1] controlled by (x, α) in L 2 or almot urely (ee Definition 2.3), and h n be a 1-increment which atifie (2.14) uniformly in n. Conider the family J t (x i ; h n ) defined by (2.12), and uppoe that ( x, J (x i ; h n ), i I ) f.d.d. (x, ω i, i I), (3.1) a n, where (ω i, i I) i a 1-increment independent of x, I := {, 1,..., l 1}, and f.d.d. tand for convergence of finite dimenional ditribution. Suppoe further that, if J (y (i) ; ω i ) i given by (2.12), we have ( ) x, J (y (i), ω i f.d.d. ), i I (x, v i, i I), (3.2) where v i, i I are V V -valued 1-increment. Then the following convergence hold true a n : (x, J (y; h n )) f.d.d. (x, v + v v l 1 ). (3.3) Remark 3.2. If we particularize our it theorem to the level i = of J (x i ; h n ), we jut get that h n f.d.d. ω a part of the tanding aumption. In return, we obtain that v = y dω in relation (3.3). Remark 3.3. A the reader might have oberved, Theorem 3.1 give a general tranfer principle from it theorem for unweighted um to it theorem for weighted um, within a rough path framework. Remark 3.4. Condition (3.1) i more demanding than condition (3.2) in Theorem 3.1. Indeed, condition (3.2) i uually reduced to the convergence of a Riemann um to an Itô or Riemann type integral. Proof. For ake of conciene we will only how the convergence of (x t, J 1 (y; h n )) for ome (, t) S 2. The convergence of the finite dimenional ditribution of (x, J (y; h n )) can be hown in a imilar way. The proof i divided into everal tep. Step 1: A decompoition of J 1 (y; h n ). Take two uniform partition on [, 1]: t k = k/n for k, n N and u j = j/m for j, m N, and m n. Set: D j = {t k : u j+1 > t k u j } and ū j = ε(u j ), (3.4) where the function ε ha been introduced in (2.11). By definition (2.13) we have n 1 n 1 J 1 (y; h n ) = δy tk h n t k t k+1 = y tk h n t k t k+1, where the econd identity i due to the fact that we have aumed y = in Definition 2.3. Next we decompoe the Riemann um thank to the et D j. We get J 1 (y; h n ) = t k D j y tk h n t k t k+1.

12 12 Y. LIU AND S. TINDEL Now we invoke relation (2.6) with = ū j and t = t k whenever t k D j. Thi yield: where ϕ i = R ϕ = J 1 (y; h n ) = ϕ + + ϕ l 1 + R ϕ, (3.5) y (i) t k D j r () t k D j ū j x ī u j t k h n t k t k+1 = ū j t k h n t k t k+1 = y (i) ū j J u j+1 u j (x i ; h n ), J u j+1 u j (r () ; h n ), (3.6) and where we have et x t = 1 by convention. Let u further decompoe ϕ i a follow: ϕ i = := ϕ i 1 + ϕ i 2. y u (i) j J u j+1 u j (x i ; h n ) + (y (i) ū j We now tudy the convergence of ϕ i 1 and ϕ i 2 eparately. y (i) u j )J u j+1 u j (x i ; h n ) Step 2: Convergence of ϕ i 2. In thi tep we how that for i = 1,..., l 1, the random variable ϕ i 2 converge to zero in probability a n. To thi aim, it uffice to conider the cae when: (y (),..., y (l 1) ) i controlled by (x, α) in L p, for an arbitrary p > 1. (3.7) Indeed, for ε >, we can find a contant K uch that P (G y > K) ε (ee Definition 2.3 for the definition of G y ). Define (ȳ (),..., ȳ (l 1) ) uch that ȳ i = y i on {G y K} and ȳ i on {G y > K}. Then (ȳ (),..., ȳ (l 1) ) atifie the condition (3.7), and we can write P ( ϕ i 2 > ε) = P ( ϕ i 2 > ε, G y K) + P ( ϕ i 2 > ε, G y > K) P ( ϕ i 2 > ε) + ε, where ϕ i 2 = ϕ i 2 when G y K and ϕ i 2 = when G y > K. So if we can how that ϕ i 2 in probability, then the ame convergence hold for ϕ i 2. Aume now that (3.7) i true. In thi cae we have ϕ i 2 y (i) ū j y (i) u j J u j+1 u j (x i ; h n ). (3.8) Taking the L 1 -norm on both ide of the inequality (3.8), invoking the fact that h n atifie relation (2.14) uniformly in n, and uing the continuity of y (i) given by (3.7), we eaily obtain the following convergence in probability: n ϕi 2. Step 3: Convergence of ϕ i 1. The convergence of l 1 i= ϕi 1 follow immediately from the aumption of the theorem. Indeed, fixing m and ending n to, our aumption (3.1)

