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Ann. 1 equence , nthe In . dNu- nd rocesses rocess mples cular a flength of val rder ters and cs incre- ocess aw nd 37 W tional con- = 2 D. HARNETT AND D. NUALART s is bounded by C√s, and the covariance between the increments in the intervals [t s,t], and [r s, r] is bounded by − − 2 α β 2 3/2 s t r − (r s)− + s t r − , | − | − | − | 3 3 if 0 < 2s r nt/2 ⌊ ⌋ Φn(t) := f ′(W(2j 1)/n)(W2j/n W(2j 2)/n). − − − Xj=1 The limit of this sum as n tends to infinity is the Stratonovich midpoint t integral, denoted by 0 f ′(Ws)◦ dWs. We show that the couple of processes 2 (Wt, Φn(t)),t 0 converges in distribution in the Skorohod space (D[0, )) to{ (W , Φ(t)),t≥ }0 ,R where ∞ { t ≥ } 1 t Φ(t)= f(W ) f(W ) f ′′(W ) dB , t − 0 − 2 s s Z0 and B = Bt,t 0 is a Gaussian martingale independent of W with vari- ance η(t),{ depending≥ } on the covariance properties of W . This limit theorem can be reformulated by saying that the following Itˆoformula in distribution holds: t 1 t (1) f(W ) =L f(W )+ f ′(W )◦ dW + f ′′(W ) dB . t 0 s s 2 s s Z0 Z0 The above mentioned convergence is proven by showing the stable con- vergence of a d-dimensional vector (Φn(t1),..., Φn(td)) and a tightness ar- gument. To show the convergence in law of the finite-dimensional distribu- tions, we show first, using the techniques of Malliavin calculus, that Φn(t) is asymptotically equivalent to a sequence of iterated Skorohod integrals involving f ′′(Wt). We then apply our d-dimensional version of the central limit theorem for multiple Skorohod integrals proved by Nourdin and Nu- alart in [5]. Recent papers by Swanson [10], Nourdin and R´eveillac [6] and Burdzy and Swanson [2] presented results comparable to (1) for a specific stochastic process. In [10], a change-of-variable form was found for a process equivalent to the bifractional Brownian motion with parameters H = K = 1/2, arising as the solution to the one-dimensional stochastic heat equation with an ad- ditive space–time white noise. This result was proven mostly by martingale methods. In [2] and [6], the respective authors considered fractional Brown- ian motion with Hurst parameter 1/4. In [2], the authors covered integrands of the form f(t,Wt), which can be applied to fBm on [ε, ). The authors of [6] proved a change-of-variable formula that holds on [0∞, ) in the sense ∞ CLT FOR A STRATONOVICH INTEGRAL 3 of marginal distributions. The proof in [6] uses Malliavin calculus; several similar methods were used in the present paper. More recently, Nourdin, R´eveillac and Swanson [7] studied the case of fractional Brownian motion with H = 1/6. In that paper, weak convergence was proven in the Skorohod space, and the Riemann sums are based on the trapezoidal approximation. It happens that the conditions on the process W are satisfied by a bifrac- tional Brownian motion with parameters H 1/2, HK = 1/4. In this case, η(t)= Ct and the process B is a Brownian motion.≤ This includes both cases studied in [6] and [10], and extends to a larger class of processes. For an- other example, we consider a class of centered Gaussian processes with twice- differentiable covariance function of the form t E[W W ]= rφ , t r, r t r ≥ where φ is a bounded function on [1, ) such that ∞ κ ψ(x) φ′(x)= + , √x 1 √x − 1/2 and ψ is bounded, differentiable and ψ′(x) C(x 1)− . This class of Gaussian processes includes the process| arising|≤ as the− limit of the median of a system of independent Brownian motions studied by Swanson in [9]. For this process, 1 φ(x)= √x arctan . √x 1 − It is surprising to remark that in this case η(t)= Ct2. This is related to the fact that the variance of the increments of W on the interval [t s,t] behaves as C√s, when s is small, although the variance of W (t) behaves− as Ct. Our third example is another Gaussian process studied by Swanson in [11]. This process also arises from the empirical quantiles of a system of independent Brownian motions. Let B = B(t),t 0 be a Brownian motion, where B(0) { ≥ } is a random variable with density f ∞. Given certain growth conditions on f, Swanson proves there is a Gaussian∈C process F = F (t),t 0 with covariance given by { ≥ } P(B(r) q(r),B(t) q(t)) α2 E[F (r)F (t)] = ρ(r, t)= ≤ ≤ − , u(q(r), r)u(q(t),t) where α (0, 1) and q(t) are defined by P(B(t) q(t)) = α. It is shown that this∈ family of processes satisfies the required≤ conditions, where η(t) is determined by f and α. The outline of this paper is as follows: In Section 2, we introduce the basic environment and recall some aspects of Malliavin calculus that will be 4 D. HARNETT AND D. NUALART used. In Section 3, a multi-dimensional version of a central limit theorem that appears in [5] is given. In Section 4, the theorem is applied to prove convergence of Φn(t). Section 5 discusses three examples of suitable process families. Finally, Section 6 contains proofs of three of the longer lemmas from Section 4. Most of the notation in this paper follows that of [5]. 2. Preliminaries and notation. Let W = W (t),t 0 be a centered Gaussian process defined on a probability space{ (Ω, ,≥ P )} with continuous covariance function F E[W (t)W (s)] = R(t,s). We will always assume that is the σ-algebra generated by W . Let denote the set of step functions onF [0, T ] for T > 0; and let H be the HilbertE space defined as the closure of with respect to the scalar product E 1 , 1 H = R(t,s). h [0,t] [0,s]i 1 The mapping [0,t] W (t) can be extended to a linear isometry between H and the Gaussian7→ space spanned by W . We denote this isometry by h W (h). In this way, W (h), h H is an isonormal Gaussian process. For7→ {q ∈ } q integers q 1, let H⊗ denote the qth tensor product of H. We use H⊙ to denote the≥ symmetric tensor product. For integers q 1, let q be the qth Wiener chaos of W , that is, the closed linear subspace of≥L2(Ω)H generated by the random variables H (W (h)), h { q ∈ H, h H = 1 , where H (x) is the qth Hermite polynomial, defined as k k } q q q x2/2 d x2/2 H (x) = ( 1) e e− . q − dxq For q 1, it is known that the map ≥ q (2) Iq(h⊗ )= Hq(W (h)) q provides an isometry between the symmetric product space