13 directly yield: ( l 1 x t, i= DISCRETE ROUGH PATH 13 ϕ i 1 ) d ( l 1 x t, yu i j ωu i j u j+1 ). (3.9) We now end m in (3.9) and recall the convergence (3.2). convergence: a n and m. ( x t, l 1 ϕ i 1 i= ) d i= ( x t, l 1 v ), i i= Thi yield the weak Step 4: Convergence of the remainder term R ϕ. Going back to equation (3.5) and ummarizing our computation, our claim (3.3) i now reduced to how that we have m up R ϕ =, (3.1) n in probability. Moreover, a in Step 2 it uffice to how the convergence (3.1) under condition (3.7). Eventually, applying Propoition 2.9 to (3.6) we obtain: R ϕ L1 K m νl α Km 1 νl α. (3.11) The convergence (3.1) then follow from (3.11) and the fact that νl + α > 1. Let u tate a more practical verion of Theorem 3.1, for which we take advantage of certain cancellation. Theorem 3.5. Let y = (y (),..., y (l 1) ) be a dicrete rough path controlled by (x, α) in L 2 or almot urely, and aume that h n atifie the inequality (2.14) uniformly in n. Suppoe that the following weak convergence hold true a n : ( ) x, J (x i ; h n ), i I f.d.d. (x, ω i, i I ), (3.12) where J (x i ; h n ) i defined by (2.12). Aume that for i I we have m up n u j (x i ; h n ) = (3.13) y (i) u j J u j+1 in probability. Here I, I are dijoint ubet of I := {, 1,..., l 1} uch that I I = I. Suppoe further that ( ) ) x, J (y (i), ω i ), i I f.d.d. (x, v i, i I, (3.14) where v i, i I are random variable. Then the following convergence hold true a n : ( ) ) x, J (y; h n f.d.d. ) (x, i I v i. Proof. The proof i imilar to Theorem 3.1 and i omitted.

14 14 Y. LIU AND S. TINDEL Theorem 3.5 allow u to ditinguih two predominant cae in it Theorem 3.1: (i) A uual aymptotic regime, for which only one level v i remain. (ii) A critical cae, for which more than one level urvive a n goe to. The following definition capture thoe different behavior. Definition 3.6. We will call a it theorem ingle if I in Theorem 3.5 ha only one element. Similarly, a it theorem i called double when I ha two element. Remark 3.7. If the convergence in (3.12) and (3.14) hold true in probability, then in a imilar way one can how that J t (y () ; h n ) converge to i I v i t in probability. Remark 3.8. In the cae ν > 1 and α 1, we have l = 1 and I = {}. Therefore, condition 2 2 (2.14), (3.12), (3.13) are reduced to h n L2 K(t ) α and (h n, x) f.d.d. (ω, x). If h n ω in L p for all p 1, then J 1 (y; h n ) alo converge in L q for q 1. Thi ituation allow to recover the reult in [8] and [24, Propoition 7.1]. A more pecific tatement will be given in Propoition 4.9 below Single it theorem I. An important cae in Theorem 3.1 i when h n converge in ditribution to a Brownian motion and the dicrete integral J 1 (y () ; h n ) converge to the Wiener integral y t dw t. In thi ubection we invetigate thi type of it theorem. Theorem 3.9. Let x be a (L p, ν, l)-rough path for p = 4, ν (, 1) and l uch that νl+ 1 > 1. 2 Let y = (y (),..., y (l 1) ) be a proce on [, 1] controlled by (x, 1) in L 2 2 or almot urely, and aume that h n atifie the inequality (2.14) uniformly in n. Suppoe that the following aumption are fulfilled: (i) We have the convergence (x, h n ) f.d.d. (x, W ) a n, where W i a tandard Brownian motion independent of x. (ii) For any partition < 1 < < m 1 of [, 1] uch that m n and +1, we have 1 m m in probability for i = 1,..., l 1. up n (x i ; h n ) = (3.15) J +1 Then we have the following convergence in ditribution for the proce y: (x, J (y; h n )) f.d.d. (x, v), (3.16) where the integral v t := t y u dw u ha to be undertood a a conditional Wiener integral. Remark 3.1. Theorem 3.9 can be generalized to ome other intereting ituation. For example, uppoe that (x, h n ) f.d.d. (x, ω), where ω i a continuou Gauian proce independent of x. Let H be the Hilbert pace correponding to ω and aume that C γ H for γ < ν. We aume that for any f C γ, we have the following convergence for a generic partition < < m 1: m j,j = m δf j,, δf j, H = and m f tj 1 [tj,t j+1 ) = f (3.17)

15 DISCRETE ROUGH PATH 15 where the econd it i a it in H. Then following the line of the proof of Theorem 3.9 one can how that J 1 (y; h n ) d y dω. Proof of Theorem 3.9. Take I = {} and I = {1,..., l}. The theorem will be proved by applying Theorem 3.5 and verifying the convergence (3.12), (3.13) and (3.14). The proof i divided into everal tep. Step 1: We will how by induction that J t (x i ; h n ) d ω i t x i u dw u, i =, 1,..., l 1. (3.18) Since h n W in f.d.d. ene, convergence (3.18) hold true when i =. Now aume that the convergence hold for i =, 1,..., τ 1 with τ < l. Take m n and u j = j/m, and et D j = {t k : u j+1 > t k u j } a in (3.4). Take j 1 uch that D j1 and j 2 uch that t D j2. Then a mall variant of (2.4) how that for all t k D j, τ x τ ε()t k = δx τ ε(),ū j ε(),t k + x τ ε(),ū j ε() + x τ ū j ε(),t k = x τ l ε(),ū j ε() xl ū j ε(),t k, where recall that the function ε i defined in (2.11) and ū j i given by (3.4). Hence it i readily checked that: J t (x τ ; h n ) = τ l= j 2 l= j=j 1 x τ l ε(),ū j ε() J u j+1 t u j (x l ; h n ) := τ A l. (3.19) Let u change the name of our variable in order to match the notation of our theorem and ue relation (3.15). Namely, et =, 1 = u j1 +1,..., 2 j 1 = u j2, 2 j 1 +1 = t. Then it i readily checked that A τ = j 2 j 1 J +1 (x τ ; h n ). Thu invoking aumption (3.15), we directly have the following convergence in probability: m up A τ =. (3.2) n In order to tudy the convergence of A l for l < τ, we firt check that we can replace x τ l ε(),ū j ε() by x τ l,u j. Indeed, we have the identity: x τ l ε(),ū j ε() xτ l,u j = δx τ l,u j,ū j ε() δxτ l,ε(),ū j ε() + xτ l u j,ū j ε() xτ l,ε(). Therefore, if we et: Ã l := j 2 then it i readily checked that: j=j 1 x τ l,u j J u j+1 t u j (x l ; h n ) = m j 2 j 1 l= x τ l J +1 (x l ; h n ), up A l Ãl = in probability. (3.21) n

16 16 Y. LIU AND S. TINDEL Let u now check the convergence for Ãl. aumption (3.18) with l < τ, we get à l d j 2 j 1 x τ l We now eparate the analyi of Ãl in two cae. Sending n and applying the induction j+1 x l u dw u. (3.22) (a) For < l < τ the quare of the L 2 -norm of the right-hand ide of (3.22) can be bounded thank to Itô iometry by: KE [ j 2 j 1 x τ l 2 j+1 which by property (2.2) applied to p = 4 and l 1 i le than K (+1 ) 2ν+1. ] x l u 2 du, (3.23) Owing to the fact that 2ν + 1 > 1, it i now trivially een that a m, the right-hand ide of (3.22) converge to zero. Thu we get: m up Ãl = (3.24) n in probability. In ummary of (3.21) and (3.24), we have the convergence in probability for < l < τ. m up A l =, (3.25) n (b) When l =, the convergence (3.22) implie that, a n and m we obtain à d Taking into account (3.21), the convergence (3.26) implie that A d t t x τ r dw r. (3.26) x τ r dw r. (3.27) Putting together (a), (b) and the cae l = τ, we can now propagate our induction hypothei. Indeed, applying (3.2), (3.25) and (3.27) to (3.19), we obtain J t (x τ ; h n ) d t x τ r dw r. Thi complete the proof of (3.18) for i =,..., l 1. In a imilar way, we can get a f.d.d. verion of (3.18). Namely, we can how that (x, J (x i ; h n )) f.d.d. (x, ω i ) (3.28) for ω i t = t xi r dw r, i I. Note that thi how that condition (3.12) in Theorem 3.5 hold true.