H⊙ (equipped 1 with the modified norm H⊗q ) and . By convention, = R and √q! k · k Hq H0 I0(x)= x. 2.1. Elements of Malliavin calculus. Following is a brief description of some identities that will be used in the paper. The reader may refer to [5] for a brief survey, or to [8] for detailed coverage of this topic. Let be the set of all smooth and cylindrical random variables of the form SF = g(W (φ ),...,W (φ )), where n 1; g : Rn R is an infinitely differentiable 1 n ≥ → function with compact support, and φi H. The Malliavin derivative of F with respect to W is the element of L2(Ω∈, H) defined as n ∂g DF = (W (φ1),...,W (φn))φi. ∂wi Xi=1 CLT FOR A STRATONOVICH INTEGRAL 5 In particular, DW (h)= h. By iteration, for any integer q> 1, we can define q 2 q the qth derivative D F , which is an element of L (Ω, H⊙ ). For example, if 2 1 2 F = g(W (t)), then D F = g′′(W (t)) [0⊗,t]. For any integer q 1 and real number p 1, let Dq,p denote the closure ≥ ≥ of with respect to the norm Dq,p defined as S k · k q p E p E i p F Dq,p = [ F ]+ [ D F ⊗ ]. k k | | k kH i Xi=1 We denote by δ the Skorohod integral, which is defined as the adjoint of the operator D. This operator is also referred to as the divergence operator in [8]. A random element u L2(Ω, H) belongs to the domain of δ, Dom δ, if and only if ∈ 2 E[ DF, u H] c E[F ] | h i |≤ u 1,2 q for any F D , where cu is a constant which depends only on u. If u Dom δ, then∈ the random variable δ(u) L2(Ω) is defined for all F D1,2 by∈ the duality relationship, ∈ ∈ E[F δ(u)] = E[ DF, u H]. h i This is sometimes called the Malliavin integration by parts formula. We q 1 iteratively define the multiple Skorohod integral for q 1 as δ(δ − (u)), with δ0(u)= u. For this definition we have ≥ q q E[F δ (u)] = E[ D F, u ⊗q ], h iH where u Dom δq and F Dq,2. Moreover, if h H q, then we have δq(h)= ∈ ∈ ∈ ⊙ Iq(h). For f, g H p, the following integral multiplication formula holds: ∈ ⊗ p p p p 2p 2r (3) δ (f)δ (g)= r! δ − (f g), r ⊗r r=0 X where r is the contraction operator; see, for example, [5], Section 2. We will⊗ use the Meyer inequality for the Skorohod integral; see, for exam- k,p k ple, Proposition 1.5.7 of [8]. Let D (H⊗ ) denote the corresponding Sobolev k space of H⊗ -valued random variables. Then for p 1 and integers k q 1, we have ≥ ≥ ≥ q (4) δ (u) Dk−q,p c u Dk,p ⊗q k k ≤ k,pk k (H ) k,p k for all u D (H⊗ ) and some constant ck,p. The following∈ three results will be used in the proof of Theorem 4.3. The reader may refer to [5] and [8] for details. 6 D. HARNETT AND D. NUALART Lemma 2.1. Let q 1 be an integer. ≥ (1) Assume F Dq,2, u is a symmetric element of Dom δq, and DrF , j 2 ∈ q r j r h r δ (u) H⊗r L (Ω, H⊗ − − ) for all 0 r +j q. Then D F, u H⊗r Dom δ and i ∈ ≤ ≤ h i ∈ q q q q r r F δ (u)= δ − ( D F, u ⊗r ). r h iH Xr=0 j+k,2 j (2) Suppose that u is a symmetric element of D (H⊗ ). Then we have j k ∧ k j Dkδj (u)= i!δj i(Dk iu). i i − − Xi=0 2q,2 q (3) Let u, v be symmetric functions in D (H⊗ ). Then q 2 q q q q i q i E[δ (u)δ (v)] = E[ D − u, D − v ⊗(2q−i) ]. i h iH Xi=0 In particular, q 2 q 2 q 2 q q i 2 δ (u) 2 = E[δ (u) ]= E[ D − u ⊗(2q−i) ]. k kL (Ω) i k kH Xi=0 Proof of (1). This is proved in [5]; see Lemma 2.1. It follows by induction from the relation F δ(u)= δ(F u)+ DF, u H; see [8], Proposition 1.3.3. h i Proof of (2). This follows from repeated application of the relation Dδ(u)= u + δ(Du); see [8], Proposition 1.3.2. Proof of (3). This follows from repeated application of the duality prop- erty; see [5], equation (2.12). 3. A central limit theorem for multiple Skorohod integrals. Let X = X(h), h H be an isonormal Gaussian process associated with a real- separable{ ∈ Hilbert} space H, defined on a probability space (Ω, , P ). We assume that is generated by X. The purpose of this section isF to prove a multi-dimensionalF version of a theorem proved in [5]; see Theorem 3.1. We begin by defining the notion of stable convergence. Definition 3.1. Assume Fn is a sequence of d-dimensional random variables defined on a probability space (Ω, , P ), and F is a d-dimensional F random variable defined on (Ω, , P ), where . We say that Fn converges stably to F as n , if, for anyG continuous andF ⊂ bounded G function f : Rd R and bounded, R→∞-valued, -measurable random variable Z, we have → F lim E(f(Fn)Z)= E(f(F )Z). n →∞ CLT FOR A STRATONOVICH INTEGRAL 7 Theorem 3.2. Let q 1 be an integer, and suppose that Fn is a sequence Rd≥ q q 1 q d of random variables in of the form Fn = δ (un) = (δ (un),...,δ (un)), for d 2q,2q q a sequence of R -valued symmetric functions un in D (H⊗ ). Suppose that 1 the sequence Fn is bounded in L (Ω, H) and that: j m a jℓ 1 (a) un, (D ℓ Fn ) h H⊗q converges to zero in L (Ω) for all integers h ℓ=1 ⊗ i 1 j, jℓ d, all integers 1 a1, . . ., am, r q 1 such that a1 + +am +r = ≤ ≤ N r ≤ ≤ − · · · q; and all h H⊗ . ∈ i q j 1 (b) For each 1 i, j d, u , D Fn H⊗q converges in L (Ω, H) to a ran- ≤ ≤ h n i dom variable sij, such that the matrix Σ := (sij)d d is nonnegative definite (i.e., λT Σλ 0 for all nonzero λ Rd). × ≥ ∈ d Then Fn converges stably to a random variable in R with conditional Gaus- sian law (0, Σ) given X. N Remark 3.3. Conditions (a) and (b) mean that for q 1, some com- binations of lower-order derivative products are negligible. For≥ example, for q = 2, then the following scalar products will converge to zero in L1(Ω, H): i un, h1 h2 H⊗2 for all h1, h2 H; • h i ⊗j i ∈ un, DFn h H⊗2 for all h H and all j (including i = j); • h i j ⊗ i k ∈ u , DFn DF H⊗2 for all 1 k, j d. • h n ⊗ n i ≤ ≤ Only the qth-order derivative products converge to a nontrivial random vari- i q j able. Usually (see Section 6), the term un, D Fn H⊗q has the same asymp- i j h i totic behavior as u , un H⊗q . h n i Remark 3.4. It suffices to impose condition (a) for h 0, where 0 is r ∈ S S a total subset of H⊗ . Proof of Theorem 3.2. As in the one-dimensional case considered in [5], we will use the conditional characteristic function. Given any h1,..., h H, we want to show that the sequence m ∈ 1 d ξn = (Fn ,...,Fn , X(h1),...,X(hm)) 1 d converges in distribution to a vector (F ,...,F , X(h1),...,X(hm)), where, for