17 DISCRETE ROUGH PATH 17 Step 2: In thi tep, we conider the convergence of J 1 (y (i) ; ω i ), which will yield condition (3.13) in Theorem 3.5. We firt how that the dicrete integral J 1 (y (i) ; ω i ), l > i > converge to zero in probability. A in the proof of Theorem 3.1, by a truncation argument, it uffice to how the convergence when (y (),..., y (l 1) ) i controlled by (x, 1 2 ) in L p for p large enough. In thi cae, imilarly to (3.23), we have J 1 (y (i) ; ω i ) 2 L 2 KE K [ y (i) u j 2 uj+1 u j ] x i u j u 2 du m 2ν 1. (3.29) Therefore, we have J 1 (y (i) ; ω i ) in probability. Combining thi convergence with (3.28) for i = 1,..., l 1, we obtain the convergence (3.13). On the other hand, for the quantity J 1 (y (i) ; ω i ) with i =, thank to the convergence of Riemann um related to Wiener integral the following convergence hold in L 2 : J 1 (y; ω ) = J 1 (y; W ) y t dw t. So the condtion (3.14) hold true with v = y t dw t. Summarizing our conideration, we can now apply Theorem 3.5 to J 1 (y; h n ) and we obtain the convergence (3.16) Single it theorem II. In Section 3.2 we have invetigated poible it theorem under the aumption (x, h n ) (x, W ), which implie in particular J t (x ; h n ) δw t. In the current ection we analyze ituation for which the convergence of J t (x i ; h n ) i aumed for a more general i. Our reult are ummarized in the following theorem. Theorem Let y = (y (),..., y (l 1) ) be a V-valued rough path on [, 1] controlled by (x, α) in L 2 or almot urely, and conider h n atifying the inequality (2.14). Recall that the increment J (y; h n ) = {J t (y; h n ); (, t) S 2 } i defined by (2.12). Suppoe that x and h n verify the following aumption: (i) There i ome τ I uch that J t (x τ ; h n ) (t )ϱ in probability for all (, t) S 2, where ϱ (R d ) τ V i a contant matrix, and J t (x i ; h n ) in probability for all i < τ and (, t) S 2. (ii) For any partition < 1 < < m 1 on [, 1] uch that m n and i+1 i, we have 1 m m up n in probability for i = τ + 1,..., l 1. (x i ; h n ) =, (3.3) J +1

18 18 Y. LIU AND S. TINDEL Then the following convergence hold true for y: in probability. J (y; n hn ) ( y (τ) t dt ) ϱ (3.31) Proof. A for Theorem 3.9, we will prove our claim thank to Theorem 3.5, and we are reduced to check (3.12), (3.13), and (3.14). The difference with Theorem 3.9 i that we now conider I = {τ} and I = I \ {τ}. We divide the proof in everal tep. Step 1: Cae i < τ. In thi firt ituation it i immediate from our aumption that (3.13) hold true for i < τ. Step 2: Cae i τ. Similarly to the proof of Theorem 3.9 (Step 1), we prove by induction the following convergence in probability for all l > i τ: ( t J (x i ; h n ) ω i where ωt i = ) xu i τ du ϱ. (3.32) To thi aim, notice that (3.32) i true for i = τ by aumption. Next aume that (3.32) hold for i = τ,..., τ 1. We decompoe the dicrete interval, t into the ubinterval D j again (ee (3.4)), with m n and t k = k, u n j = j. Let m,..., 2 j 1 +1 be a in Theorem 3.9 (Step 1). Then an approximation procedure imilar to (3.21) allow to replace each x τ l ε()t k by an expreion of the form: τ l= x τ l x l t k, in the um defining J t (x τ ; h n ). Therefore, we get an equivalent of (3.21) in our context: with m up J t (x τ ; h n ) n à l = j 2 j 1 τ l= x τ l J +1 (x l ; h n ). à l = (3.33) We now handle each Ãl. For l < τ, each J +1 (x l ; h n ) converge to in probability a n for all j, according to our aumption (i). Hence Ãl in probability a n and m. When τ < l < τ, we proceed along the( ame line a for (3.22) and (3.23). Namely, we invoke the fact that n J +1 ) (x l ; h n j+1 ) = x l τ u du ϱ for each j 1 j j 2 and then ue the extra regularity given by x l u on each [, +1 ]. Thi yield the following it in probability: m up Ãl. (3.34) n

19 Let now l = τ. Then DISCRETE ROUGH PATH 19 Ã τ = j 2 j 1 J +1 (x τ ; h n ), and it i immediate from identity (3.3) that (3.34) hold true for l = τ. In ummary, we have proved that for all l {,..., τ } \ {τ} we have: in probability. m up Ãl = (3.35) n In the cae l = τ, by ending n, our aumption (i) allow to write: Ã τ j 2 j 1 x τ τ (+1 ) ϱ, (3.36) in probability, and thu a n and m, we obtain ( t ) Ã τ x τ τ u du ϱ (3.37) in probability. Combining (3.35) and (3.37) and taking into account relation (3.33), we end up with: ( t ) J t (x τ ; h n ) x τ τ u du ϱ in probability. Thi complete our induction and the proof of (3.32). Step 3: Convergence of J 1 (y (i) ; ω i ) In a imilar way a in the proof of Theorem 3.9 (ee relation (3.29)), we can how the convergence J 1 (y (i) ; ω i ), in probability for i τ, o the convergence (3.13) hold true. On the other hand, it i clear from claical integration theory that: ( ) J 1 (y (τ) ; ω τ ) y u (τ) du ϱ, which implie the convergence (3.14). Summarizing, we have proved (3.12)-(3.14) and the convergence (3.31) follow immediately from Theorem 3.5. Remark A in Remark 3.1, one can generalize Theorem 3.11 to ome other intereting cae. For example, uppoe that (x, J (x τ ; h n )) f.d.d. (x, ω), where ω i a continuou Gauian proce independent of x and with value in (R d ) τ V. A before, let H be the Hilbert pace correponding to ω and uppoe that C γ H for all γ < ν. Suppoe that (3.17) hold true for any f C γ. Then one can how in a imilar way a in Theorem 3.11 that J 1 (y; h n ) d y (τ) dω.

20 2 Y. LIU AND S. TINDEL 3.4. Double it theorem. In thi ubection, we conider the double it theorem cae, which ha been introduced in Definition 3.6. Thi uually correpond to a tranition in term of roughne for the underlying noie x. Theorem Let y = (y (),..., y (l 1) ) be a rough path on [, 1] controlled by (x, α) in L 2 or almot urely, and uppoe that h n atifie the inequality (2.14) uniformly in n. Furthermore, we aume that x and h n fulfill the following condition: (i) We have the convergence (x, h n ) f.d.d. (x, W ) a n, where W i a tandard Brownian motion independent of x. (ii) There exit a contant matrix ϱ (R d ) τ V and ome τ : < τ < l uch that for any partition < 1 < < m 1 on [, 1] uch that m n, i+1 i 1 m, and =, m = t, we have J +1 m n j (x τ ; h n ) = (t )ϱ, (3.38) where the it ha to be undertood a a it in probability for all (, t) S 2 and where J +1 (x i ; h n ) i defined by (2.12). (iii) For any i {1,..., l 1} \ {τ} and any partition < < m 1 in (ii) above, we have the following convergence in probability: m up n (x i ; h n ) =. (3.39) J +1 Then the following it hold true for J (y; h n ): (x, J (y; h n )) f.d.d. (x, y t dw t + ( ) δy (τ) t dt ϱ). Proof. A in Theorem 3.9 and Theorem 3.11, we apply Theorem 3.5 and we are reduced to how relation (3.12), (3.13), and (3.14). In the current ituation, we conider I = {, τ} and I = I \ I. We divide again the proof in everal tep. Step 1: Cae i < τ. A in the proof of Theorem 3.9, by induction we can how that for i < τ we have the convergence J t (x i ; h n ) d ω i t t x i u dw u. (3.4) Step 2: Cae i = τ. An approximation argument imilar to (3.21) and (3.33) yield: with m up J t (x i ; h n ) n à l,i = j 2 j 1 i à l,i = (3.41) l= x i l J +1 (x l ; h n ).