any vector λ Rd, F satisfies ∞ ∞ ∈ ∞ iλ F∞ 1 T (5) E(e · X(h ),...,X(h )) = exp( λ Σλ), | 1 m − 2 d j Rd where λ Fn = j=1 λjFn denotes the usual scalar product in , and we use this notation· to avoid confusion with the scalar product in H. P 1 Since Fn is bounded in L (Ω, H), the sequence ξn is tight in the sense that d for any ε> 0, there is a K> 0 such that P (Fn [ K, K] ) > 1 ε, which follows from Chebyshev inequality. Dropping to∈ a subsequen− ce if− necessary, 1 d we may assume that ξn converges in distribution to a limit (F ,...,F , ∞ ∞ 8 D. HARNETT AND D. NUALART Rm X(h1),...,X(hm)). Let Y := g(X(h1),...,X(hm)), where g b∞( ), and iλ Fn d ∈C consider φn(λ)= φ(λ, ξn) := E(e · Y ) for λ R . The convergence in law of ξ implies that for each 1 j d, ∈ n ≤ ≤ ∂φn j iλ Fn j iλ F∞ (6) lim = lim iE(F e · Y )= iE(F e · Y ), n ∂λ n n →∞ j →∞ ∞ where convergence in distribution follows from a truncation argument ap- j plied to Fn. On the other hand, using the duality property of the Skorohod integral and the Malliavin derivative, ∂φ n n E q j iλ Fn E j q iλ F = i (δ (un)e · Y )= i ( un, D (e · Y ) H⊗q ) ∂λj h i q q j a iλ Fn q a (7) = i E( u , D (e · ) D − Y ⊗q ) a h n ⊗ iH a=0 X q 1 −e j q iλ Fn q j a iλ Fn q a = i E u , Y D e · ⊗q + E u , D e · D − Y ⊗q . h n iH a h n ⊗ iH ( a=0 ) X j a iλ Fn q a By condition (a), we have that un, D e · D − Y He⊗q converges to zero in L1(Ω) when a ∂ iλ F∞ E(e · X(h1),...,X(hm)) ∂λj | d iλ F∞ = λ s E(e · X(h ),...,X(h )), − k kj | 1 m Xk=1 which has unique solution (5). CLT FOR A STRATONOVICH INTEGRAL 9 4. Central limit theorem for the Stratonovich integral. Suppose that W = W ,t 0 is a centered Gaussian process, as in Section 2, that meets { t ≥ } conditions (i) through (v), below, for any T > 0, where the constants Ci may depend on T : (i) For any 0 Then for each t 0, ≥ + + lim η (t)= η (t) and lim η−(t)= η−(t) n n n n →∞ →∞ + both exist, where η (t), η−(t) are nonnegative and nondecreasing functions. Consider a real-valued function f 9(R), such that f and all its deriva- tives up to order 9 have at most exponential∈C growth, that is, f (k)(x) < K exp(K x α), x R,α< 2, | | 1 2| | ∈ for k = 0,..., 9, and positive constants K1, K2. We will refer to this as con- dition (0). In the following, the term C represents a generic positive constant, which may change from line to line. The constant C may depend on T and the constants in conditions (0) and (i)–(v), listed above. The results of the next lemma follow from conditions (i) and (ii). Lemma 4.1. Using the notation described above, for integers 0 a < b and integers r, n 1, we have the estimate ≤ ≥ b r r/2 β (j, k) C(b a + 1)n− . | n | ≤ − j,kX=a Proof. Suppose first that r = 1. Let I = (j, k) : a j, k b, k j 2, j k 2 and J = (j, k) : a j, k b, (j, k) /{I . Consider≤ the≤ decompo-| − |≥ sition∧ ≥ } { ≤ ≤ ∈ } b β (j, k) = β (j, k) + β (j, k) . | n | | n | | n | j,k=a (j,k) I (j,k) J X X∈ X∈ Then by condition (ii), the first sum is bounded by 1/2 α 1/2 n− j k − Cn− (b a + 1), | − | ≤ − (j,k) I X∈ and the second sum, using condition (i) and Cauchy–Schwarz, is bounded 1/2 by Cn− (b a + 1). For the case r> 1, condition (i) implies βn(j, k) 1/2 − | |≤ C1n− for all j, k. It follows that we can write b b r (r 1)/2 β (j, k) C n− − β (j, k) | n | ≤ 1 | n | j,kX=a j,kX=a r/2 C(b a + 1)n− . ≤ − CLT FOR A STRATONOVICH INTEGRAL 11 Corollary 4.2. Using the notation of Lemma 4.1, for each integer r 1, ≥ nt/2 ⌊ ⌋ ( β (2j 1, 2k 1) r + β (2j 1, 2k 2) r | n − − | | n − − | j,kX=1 + β (2j 2, 2k 1) r + β (2j 2, 2k 2) r) | n − − | | n − − | nt r/2 C n− . ≤ 2 Proof. Note that nt/2 ⌊ ⌋ ( β (2j 1, 2k 1) r + β (2j 1, 2k 2) r | n − − | | n − − | j,kX=1 + β (2j 2, 2k 1) r + β (2j 2, 2k 2) r) | n − − | | n − − | 2 nt/2 1 ⌊ ⌋− r = βn(j, k) . | | j,kX=0 Consider a uniform partition of [0, ) with increment length 1/n. The ∞ Stratonovich midpoint integral of f ′(W ) will be defined as the limit in dis- tribution of the sequence (see [10]) nt/2 ⌊ ⌋ (8) Φn(t) := f ′(W(2j 1)/n)(W2j/n W(2j 2)/n). − − − Xj=1 1 We introduce the following notation, as used in [5]: εt := [0,t]; and ∂j/n := 1 [j/n,(j+1)/n]. The following is the major result of this section. Theorem 4.3. Let f be a real function satisfying condition (0), and let W = Wt,t 0 be a Gaussian process satisfying conditions (i) through (v). Then{ ≥ } 1 t (W , Φ (t)) L W ,f(W ) f(W ) f ′′(W ) dB t n −→ t t − 0 − 2 s s Z0 as n in the Skorohod space (D[0, ))2, where η(t)= η+(t) η (t) for →∞ ∞ − − the functions defined in condition (v); and B = Bt,t 0 is scaled Brown- { E ≥2 } ian motion, independent of W , and with variance [Bt ] = 2η(t). The rest of this section consists of the proof of Theorem 4.3, and is pre- sented in a series of lemmas. The proofs of Lemmas 4.4, 4.5 and 4.9, which 12 D. HARNETT AND D. NUALART are rather technical, are deferred to Section 6. We begin with an expansion of f(Wt), following the methodology used in [10]. Consider the telescoping series nt/2 ⌊ ⌋ f(Wt)= f(W0)+ [f(W2j/n) f(W(2j 2)/n)] − − Xj=1 + f(Wt) f(W(2/n) nt/2 ), − ⌊ ⌋ nt where the sum is zero by convention if 2 = 0. Using a Taylor series ex- pansion of order 2, we obtain ⌊ ⌋ Φ (t)= f(W ) f(W ) n t − 0 nt/2 ⌊ ⌋ 1 2 2 f ′′(W(2j 1)/n)(∆W2j/n ∆W(2j 1)/n) − 2 − − − Xj=1 nt/2 nt/2 ⌊ ⌋ ⌊ ⌋ R0(W2j/n)+ R1(W(2j 2)/n) − − Xj=1 Xj=1 (f(Wt) f(W(2/n) nt/2 )), − − ⌊ ⌋ where R0,R1 represent the third-order remainder terms in the Taylor ex- pansion, and can be expressed in integral form as 1 W2j/n (9) R (W )= (W u)2f (3)(u) du 0 2j/n 2 2j/n − ZW(2j−1)/n and W(2j−1)/n 1 2 (3) (10) R1(W(2j 2)/n)= (W(2j 2)/n u) f (u) du. − −2 − − ZW(2j−2)/n By condition (0) we have for any T > 0 that E lim sup f(Wt) f(W(2/n) nt/2 ) = 0, n 0 t T | − ⌊ ⌋ | →∞ ≤ ≤ so this term vanishes uniformly on compacts in probability (ucp), and may be neglected. Therefore, it is sufficient to work with the term (11) ∆ (t) := f(W ) f(W ) 1 Ψ (t)+ R (t), n t − 0 − 2 n n where nt/2 ⌊ ⌋ 2 2 Ψn(t)= f ′′(W(2j 1)/n)(∆W2j/n ∆W(2j 1)/n) − − − Xj=1 CLT FOR A STRATONOVICH INTEGRAL 13 and nt/2 ⌊ ⌋ Rn(t)= (R1(W(2j 2)/n) R0(W2j/n)). − − Xj=1 We will first decompose the term Ψn(t), using a Skorohod integral rep- resentation. Using (2) and the second Hermite polynomial, one can write 2 2 2 ∆W (h)= H2(∆W (h))+1= δ (h⊗ ) + 1 for any h H with h H =1. It follows that ∈ k k nt/2 ⌊ ⌋ 2 2 2 Ψn(t)= f ′′(W(2j 1)/n)δ (∂(2⊗j 1)/n ∂(2⊗j 2)/n). − − − − Xj=1 From Lemma 2.1, we have for random variables u, F , 2 2 2 F δ (u)= δ (F u) + 2δ( DF, u H)+ D F, u ⊗2 , h i h iH so we can write nt/2 ⌊ ⌋ 2 2 2 Ψn(t)= δ (f ′′(W(2j 1)/n)(∂(2⊗j 1)/n ∂(2⊗j 2)/n)) − − − − Xj=1 nt/2 ⌊ ⌋ (3) 2 2 + 2δ(f (W(2j 1)/n) ε(2j 1)/n, ∂(2⊗j 1)/n ∂(2⊗j 2)/n H) − h − − − − i Xj=1 nt/2 ⌊ ⌋ (4) 2 + f (W(2j 1)/n)( ε(2j 1)/n, ∂(2j 1)/n H − h − − i Xj=1 2 ε(2j 1)/n, ∂(2j 2)/n H) − h − − i := Fn(t)+ Bn(t)+ Cn(t). Hence, we have ∆ (t)= f(W ) f(W ) 1 (F (t)+ B (t)+ C (t)) + R (t). n t − 0 − 2 n n n n In the next two lemmas, we show that the terms Bn(t),Cn(t) and Rn(t) converge to zero in probability as n . The proofs of these lemmas are deferred to Section 6. →∞ Lemma 4.4. Let 0 r Lemma 4.5. Let 0 r 2 nt nr 3/2 E[(B (t) B (r)) ] C n− n − n ≤ B 2 − 2 and 2 nt nr 3/2 E[(C (t) C (r)) ] C n− . n − n ≤ C 2 − 2 It follows that for any 0 t T , Bn(t) and Cn(t) converge to zero in prob- ability as n . ≤ ≤ →∞ 1 1 Corollary 4.6. Let Zn(t) := Rn(t) 2 Bn(t) 2 Cn(t). Then given 0 t Next, we will develop a comparable estimate for differences of the form Fn(t) Fn(r). In order to prove this estimate, we need a technical lemma which− will be used here and also in Section 6. Lemma 4.7. Suppose a, b are nonnegative integers such that a + b 9. For fixed T > 0 and interval [t ,t ] [0, T ], let ≤ 1 2 ⊂ nt2/2 ⌊ ⌋ (a) 2 2 ga = f (W(2ℓ 1)/n)(∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n). − − − − ℓ= nt1/2 +1 ⌊ X ⌋ CLT FOR A STRATONOVICH INTEGRAL 15 Then we have for 1 p< ≤ ∞ p/2 b p nt2 nt1 p/2 E[ D g ] C n− . k akH⊗2+b ≤ 2 − 2 Proof. We may assume t = 0 with t T . For each b we can write 1 2 ≤ b 2 p/2 E[( D g ⊗2+b ) ] k akH nt2/2 ⌊ ⌋ E (a+b) (a+b) = f (W(2ℓ 1)/n)f (W(2m 1)/n) " − − ℓ,mX=1 b b ε(2⊗ℓ 1)/n, ε(2⊗m 1)/n H⊗b × h − − i p/2 2 2 2 2 ∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n, ∂(2⊗m 1)/n ∂(2⊗m 2)/n H⊗2 × h − − − − − − i ! # p/2 E (a+b) p b sup f (Ws) sup ε(2ℓ 1)/n, ε(2m 1)/n H ≤ 0 s t| | ℓ,m |h − − i | h ≤ ≤ i nt2/2 p/2 ⌊ ⌋ 2 2 2 2 ∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n, ∂(2⊗m 1)/n ∂(2⊗m 2)/n H⊗2 . × |h − − − − − − i |! ℓ,mX=1 Recall that condition (0) holds for f and its first 9 derivatives, so the first two terms are bounded. For the last term, note that by Corollary 4.2 with r = 2, nt2/2 ⌊ ⌋ 2 2 2 2 ∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n, ∂(2⊗m 1)/n ∂(2⊗m 2)/n H⊗2 |h − − − − − − i | ℓ,mX=1 nt2/2 ⌊ ⌋ = β (2ℓ 1, 2m 1)2 β (2ℓ 1, 2m 2)2 | n − − − n − − ℓ,mX=1 β (2ℓ 2, 2m 1)2 + β (2ℓ 2, 2m 2)2 − n − − n − − | nt2 1 C n− . ≤ 2 Lemma 4.8. For 0 s Then given 0 t nt2/2 ⌊ ⌋ 2 2 un = f ′′(W(2j 1)/n)(∂(2⊗j 1)/n ∂(2⊗j 2)/n) − − − − j= nt1 +1 ⌊X2 ⌋ and 4 4 4 2 4 u D2,4 ⊗2 = E u ⊗2 + E Du ⊗3 + E D u ⊗4 . k nk (H ) k nkH k nkH k nkH From Lemma 4.7 we have E u 4 , E Du 4 , E D2u 4 C( nt2 k nkH⊗2 k nkH⊗3 k nkH⊗4 ≤ ⌊ 2 ⌋− nt1 )2n 2, and so it follows that ⌊ 2 ⌋ − 2 2 4 nt2 nt1 2 E[(δ (u )) ] C n− . n ≤ 2 − 2 From this result, given 0 t1 By Corollary 4.6 and Lemma 4.8, it follows that ∆n(t)= f(Wt) f(W0) 1 − − 2 Fn(t)+ Zn(t) is tight, since both sequential parts Fn(t),Zn(t) are tight. Further, we have that Zn(t) tends to zero in probability, and Fn(t) is in a form suitable for Theorem 3.2. In the next lemma, we show that the conditions of Theorem 3.2 are satisfied by Fn(t) evaluated at a finite set of points. i Lemma 4.9. Fix 0= t0 Fn converges stably as n to a random variable ξ = (ξ1,...,ξd) with distribution (0, Σ), where→∞Σ is a diagonal d d matrix with the following entries: N × ti 2 2 si = f ′′(Ws) η(ds), Zti−1 where η(t)= η+(n) η (t) is as defined in condition (v). − − Remark 4.10. As we will see later, η(t) is continuous, nonnegative and nondecreasing. It follows from the structure of Σ that, given W , Fn converges stably to a d-dimensional vector with conditionally independent components of the form ti i 2 F = ζi f ′′(Ws) η(ds), ∞ sZti−1 where each ζ (0, 1). Thus, we may conclude that for each i, i ∼N ti i F L f ′′(W ) dB n −→ s s Zti−1 for a scaled Brownian motion B = Bt,t 0 that is independent of Wt, E 2 { ≥ } with [Bt ]= η(t). Proof of Theorem 4.3. To prove Theorem 4.3, it is enough to show that for any finite set of times 0 = t ti 2 2 si = f ′′(Ws) η(ds). Zti−1 18 D. HARNETT AND D. NUALART It follows that, conditioned on W , each component may be expressed as an independent Gaussian random variable, equivalent in law to ti f ′′(Ws) dBs, Zti−1 where B = Bt,t 0 is a scaled Brownian motion independent of W with E 2 { ≥ } [Bt ]= η(t). Finally, tightness follows from Lemma 4.8 and Corollary 4.6. Theorem 4.3 is proved. 5. Examples. 5.1. Bifractional Brownian motion. The bifractional Brownian motion is a generalization of fractional Brownian motion, first introduced by Houdr´e and Villa [3]. It is defined as a centered Gaussian process BH,K = BH,K(t), t 0 , with covariance defined by { ≥ } 1 1 E[BH,KBH,K]= (t2H + s2H )K + t s 2HK , t s 2K 2K | − | where H (0, 1), K (0, 1]. (Note that the case K = 1 corresponds to frac- tional Brownian∈ motion∈ with Hurst parameter H.) The reader may refer to [4] and its references for further discussion of properties. In this section, we show that the results of Section 4 are valid for bifrac- tional Brownian motion with parameter values H, K such that H 1/2 and 2HK = 1/2. In particular, this includes the endpoint cases H = 1≤/4, K = 1 studied in [6], and H = 1/2, K = 1/2 studied in [10]. H,K Proposition 5.1. Let Bt ,t 0 denote a bifractional Brownian motion. The covariance conditions{ (i)–(iv)≥ } of Section 4 are satisfied for val- ues of 0 1 1 + √t s (t2H + (t s )2H )K + s1/2 − − 2K − 2K Cs1 /2, ≤ where we used the inequality am bm (a b)m for a>b> 0 and m< 1. − ≤ − CLT FOR A STRATONOVICH INTEGRAL 19 Condition (ii). E H,K H,K H,K H,K [(Bt Bt s )(Br Br s )] − − − − 1 = ([t2H + r2H ]K [t2H + (r s)2H ]K 2K − − [(t s)2H + r2H ]K + [(t s)2H + (r s)2H ]K ) − − − − 1 + ( t r + s 2HK 2 t r 2HK + t r s 2HK). 