21 DISCRETE ROUGH PATH 21 In the ame way a in (3.24), and taking into account the convergence (3.4), for < l < τ we have the convergence Ãl,τ = m n in probability. On the other hand, in the ame way a for relation (3.26), the following it hold true for l = : Ã,τ m n (d) = t In addition, owing to aumption (3.38) we have x τ u dw u. à τ,τ = (t )ϱ, m n where the it hold in probability. In ummary of the convergence of à l,τ, l =,..., τ and taking into account (3.41), we obtain J t (x τ ; h n ) d t x τ u dw u + (t )ϱ. Notice that we can add up it in ditribution here, ince one of the it i determinitic. Step 3: Cae i > τ. In the following, we how by induction the convergence t ( t ) J t (x i ; h n d ) ωt i x i u dw u + xu i τ du ϱ, (3.42) for l > i τ. Indeed, we have hown that convergence (3.42) hold when i = τ. Now uppoe that the convergence hold for i = τ,..., τ 1, and we wih to propagate the induction aumption. Thank to the induction aumption and in a imilar way a in (3.24) we can how that for l {1,..., τ 1} \ {τ} we have Ãl,τ =, (3.43) m n where the it i undertood in probability. Moreover, invoking aumption (3.39) we alo have the following it in probability: m up Ãτ,τ =. (3.44) n On the other hand, we let the patient reader check that ( t ) à τ,τ xu i τ du ϱ (3.45) in probability, imilarly to what ha been done in (3.36) and (3.37). Taking into account (3.43), (3.44), (3.45) and (3.41), it i eaily checked that (3.42) for i = τ i reduced to the following convergence: ( t Ã,τ + ) x τ τ u du ϱ d t x τ u dw u + ( t ) x τ τ u du ϱ, (3.46)

22 22 Y. LIU AND S. TINDEL a n and then m. In order to prove (3.46), we firt fix m and let n go to. Then, owing to the fact that (x, h n ) f.d.d. (x, W ), we get that n ( t Ã,τ + ) x τ τ u du ϱ (d) = x τ δw j +1 + ( t ) x τ τ u du ϱ. Then, conditioning on x and conidering it of Riemann um for Wiener integral, we end up with: m x τ δw j +1 + ( t ) x τ τ u du ϱ L 2 = t x τ udw u + ( t ) x τ τ u du ϱ, from which (3.46), and thu (3.42) for i = τ are eaily deduced. Therefore, we can conclude by induction that the convergence (3.42) hold for all i = τ,..., l 1. Step 4: Proof of (3.13) and (3.14). Recall that ω i i defined by relation (3.4) when i < τ and by (3.42) when i τ. For i = 1,..., l 1, i and i τ, a in the proof of Theorem 3.9 (ee relation (3.29)) we can how that (3.13) hold. On the other hand, it i eay to how by claical integration argument that in probability and J 1 (y (τ) ; ω τ ) J 1 (y; ω ) ( ) y u (τ) du ϱ (3.47) y u dw u (3.48) in probability, by convergence of Riemann um for Wiener integral. Putting together (3.47) and (3.48) and invoking the ame argument a in Step 3, we can conclude that (3.14) i atified. In concluion, we have checked condition (3.12)-(3.14), and our reult follow directly from Theorem Breuer-Major theorem In thi ection, we conider generalization of Breuer-Major theorem [4]. Notice that recent contribution (ee e.g. [31, 32, 33, 35]) to thi area involving weighted um of tationary equence motly conider equence of functional of one-dimenional fractional Brownian motion (fbm). Thi i why we alo tick to the one-dimenional fbm cae, though multidimenional tudie for more general Gauian procee do not eem out of reach in our framework. Alo oberve that the aforementioned reference focu on equence in a fixed chao or in a finite um of chao. In contrat, we will be able to handle general equence in L 2 with repect to a Gauian meaure Weighted Breuer-Major theorem I. In thi ubection, we conider the weighted type Breuer-Major theorem in the context of our ingle it Theorem 3.9. Let u firt introduce ome additional notation. Let dγ(t) = (2π) 1/2 e t2 /2 dt be the tandard Gauian meaure on the real line, and let f L 2 (γ) be uch that f(t)dγ(t) =. R

23 DISCRETE ROUGH PATH 23 It i well-known that the function f can be expanded into a erie of Hermite polynomial a follow: f(t) = a q H q (t), q=d e t 2 dt q where d 1 i ome integer and H q (t) = ( 1) q e t2 2 dq 2 i the Hermite polynomial of order q. If a d, then d i called the Hermite rank of the function f. Note that ince f L 2 (γ), we have q=d a q 2 q! <. Our underlying proce X i a one-dimenional Gauian equence. For uch a proce the baic tool to meaure dependence are baed on correlation function. Throughout thi ubection, we aume that the following hypothei on correlation hold true. Hypothei 4.1. Let X k, k Z be a centered tationary Gauian equence uch that X k ha unit variance. Denote ρ(k) = E(X X k ). We uppoe that k Z ρ(k) d < for ome d 1. For ake of conciene we will not recall the baic notion of Gauian analyi which will be ued in thi ection. The intereted reader i referred to [34] for further detail. We now recall a claical verion of Breuer-Major theorem. Theorem 4.2. Let {X k, k Z} be a centered tationary Gauian equence atifying Hypothei 4.1 for d 1. Conider f L 2 (γ) with rank d. For n 1, let = t < < t n = 1 be the uniform partition of [, 1] defined in Section 1. we et h n t = t k <t f(x k) for all (, t) S 2. Then the following central it theorem hold true: h n / n f.d.d. σw a n, where the variance σ 2 [, ) i defined by: σ 2 = q!a 2 q ρ(k) q. (4.1) q=d In thi ubection we pecialize Theorem 4.2 to a ituation where X k = n ν δx tk t k+1, where x i a fbm with Hurt parameter ν. In thi context we are intereted in the following quetion: (1) Do we have the convergence of the weighted um k Z 1 n 1 y k f(x k ) a n, (4.2) n for a general weight y k? (2) Doe the central it theorem for (4.2) till hold in general? We will give a complete anwer to thee two quetion when the weight proce y i a controlled proce a introduced in Definition 2.3. Before we tart our dicuion, let u recall ome baic fact about fbm. (i) If x i a one-dimenional fbm with Hurt parameter ν, then x i almot urely γ-hölder continuou for all γ < ν. (ii) For a fbm x, the covariance function ρ alluded to in Hypothei 4.1 i defined by ρ(k) = E(δx 1 δx k,k+1 ). (4.3)

24 24 Y. LIU AND S. TINDEL Then, whenever ν < 1, we have 2 k Z ρ(k) =. We alo label the following notation for further ue. Notation 4.3. Let x be a one-dimenional fbm with Hurt parameter ν. We conider x a a (L p, ν, l) rough path according to Definition 2.1, where p i any real number in [1, ) and l i the mallet integer atifying νl > 1. In addition, we will chooe xi t = 1 i! (δx t) i for all (, t) S 2 and i = 1,..., l for l N. Let u recall the following identity of multiple Wiener integral. The reader i referred to e.g. [23, 34, 37] for more detail: Lemma 4.4. Let f L 2 ([, 1] p ) and g L 2 ([, 1] q ) be ymmetric function. Then we have the identity p q ( )( ) p q I p (f)i q (g) = r! I p+q 2r (f r g), (4.4) r r r= where I p (f) i the pth multiple Wiener integral of f, and ( ) p r = p! t k <t r!