2K | − | − | − | | − − | 1 This can be interpreted as the sum of a position term, 2K ϕ(t,r,s), and a distance term, 1 ψ(t r, s), where 2K − ϕ(t,r,s) = [t2H + r2H ]K [t2H + (r s)2H ]K [(t s)2H + r2H ]K − − − − + [(t s)2H + (r s)2H ]K − − and ψ(t r, s)= t r + s 2HK 2 t r 2HK + t r s 2HK . − | − | − | − | | − − | We begin with the position term. Note that if K =1, then ϕ(t,r,s)=0, so 1 we may assume K< 1 and H> 4 . Assume 0 2 2HK 2 2 3/2 Cs (t r s) − Cs t r − , ≤ − − ≤ | − | 3/2 3/2 3/2 since t r 2s implies (t r s)− 2 t r − . Condition| − |≥(iii). − − ≤ | − | E H,K H,K H,K H,K [Bt (Br+s 2Br + Br s )] | − − | 1 = [t2H + (r + s)2H ]K 2[t2H + r2H ]K + [t2H + (r s)2H ]K 2K − − 1 [ t r + s 2HK 2 t r 2HK + t r s 2HK] . − 2K | − | − | − | | − − | Take first the term, ϕ(t,r,s). If r< 2s, then [t2H + (r + s)2H ]K 2[t2H + r2H ]K + [t2H + (r s)2H ]K Cs2HK = Cs1/2, | − − |≤ based on the inequality aK bK (a b)K for a>b> 0 and K< 1. Hence, we will assume r 2s. If K−=1,≤ then−H = 1 , and we have ≥ 4 s 1 s 1 √r + s 2√r + √r s = dx dx | − − | 2√r + x − 2√r s + x Z0 Z0 − s s 1 1 = dy dx 4 (r s + x + y)3/2 Z0 Z0 − 1 2 3/2 s (r s)− ; ≤ 4 − and if K< 1, ϕ(t,r,s) | | s 2H 2H K 1 2H 1 = 2HK[t + (r + x) ] − (r + x) − dx Z0 s 2H 2H K 1 2H 1 2HK[t + (r s + x) ] − (r s + x) − dx − − − Z0 s s 4H2K(K 1) ≤ − Z0 Z0 2H 2H K 2 4H 2 [t + (r s + x + y) ] − (r s + x + y) − dy dx × − − s s 2H 2H K 1 + 2H(2H 1)K[t + (r s + x + y) ] − − − Z0 Z0 2H 2 (r s + x + y) − dy dx × − 2 2 2HK 2 2 2HK 2 4H K(1 K)s (r s) − + 2H(1 2H)Ks (r s) − ≤ − − − − 2 3/2 Cs (r s)− . ≤ − CLT FOR A STRATONOVICH INTEGRAL 21 This bound for ϕ(t,r,s) also holds in the case t r < 2s, so the bound of Cs1/2 is valid for this case. Next for the second term.| − Note| that if t r < 2s, then | − | 1 ( t r + s 2HK 2 t r 2HK + t r s 2HK ) 2(3s)2HK Cs1/2. 2K | − | − | − | | − − | ≤ ≤ If t r 2s, then we have | − |≥ t r + s 2 t r + t r s | | − | − | − | | − |− | s 1 s 1 p = p dxp dx 2 t r + x − 2 t r s + x Z0 Z0 | − | | − |− s s 1 = p pdy dx ( t r s + x + y)3/2 Z0 Z0 | − |− s2 s2 , ≤ 4( t r s)3/2 ≤ 2 t r 3/2 | − |− | − | 1 2 using the inequality t r s t r for t r 2s. This bound for ψ(t r, s) | − |− ≤ | − | | − |≥ − holds even in the case r< 2s, so the bound of Cs1/2 when r< 2s is verified as well. Condition (iv). For the first part, we have for all t s, ≥ E H,K H,K H,K [Bt (Bt+s Bt s )] | − − | 1 1 = [t2H + (t + s)2H ]K [t2H + (t s)2H ]K . 2K − 2K − This is bounded by Cs1/2 if t< 2s. On the other hand, if t 2s, ≥ 1 1 [t2H + (t + s)2H ]K [t2H + (t s)2H ]K 2K − 2K − s 1 2H 2H K 1 2H 1 = K 2HK[t + (t + x) ] − (t + x ) − dx 2 s Z− 2HK 1 Cs (t s) − ≤ − 1/2 = Cs(t s)− . − For 0 1 1 + r t + s 2HK r t s 2HK 2K | − | − 2K | − − | 1/2 1/2 Cs (t s)− + Cs r t − . ≤ − | − | 22 D. HARNETT AND D. NUALART If t< 2s or t r < 2s, then we have an upper bound of Cs1/2 by condition (i) and Cauchy–Schwarz.| − | For the third bound, if t> 2s, E H,K H,K H,K [Bs (Bt Bt s )] | − − | 1 1 [s2H + t2H ]K [s2H + (t s)2H ]K ≤ 2K − 2K − 1 2HK 1 2HK + (t s) (t 2s) 2K − − 2K − s 2 2H 2H K 2H 1 K HK[s + (t s + x) ] (t s + x) − dx ≤ 2 0 − − Z s 1 1/2 + (t 2s + x)− dx 2K+1 − Z0 1/2 1/2+γ γ Cs(t 2s)− = Cs (t 2s)− ≤ − − 1 for γ = 2 . Proposition 5.2. Let BH,K be a bifractional Brownian motion with parameters H 1/2 and HK = 1/4. Then condition (v) of Section 4 holds, ≤ + + with the functions η (t) = 2CK t and η−(t) = 2CK−t, where 1 ∞ C+ = 2+ (√2m + 1 2√2m + √2m 1)2 , K 4K − − m=1 ! X 2 (2 √2) 1 ∞ 2 C− = − + (√2m + 2 2√2m +1+ √2m) . K 22K 4K − m=1 X Proof. As in Proposition 5.1, we use the decomposition 1 j k 1 1 j k 1 β (j, k)= ϕ , , + ψ − , n 2K n n n 2K n n K 1/2 K 1/2 = 2− n− ϕ(j, k, 1) + 2− n− ψ(j k, 1). − The first task is to show that nt ⌊ ⌋ 1 2 (14) lim n− ϕ(j, k, 1) = 0. n →∞ j,kX=1 Proof of (14). We consider two cases, based on the value of H. First, 1 assume H< 2 . Then ϕ(j, k, 1) = [(j + 1)2H + (k + 1)2H ]K [(j + 1)2H + k2H ]K − [j2H + (k + 1)2H ]K + [j2H + k2H ]K − CLT FOR A STRATONOVICH INTEGRAL 23 1 2H 2H K 1 2H 1 = 2HK[(j + 1) + (k + x) ] − (k + x) − dx Z0 1 2H 2H K 1 2H 1 2HK[j + (k + x) ] − (k + x) − dx − Z0 1 1 2 2H 2H K 2 = 4H K(1 K)[(j + y) + (k + x) ] − − Z0 Z0 2H 1 2H 1 (k + x) − (j + y) − dy dx × 2HK 2H 1 2H 1 1/2 2H 2H 1 Ck − − j − = Ck− − j − . ≤ With this bound, it follows that nt nt ⌊ ⌋ ⌊ ⌋ 1 2 C 4H 2 ∞ 1 4H ϕ(j, k, 1) j − k− − n ≤ n j,kX=1 Xj=1 Xk=1 C 4H 1 nt − ≤ n ⌊ ⌋ 4H 2 Ctn − , ≤ 1 which tends to zero as n because H< 2 . →∞ 1 1 Next, we have the case H = 2 . Note that this implies K = 2 , and we have ϕ(j, k, 1) = j + k + 2 2 j + k +1+ j + k | | | − | 3/2 Cp(j + k)− . p p ≤ So with this bound, nt nt ⌊ ⌋ ⌊ ⌋ 1 2 C 3 n− ϕ(j, k, 1) (j + k)− ≤ n j,kX=1 j,kX=1 nt ⌊ ⌋ C ∞ 3 m− ≤ n Xj=1 mX=j+1 nt ⌊ ⌋ C 2 j− , ≤ n Xj=1 2 which tends to zero as n because j− is summable. Hence, (14) is proved. →∞ + From (14), it follows that to investigate the limit behavior of ηn (t), ηn−(t), it is enough to consider nt/2 nt/2 1 ⌊ ⌋ 2 ⌊ ⌋ ψ(2j 2k, 1)2 + ψ(2j 2k, 1)2 = ψ(2j 2k, 1)2 n − − n − j,kX=1 j,kX=1 24 D. HARNETT AND D. NUALART and nt/2 1 ⌊ ⌋ ψ(2j 2k + 1, 1)2 + ψ(2j 2k 1, 1)2 n − − − j,kX=1 nt/2 2 ⌊ ⌋ = ψ(2j 2k + 1, 1)2; n − j,kX=1 since the sums of ψ(2j 2k + 1, 1)2 and ψ(2j 2k 1, 1)2 are equal by symmetry. We start with− − − nt/2 1 ⌊ ⌋ ψ(2j 2k, 1)2 n − j,kX=1 nt/2 1 ⌊ ⌋ = ( 2j 2k + 1 2 2j 2k + 2j 2k 1 )2 4K n | − |− | − | | − − | j,k=1 X p p p nt/2 1 ⌊ ⌋ = 4 4K n Xj=1 nt/2 j 1 2 ⌊ ⌋ − + ( 2j 2k + 1 2 2j 2k + 2j 2k 1)2 4K n − − − − − j=1 k=1 X X p p p nt/2 j 1 4 nt/2 2 ⌊ ⌋ − = ⌊ ⌋ + (√2m + 1 2√2m + √2m 1)2 4K n 4Kn − − m=1 Xj=1 X nt/2 4 nt/2 2 ⌊ ⌋ ∞ = ⌊ ⌋ + (√2m + 1 2√2m + √2m 1)2 4K n 4Kn − − m=1 Xj=1 X nt/2 2 ⌊ ⌋ ∞ (√2m + 1 2√2m + √2m 1)2, − 4K n − − Xj=1 mX=j where the last term tends to zero since ∞ 2 ∞ 3 2 (√2m + 1 2√2m + √2m 1) (2m 1)− C(2j 1)− − − ≤ − ≤ − mX=j mX=j and nt/2 ⌊ ⌋ C 2 (2j 1)− 0 n − −→ Xj=1 CLT FOR A STRATONOVICH INTEGRAL 25 as n . We therefore conclude that →∞ nt/2 ⌊ ⌋ + 2 2 η (t) = lim (βn(2j 1, 2k 1) + βn(2j 2, 2k 2) ) n →∞ − − − − j,kX=1 nt/2 2 ⌊ ⌋ = lim ψ(2j 2k, 1)2 = 2C+t, n K →∞ n − j,kX=1 where 1 ∞ + √ √ √ 2 CK = K 2+ ( 2m + 1 2 2m + 2m 1) . 4 − − ! mX=1 For the other term, nt/2 1 ⌊ ⌋ ψ(2j 2k + 1, 1)2 n − j,kX=1 nt/2 1 ⌊ ⌋ = (2 √2)2 4K n − Xj=1 nt/2 j 1 2 ⌊ ⌋ − + ( 2j 2k + 2 2 2j 2k + 1 2j 2k)2. 