(p r)!. Let H be the completion of the pace of indicator function with repect to the inner product 1 [u,v], 1 [,t] H = E(δx uv δx t ). Let = t < < t n = 1 be the uniform partition of [, 1] alluded to the above and < < m 1 be another partition of [, 1] with m n. In the following, we take l uch that l 1 1 < l (or equivalently l i the mallet 2ν integer uch that νl + α > 1 with α = 1 ). We et 2 h n,q t = H q (n ν δx tk,t k+1 ) and ζ i,q j = J +1 (x i ; h n,q ) (4.5) for k =,..., n 1, j =,..., m 1, i =,..., l 1 and q N, where J +1 (x i ; h n,q ) i given by (2.12). We denote by ϑ(q, q, i) the following quantity ( ϑ(q, q, i) := E We will need the following auxiliary reult. j,j = ζ i,q j ζ i,q j ). (4.6) Lemma 4.5. Let x be a one-dimenional fbm on [, 1] with Hurt parameter ν 1. Take 2 i = 1,..., l 1, where we recall that l atifie l 1 1 < l. Then for 2ν q, q l the following etimate hold true: (i) When q q 2i, we have ϑ(q, q, i) K(n 1 2ν + nm 2iν + n 1 ν ) q r= 1 2 (q+q ) i where ϑ(q, q, i) i defined by (4.6) and K i a poitive univeral contant. (ii) When q q > 2i, we have ( )( ) q q r!, (4.7) r r ϑ(q, q, i) =. (4.8)

25 DISCRETE ROUGH PATH 25 (iii) When q q 2i, the following inequality hold true for all (, t) S 2 : q ( )( ) q q E(J t (x i ; h n,q )J t (x i ; h n,q )) Kn(t ) 2iν+1 r!. (4.9) r r (iv) When q q > 2i, for all (, t) S 2 we have: E(J t (x i ; h n,q )J t (x i ; h n,q )) =. r= 1 2 (q+q ) i Remark 4.6. Notice that our aumption imply in particular that q q > 1. Thi i alo 2ν the condition on the Hermite rank of f which will feature in Theorem 4.7 below. Proof of Lemma 4.5. Step 1: Without lo of generality let u aume that q q. By the definition of ζ i,q j we can write ϑ(q, q, i) = j,j = t k <+1 t k < +1 a(j, j, k, k ), (4.1) where, recalling that ε( ) i defined by (2.12), we have ( ) a(j, j, k, k ) = E x i ε( )t k x i ε( )t H k q(n ν δx tk,t k+1 )H q (n ν δx tk,t k +1 ). Now et β k = n ν 1 [tk,t k+1 ]. Recalling that H q (n ν δx tk t k+1 ) = I q (β q ) and invoking identity (4.4), we eaily obtain: q ( )( ) q q a(j, j, k, k ( ) ) = r! E x i ε( r r j )t k x i ε( )t I k q+q 2r(β q r k β q r k ) β k, β k r H. r= Now oberve that β k, β k H = ρ(k k ), where the covariance function ρ i defined by (4.3). Therefore, owing to an application of integration by part, we end up with the following identity: q ( )( ) q q a(j, j, k, k ) = r! b(r)ρ(k k ) r, (4.11) r r r= where b(r) i the coefficient defined by: b(r) = E D q+q 2r (x i ε( )t k x i ε( )t k ), β (q r) k β (q r) k H (q+q 2r). (4.12) Step 2: Conider q l. Due to the fact that x i belong to the um of the firt i chao, when q q > 2i, it i eay to ee that D q+q 2r (x i ε( )t k x i ε( )t k ) = (4.13) for all r =,..., q. Taking into account (4.11), thi implie that whenever q q > 2i we have and thu the etimate in (4.8) hold. a(j, j, k, k ) =, (4.14) In the following, we aume that q q 2i and we focu on inequality (4.7). Note firt that ince q q and q l, we have 1 2 (q + q ) i q (l 1) >.