4K n − − − − − j=1 k=1 X X p p p Hence, by a similar computation, nt/2 ⌊ ⌋ 2 2 η−(t) = lim βn(2j 1, 2k 2) + βn(2j 2, 2k 1) = 2C−t, n K →∞ − − − − j,kX=1 where 2 (2 √2) 1 ∞ 2 C− = − + (√2m + 2 2√2m +1+ √2m) . K 22K 4K − m=1 X + As a concluding remark, it is easy to show that CK >CK−, and in general we have η+(t) η (t). ≥ − 5.2. A Gaussian process with differentiable covariance function. Con- sider the following class of Gaussian processes. Let Ft, 0 t T be a mean-zero Gaussian process with covariance defined by{ ≤ ≤ } t (15) E[F F ]= rφ , t r, r t r ≥ where φ : [1, ) R is twice-differentiable on (1, ) and satisfies the follow- ing: ∞ → ∞ 26 D. HARNETT AND D. NUALART (φ.1) φ := supx 1 φ(x) cφ,0 < . (φ.2) Fork k∞ 1 Proposition 5.3. The process Ft, 0 t T described above satisfies conditions (i)–(iv) of Section 4. { ≤ ≤ } Proof. Condition (i). By conditions (φ.1) and (φ.2), 2 s E[(Ft Ft s) ]= tφ(1) + (t s)φ(1) 2(t s)φ 1+ − − − − − t s − s 2(t s) φ 1+ φ(1) + s φ(1) ≤ − t s − | | − 1+s/(t s) − 2(t s) φ′(x) dx + s φ ≤ − k k∞ Z1 1+s/(t s) c 2(t s) − φ,1 dx + s φ ≤ − √x 1 k k∞ Z1 − Cs1/2√t s + s φ ≤ − k k∞ Cs1/2, ≤ where the constant C depends on max √T , φ . Condition (ii). For 2s r t 2s we{ havek byk∞ the} mean value theorem ≤ ≤ − E[FtFr Ft sFr FtFr s + Ft sFr s] | − − − − − − | t t s t t s = r φ φ − (r s) φ φ − r − r − − r s − r s − − t t s s sup φ′′(x) − ≤ [(t s)/r,t/(r s)]| | r s − r − − − 1/2 3/2 t s − t s − ts c s − − 1 ≤ φ,2 r r − r(r s) − 2 C√Ts 2 3/2 = C√Ts t r − . ≤ (t r)3/2 | − | − CLT FOR A STRATONOVICH INTEGRAL 27 Condition (iii). By symmetry we can assume r t. Consider the following cases: First, suppose 2s r t 2s. Then we have≤ ≤ ≤ − E[Ft(Fr+s 2Fr + Fr s)] | − − | t t t = (r + s)φ 2rφ + (r s)φ r + s − r − r s − t t t t = (r + s) φ φ (r s) φ φ r + s − r − − r − r s − st t t sup φ′′(x) ≤ r [t/(r+s),t/(r s)]| | r s − r + s − − 2 2 1/2 3/2 2s t cφ,2 r + s r + s ≤ r(r s)(r + s) t t r s − − − Cs2t3/2 . ≤ r(t r)3/2 − t t There are two possibilities, depending on the value of r. If r 2 , then r 2, and we have a bound of ≥ ≤ 2 t √T 2 3/2 Cs 2C√Ts t r − . r (t r)3/2 ≤ | − | − t t On the other hand, if r< 2 , then t r 2 and r 2 t √T 2 3/2 3/2 Cs 2C√Ts [(r s)− + t r − ]. t r r√t r ≤ − | − | − − For the case t r < 2s, assume that t = r + ks for some 0 k< 2. Then | − | ≤ E[Ft(Fr+s 2Fr + Fr s)] | − − | t (r + s) t t = (t (r ))φ ∨ 2rφ + (r s)φ ∧ s t (r + s) − r − r s ∧ − t (r + s) = (t (r ))φ ∨ (r + s)φ(1) ∧ s t (r + s) − ∧ t t 2rφ + 2rφ(1) + (r s)φ (r s)φ(1) − r − r s − − − (k + 1)s 3(r + s) φ 1+ φ(1) ≤ r s − − 1+(k+1)s/(r s) − 3(r + s) φ′(x) dx ≤ Z1 28 D. HARNETT AND D. NUALART 1+(k+1)s/(r s) c 3(r + s) − φ,1 dx ≤ √x 1 Z1 − C√Ts1/2. ≤ For the last case, note that if t r< 2s, then we have an upper bound of 1/2 ∧ Cs , since E[FsFt] s φ . ≤ k k∞ Condition (iv). Take first the bound for E[Ft(Ft+s Ft s)]. Note that if t< 2s, then an upper bound of Cs1/2 is clear, so we will− assume− t 2s. We have ≥ E[FtFt+s FtFt s] | − − | t + s t = tφ (t s)φ t − − t s − t + s t t + s (t s) sup φ′(x) + s φ ≤ − | | t − t s t [(t+s)/t,t/(t s)] − − 2 s t t √T cφ,1 + cφ,0s ≤ t t + s s √t s r r − 1/2 Cs√T (t s)− . ≤ − For the case r = t, first assume r t 2s. By condition (φ.2), 6 ≤ − t + s t s E[FrFt+s FrFt s] = rφ rφ − | − − | r − r 2 s sup φ′(x) ≤ [(t s)/r,(t+s)/r]| | − 2s√rc C√Ts φ,1 . ≤ √t r s ≤ √t r − − − If r t + 2s, then ≥ r r E[FrFt+s FrFt s] = (t + s)φ (t s)φ | − − | t + s − − t s − 2s r t φ′ dx + 2s φ ≤ t s + x k k∞ Z0 − 2stc √t + s 2sc √T φ, 1 + φ, 0 ≤ √r t √t s − − 1/2 1/2 Cs(r t)− + Cs(t s)− . ≤ − − For the case t< 2s or r t < 2s, the bound follows from condition (i) and Cauchy–Schwarz. | − | CLT FOR A STRATONOVICH INTEGRAL 29 For the third part of condition (iv), we have for t> 2s, t t s E[FsFt FsFt s]= sφ sφ − − − s − s t t s cφ,1s s sup φ′(x) − ≤ [(t s)/s,t/s]| | s − s ≤ (t s)/s 1 − − − 3/2 1/2 1/2+γ γ Cs (t 2s)− = Cs (t 2ps)− , ≤ − − 1 where γ = 2 . Proposition 5.4. Suppose φ(x) satisfies conditions (φ.1), (φ.3), and in addition, φ(x) satisfies κ ψ(x) (φ.4): φ′(x)= + , √x 1 √x − where κ R and ψ : (1, ) R is a bounded differentiable function satisfying ∈ 1/2 ∞ → ψ′(1 + x) Cψx− for some positive constant Cψ. Then condition (v) | |≤ + + 2 2 of Section 4 is satisfied, with η (t)= Cβ t , and η−(t)= Cβ−t for positive + constants Cβ ,Cβ−. Remark 5.5. Observe that condition (φ.4) implies (φ.2), but not (φ.3). Proof of Proposition 5.4. We want to show nt/2 ⌊ ⌋ (16) β (2j 1, 2k 1)2 C t2; n − − −→ β,1 j,kX=1 nt/2 ⌊ ⌋ (17) β (2j 2, 2k 2)2 C t2; n − − −→ β,2 j,kX=1 nt/2 ⌊ ⌋ (18) β (2j 1, 2k 2)2 C t2, n − − −→ β,3 j,kX=1 + so that Cβ = Cβ,1 + Cβ,2 and Cβ− = 2Cβ,3. We will show computations for (16), with the others being similar. As in Proposition 5.2, nt/2 nt/2 ⌊ ⌋ ⌊ ⌋ β (2j 1, 2k 1)2 = β (2j 1, 2j 1)2 n − − n − − j,kX=1 Xj=1 nt/2 j 1 ⌊ ⌋ − + 2 β (2j 1, 2k 1)2, n − − Xj=1 Xk=1 30 D. HARNETT AND D. NUALART so it is enough to show nt/2 j 1 ⌊ ⌋ − 2 2 (19) lim βn(2j 1, 2k 1) = C1t n →∞ − − Xj=1 Xk=1 and nt/2 ⌊ ⌋ 2 2 (20) lim βn(2j 1, 2j 1) = C2t . n →∞ − − Xj=1 Proof of (19). For 1 k j 1, we have ≤ ≤ − 2k 2j 2j 1 β (2j 1, 2k 1) = φ φ − n − − n 2k − 2k 2k 1 2j 2j 1 − φ φ − − n 2k 1 − 2k 1 − − 2k 2j/(2k) = φ′(x) dx n (2j 1)/(2k) Z − 2j/(2k 1) 2k 1 − − φ′(x) dx. − n (2j 1)/(2k 1) Z − − Using the change of index j = k + m and a change of variable for the two integrals, this becomes 1 2m y βn(2j 1, 2k 1) = φ′ 1+ dy − − n 2m 1 2k (21) Z − 1 2m+1 y φ′ 1+ dy. − n 2k 1 Z2m − With the decomposition of (φ.4), we will address (21) in two parts. Using the first term, we have κ 2m 2k κ 2m+1 2k 1 dy − dy n 2m 1 s y − n 2m s y Z − Z 2κ = [√2k(√2m √2m 1) √2k 1(√2m + 1 √2m)]. n − − − − − We are interested in the sum nt/2 nt/2 k ⌊ ⌋ ⌊ ⌋− 4κ2 (22) [√2k(√2m √2m 1) √2k 1(√2m + 1 √2m)]2. n2 − − − − − m=1 Xk=1 X CLT FOR A STRATONOVICH INTEGRAL 31 We can write √2k(√2m √2m 1) √2k 1(√2m + 1 √2m) − − − − − = √2k 1(√2m + 1 2√2m + √2m 1) − − − − + (√2k √2k 1)(√2m √2m 1). − − − − Observe that 1 [(√2k √2k 1)(√2m √2m 1)]2 − − − − ≤ (2k 1)(2m 1) − − and so nt/2 nt/2 k nt/2 2 4κ2 ⌊ ⌋ ⌊ ⌋− 1 4κ2 ⌊ ⌋ 1 C log(nt)2 . n2 (2k 1)(2m 1) ≤ n2 2k 1 ≤ n2 m=1 ! Xk=1 X − − Xk=1 − Therefore the contribution of this term is zero, and it follows by Cauchy– Schwarz that the only significant term is nt/2 nt/2 k 4κ2 ⌊ ⌋ ⌊ ⌋− (2k 1)(√2m + 1 2√2m + √2m 1)2 n2 − − − m=1 Xk=1 X nt/2 nt/2 m ⌊ ⌋ ⌊ ⌋− 2k 1 = 4κ2 (√2m + 1 2√2m + √2m 1)2 − − − n2 mX=1 Xk=1 nt/2 ⌊ ⌋ ( nt/2 m)2 = 4κ2 (√2m + 1 2√2m + √2m 1)2 ⌊ ⌋− , − − n2 m=1 X which converges as n to →∞ ∞ κ2t2 (√2m + 1 2√2m + √2m 1)2. − − m=1 X 1 Next, we consider the term √x ψ(x). The contribution of this term to (21) is 1 2m 2k y ψ 1+ dy n 2k + y 2k 2m 1 s (23) Z − 1 2m+1 2k 1 y − ψ 1+ dy. − n 2k 1+ y 2k 1 Z2m s − − We can bound (23) by 1 2m 2k y 2m+1 2k 1 y ψ 1+ dy − ψ 1+ dy n 2k + y 2k − 2k 1+ y 2k 1 Z2m 1 s Z2m s − − − 32 D. HARNETT AND D. NUALART 1 √2k √2k 1 sup ψ(x) − − ≤ n (1, )| | √2k + 2m 1 ∞ − 2k 2m y + ψ 1+ dy 2k + 2m 1 2k r Z2m 1 − − 2m+1 y ψ 1+ dy − 2k 1 Z2m − 1 = (Ak,m + Bk,m). n Since ψ(x) is bounded, we have | | C C (24) Ak,m . ≤ √2k 1√2k + 2m 1 ≤ √2k 1√2m 1 − − − − For B using that ψ (x + 1) Cx 1/2, k,m | ′ |≤ − 2m y 2m+1 y ψ 1+ dy ψ 1+ dy 2k − 2k 1 Z2m 1 Z2m − − 2m u u + 1 = ψ 1+ ψ 1+ du 2k − 2k 1 Z2m 1 − − 2m u/(2k) ψ′(1 + v) dv du ≤ Z2m 1 Z(u+1)/(2k 1) − − 2m (u+1)/(2k 1) − 1/2 C v− dv du ≤ 2m 1 u/(2k) Z − Z C (√2m + 1 √2m) ≤ √2k 1 − − C ≤ √2k 1√2m 1 − − so that 2k C C (25) Bk,m . ≤ 2k + 2m 1 · √2k 1√2m 1 ≤ √2k 1√2m 1 r − − − − − Hence, from (24) and (25), we obtain nt/2 nt/2 k ⌊ ⌋ ⌊ ⌋− C 1 2 n2 √2k 1√2m 1 Xk=1 mX=1 − − nt/2 C ⌊ ⌋ 1 C log(n)2 , ≤ n2 (2m 1)(2k 1) ≤ n2 k,mX=1 − − CLT FOR A STRATONOVICH INTEGRAL 33 so the portion represented by (23) tends to zero as n . Since this term is not significant, it follows by Cauchy–Schwarz that the→∞ behavior of nt/2 j 1 ⌊ ⌋ − β (2j 1, 2k 1)2 n − − Xj=1 Xk=1 is dominated by equation (22), and we have result (19), with ∞ C = κ2 (√2m + 1 2√2m + √2m 1)2. 1 − − m=1 X Proof of (20). For each j, β (2j 1, 2j 1)2 n − − 2j 2j 1 2j 2j 1 2 = φ(1) 2 − φ + − φ(1) n − n 2j 1 n − 1 1 2 = φ(1) + (4j 2) φ(1) φ 1+ n2 − − 2j 1 − φ(1)2 4(2j 1)φ(1) 1 = + − φ(1) φ 1+ n2 n2 − 2j 1 − 4(2j 1)2 1 2 + − φ(1) φ 1+ . n2 − 2j 1 − 1 cφ,3 Since φ(1) φ(1 + 2j 1 ) √2j 1 by (φ.3), we see that | − − |≤ − nt/2 ⌊ ⌋ 2 φ(1) 4(2j 1)φ(1) 1 1/2 + − φ(1) φ 1+ Cn− , n2 n2 − 2j 1 ≤ j=1 X − which implies only the last term is significant in the limit. Again we use (φ.4) to obtain 1 φ(1) φ 1+ − 2j 1 − 1+1/(2j 1) − = φ′(x) dx − Z1 1+1/(2j 1) 1 1+1/(2j 1) 1 = κ − dx − ψ(x) dx − √x 1 − √x Z1 − Z1 2κ 1 = + O ; −√2j 1 2j 1 − − 34 D. HARNETT AND D. NUALART hence 4(2j 1)2 1 2 16κ2(2j 1)2 j1/2 − φ(1) φ 1+ = − + O n2 − 2j n2(2j 1) n2 − and taking n , →∞ nt/2 ⌊ ⌋ 16κ2(2j 1) j1/2 lim − + O = 4κ2t2, n 2 2 →∞ n n Xj=1 2 2 which gives (20). Thus (16) is proved with C = 4κ +2κ ∞ (√2m + 1 β,1 m=1 − 2√2m + √2m 1)2. P By similar computations,− ∞ C = 4κ2 + 2κ2 (√2m + 1 2√2m + √2m 1)2 β,2 − − m=1 X and ∞ C = 4κ2 + 2κ2 (√2m + 2 2√2m +1+ √2m)2 β,3 − m=1 X and so ∞ C+ = C + C = 8κ2 + 4κ2 (√2m + 1 2√2m + √2m 1)2, β β,1 β,2 − − m=1 X 2 2 ∞ 2 C− = 2C = 8κ + 4κ (√2m + 2 2√2m +1+ √2m) . β β,3 − m=1 X + + Note that Cβ Cβ−, and it follows that η(t)= η (t) η−(t) is nonnegative, and strictly positive≥ if κ = 0. − 6 For a particular example, we consider a mean-zero Gaussian process Ft, t 0 , with covariance given by { ≥ } 1 r t E[FrFt]= √rt sin− ∧ . √rt This process was studied by Swanson in a 2007 paper [9], and it appears in the limit of normalized empirical quantiles of a system of independent Brownian motions. Corollary 5.6. The process Ft, 0 t T , with the covariance de- { ≤ ≤ } + 2 scribed above, satisfies the conditions of Section 4, with η(t) = (Cβ Cβ−)t , + 2 − where Cβ , Cβ− are as given in Proposition 5.4, with κ = 1/4. CLT FOR A STRATONOVICH INTEGRAL 35 Proof. Assume 0 r 1 1 √x tan− , if x> 1, (26) φ(x)= √x 1 π − , if x = 1. 2 Condition (φ.1) is clear by continuity and L’Hˆopital. Conditions (φ.2) and (φ.3) are easily verified by differentiation. For (φ.4) we can write, 1 1 √x 1 1 1 φ′(x)= + − tan− −2√x 1 2√x √x 1 − √x 1 − − − so that κ = 1/2, and − 1 √x 1 1 1 ψ(x)= − tan− 2 √x 1 − √x 1 − − satisfies (φ.4). 5.3. Empirical quantiles of independent Brownian motions. For our last example, we consider a family of processes studied by Swanson in [11]. Sim- ilar to [9], this Gaussian family arises from the empirical quantiles of inde- pendent Brownian motions, but this case is more general, and does not have a covariance representation (15). Let B = B(t),t 0 be a Brownian motion with random initial position. Assume B(0){ has a≥ density} function f (R) such that ∈C∞ sup(1 + x m) f (n)(x) < x R | | | | ∞ ∈ for all nonnegative integers m and n. It follows that for t> 0, B has density u(x,t)= f(y)p(t,x y) dy, R − Z 1/2 x2/(2t) where p(t,x) = (2πt)− e− . For fixed α (0, 1), define the α-quantile q(t) by ∈ q(t) u(x,t) dx = α, Z−∞ where we assume f(q(0)) > 0. It is proved in [11] (Theorem 1.4) that there exists a continuous, centered Gaussian process F (t),t 0 with covariance { ≥ } P(B(r) q(r),B(t) q(t)) α2 (27) E[F F ]= ρ(r, t)= ≤ ≤ − . r t u(q(r), r)u(q(t),t) 36 D. HARNETT AND D. NUALART In [11], the properties of ρ are studied in detail, and we follow the notation and proof methods given in Section 3 of that paper. Swanson defines the following factors: 2 1 ρ˜(r, t)= P(B(r) q(r),B(t) q(t)) α and θ(t) = (u(q(t),t))− ≤ ≤ − so that ρ(r, t)= θ(r)θ(t)˜ρ(r, t). For fixed T > 0 and 0 ∂ 1/2 ρ(r, t) C t r − ; ∂r ≤ 1| − | (b) 2 ∂ 3/2 ρ(r, t) C t r − ; ∂r2 ≤ 2| − | (c) ∂ 1/2 ρ(r, t) C t r − ; ∂t ≤ 3| − | (d) 2 ∂ 3/2 ρ(r, t) C t r − . ∂t2 ≤ 4| − | Proof. Results (a) and (c) are proved in Theorem 3.1 of [11]. Bounds for (b) and (d) follow by differentiating the expressions for ∂rρ(r, t) and ∂tρ(r, t) given in the proof of that theorem. Proposition 5.8. Let T > 0, 0 Proof. Conditions (i) and (ii) are proved in [11] (Corollaries 3.2, 3.5 and Remark 3.6). For condition (iii), there are several cases to consider. Case 1: s r t 2s. Using Lemma 5.7(a), ≤ ≤ − E[Ft(Fr+s 2Fr + Fr s)] ρ(r + s,t) ρ(r, t) + ρ(r, t) ρ(r s,t) | − − | ≤ | − | | − − | s ∂ 0 ∂ ρ(r + x,t) dx + ρ(r + y,t) dy ≤ 0 ∂r s ∂r Z Z− s 1/2 1/2 2 C t r x − dx Cs . ≤ 1| − − | ≤ Z0 Case 2: If t r < 2s, the computation is similar to case 1, where we use the fact that | − | s 1/2 1/2 x− dx = 2s . Z0 Case 3: For r, t 2s and t r 2s, the results follow from Lemma 5.7(b) and (d) for r E[Ft(Ft+s Ft s)] = ρ(t,t + s) ρ(t,t s) − − − − = θ(t)[θ(t + s)˜ρ(t,t + s) θ(t s)˜ρ(t,t s)] − − − = θ(t)[˜ρ(t,t + s)∆θ + θ(t s)∆˜ρ], − where ∆θ = θ(t) θ(t s) and ∆˜ρ =ρ ˜(t,t + s) ρ˜(t,t s). First, note that − − − − ∂ ∂ u′(q(t),t) = u(q(t),t)q′(t)+ u(q(t),t) C, | | ∂x ∂t ≤ where we used that q (t) is bounded; see Lemma 1.1 of [11 ]. Since u(q(t),t) ′ is continuous and strictly positive on [0, T ], it follows that θ(t) is bounded and u′(q(t),t) (30) θ′(t) = | | C, | | u2(q(t),t) ≤ hence, s ∆θ θ′(t + x) dx Cs. | |≤ s| | ≤ Z− For ∆˜ρ, we have ∆˜ρ = P(B(t) q(t),B(t + s) q(t + s)) | | | ≤ ≤ P(B(t) q(t),B(t s) q(t s)) − ≤ − ≤ − | 38 D. HARNETT AND D. NUALART q(t) q(t+s) = p(s,x y)u(x,t) dy dx − Z−∞ Z−∞ q(t s) q(t) − p(s,x y)u(x,t s) dy dx − − − Z−∞ Z−∞ q(t s) q(t) − p(s,x y)u(x,t) p(s,x y)u(x,t s) dy dx + Cs ≤ − − − − Z Z −∞ −∞ q(t s) − u(x,t s) u(x,t) dx + Cs ≤ | − − | Z−∞ t ∂ ∞ u(x, r) dr dx + Cs ≤ ∂r Z Zt s −∞ − 1 t ∂2 ∞ = 2 u(x, r) dr dx + Cs 2 t s ∂x Z−∞ Z − t 1 ∞ ∞ f ′′(y) p(r, x y) dr dy dx + Cs Cs. ≤ 2 t s| | − ≤ Z−∞ Z−∞ Z − When t< 2s, we write E[Ft(Ft+s Ft s)] ρ(t,t + s) ρ(t,t) + ρ(t,t) ρ(t s,t) | − − | ≤ | − | | − − | s ∂ 0 ∂ ρ(t,t + x) dx + ρ(t + y,t) dy ≤ ∂t ∂r Z0 Z s − Cs1 /2, ≤ using Lemma 5.7 and the fact that s 1/2 1/2 x− dx = 2s . Z0 For the second part of condition (iv), we consider E[Fr(Ft+s Ft s)] and E[Fs(Ft Ft s)] . | − − | | − − | 1/2 When r The rest of this section is dedicated to verifying condition (v). We start with two useful estimates. As in Proposition 5.8, suppose 0 a b Using this estimate and the fact that e− e− b a for 0 a b, we obtain − ≤ − ≤ ≤ (q(t) q(r))2 /(2(t r)) (q(t) q(r s))2 /(2(t r+s)) (32) e− − − e− − − − Cs 1. | − |≤ ≤ Recalling the definitions in condition (v), we can write for t [0, T ], ∈ 2 nt/2 ⌊ ⌋ + 2 2 η (t) η−(t)= β (ℓ 1,ℓ 1) + 2 β (2k 1, 2j 1) n − n n − − n − − ℓ=1 k j 1 X ≤X− + 2 β (2k 2, 2j 2)2 2 β (2k 2, 2j 1)2 n − − − n − − k j 1 k j 1 ≤X− ≤X− 2 β (2k 1, 2j 2)2. − n − − k j 1 ≤X− For the first sum, since Fℓ/n F(ℓ 1)/n is Gaussian, we have − − 2 E 2 2 1 E 4 βn(ℓ 1,ℓ 1) = ( [(Fℓ/n F(ℓ 1)/n) ]) = 3 [(Fℓ/n F(ℓ 1)/n) ]. − − − − − − By Theorem 3.7 of [11], nt ⌊ ⌋ t 4 6 2 (Fℓ/n F(ℓ 1)/n) (u(q(s),s))− ds − − −→ π 0 Xℓ=1 Z in L2 as n . For the second sum, assume 1 k < j, and we study the term →∞ ≤ β (2k 1, 2j 1) n − − 2k 2j 2k 1 2j 2k 2j 1 2k 1 2j 1 = ρ , ρ − , ρ , − + ρ − , − n n − n n − n n n n 2j 2k/n 2j 2j = θ θ′(r)˜ρ r, + θ(r)∂rρ˜ r, dr n (2k 1)/n n n Z − 2j 1 2k/n 2j 1 2j 1 θ − θ′(r)˜ρ r, − + θ(r)∂rρ˜ r, − dr. − n (2k 1)/n n n Z − We can write this as 2j 2k/n 2j 2j 1 (33) θ θ(r) ∂rρ˜ r, ∂rρ˜ r, − dr n (2k 1)/n n − n Z − 2j 2j 1 2k/n 2j 1 (34) + θ θ − θ(r) ∂rρ˜ r, − dr n − n (2k 1)/n n Z − 2k/n 2j 2j 2j 1 2j 1 (35) + θ′(r) θ ρ˜ r, θ − ρ˜ r, − dr. (2k 1)/n n n − n n Z − 40 D. HARNETT AND D. NUALART The first task is to show that components (34) and (35) have a negligible contribution to η(t). For (34), it follows from (30) that 2j 2j 1 1 (36) θ θ − Cn− , n − n ≤ and using (29), we have 2k/n 2j 1 θ(r)∂rρ˜ r, − dr (2k 1)/n n Z − 2k/n 2j 1 2j 1 = p − r, q − q(r) dr (2k 1)/n n − n − Z − 1/2 Cn− . ≤ 2 Hence, the contribution of (34) to the sum of βn(2k 1, 2j 1) is bounded by C(n 3/2)2 n2 Cn 1. We can write component− (35) as− − · ≤ − 2k/n 2j 2j 2j 1 θ′(r) θ ρ˜ r, ρ˜ r, − (2k 1)/n n n − n Z − 2j 2j 1 2j 1 + θ θ − ρ˜ r, − dr. n − n n 2k 1 2k Using (29), we have for each r [ − , ], ∈ n n 2j 2j 1 1/2 1/2 ρ˜ r, ρ˜ r, − Cn− (2j 2k 1)− . n − n ≤ − − Then, using (36) and (30), we have ( 35) bounded by 1/2 1/2 1 1 C[n− (2j 2k 1)− + n− ]n− . − − Hence, the contribution of (35)to the sumof β (2k 1, 2j 1)2 is bounded by n − − nt/2 j 1 ⌊ ⌋ − 2 1 1 2 1 Cn− [n− (2j 2k 1)− + n− ] Cn− . − − ≤ Xj=1 Xk=1 We now turn to component (33). By (29), ∂ 2j 1 2j 2j θ(r) ρ˜ r, = p r, q q(r) . ∂r n 2 n − n − To simplify notation, define (q(j/n) q(r))2/(2(j/n r)) ψn(j, r)= e− − − . 2k 1 2k By (31), we have for the interval I2k = [ n− , n ], ((q(2j/n) q(r))2 C(2j 2k + 1) sup − − . r I 2(2j/n r) ≤ n ∈ 2k − CLT FOR A STRATONOVICH INTEGRAL 41 C(2j 2k+1)/n This implies that inf ψn(2j, r), r I2k e− − , and hence when j, k are small compared{ to n, ψ is∈ close}≥ to unity. We can write | | 2k/n 2j 2j 1 θ(r) ∂rρ˜ r, ∂rρ˜ r, − dr (2k 1)/n n − n (37) Z − 1 2k/n 1 1 = dr 2√2π (2k 1)/n 2j/n r − (2j 1)/n r Z − − − − 1 2k/n p p (1 ψn(2j 1, r)) − 2√2π (2k 1)/n − − (38) Z − 1 1 dr × 2j/n r − (2j 1)/n r − − − 1 2k/n ψ (2pj, r) ψ (2j p1, r) (39) + n − n − dr. 2√2π (2k 1)/n 2j/n r Z − − For component (38), by the above estimatep for inf ψn(2j, r), r I2k , we have { ∈ } 1 sup 1 ψ(2j, r) Cn− (2j 2k + 1) 1, r I | − |≤ − ≤ ∈ 2k hence (38) is bounded by 3/2 Cn− (2j 2k + 1)( 2j 2k + 1 2 2j 2k + 2j 2k 1). − − − − − − Given ε> 0, we can find anpM > 1 such thatp p ∞ (√2m + 1 2√2m + √2m 1)2 < ε. − − mX=M 2 The contribution of (38) to the sum of βn(2k 1, 2j 1) is thus bounded by − − nt/2 j 1 ⌊ ⌋ − 1 2 2j 2 (2πn)− θ sup (1 ψn(2j, r)) n r I2k − Xj=1 Xk=1 ∈ ( 2j 2k + 1 2 2j 2k + 2j 2k 1)2 × − − − − − nt/2 j M 1 ⌊ ⌋ − − p p p 1 2 Cn− ( 2j 2k + 1 2 2j 2k + 2j 2k 1) ≤ − − − − − j=1 k=1 X X p p p nt/2 j 1 ⌊ ⌋ − 1 1 + Cn− Cn− (2j 2k + 1) − j=1 k=j M X X− 42 D. HARNETT AND D. NUALART ( 2j 2k + 1 2 2j 2k + 2j 2k 1)2 × − − − − − nt/2 nt/2 ⌊ ⌋ ⌊ p⌋ 2 p p 1 1 M Cn− ε + Cn− , ≤ n2 Xj=1 Xj=1 which is less than Cε as n , since θ(t) is bounded. →∞ 1 For (39), by we have sup ψn(2j, r) ψn(2j 1, r) , r I2k Cn− , and {| 3/2 − −1/2 | ∈ }≤ hence (39) is bounded by Cn− (2j 2k 1)− . Therefore the contribu- − − 2 tion of the term including (39) to the sum of βn(2k 1, 2j 1) is bounded by − − nt/2 j 1 ⌊ ⌋ − 3 1 2 Cn− (2j 2k 1)− Cn− log(nt), − − ≤ Xj=1 Xk=1 because θ(t) is bounded. 2 It follows that the sum of βn(2k 1, 2j 1) is dominated by (33), and the significant term in (33) is given− by (37).− Hence, it is enough to consider 2 2j θ2 ( 2j 2k + 1 2 2j 2k + 2j 2k 1)2. nπ n − − − − − j k 1 ≤X− p p p Using the change of index j = k + m, this is nt/2 j 1 2 ⌊ ⌋ 2j − θ2 (√2m + 1 2√2m + √2m 1)2. nπ n − − m=1 Xj=1 X Taking n , this behaves like →∞ a t θ2(s) ds, π Z0 where ∞ a = (√2m + 1 2√2m + √2m 1)2. − − m=1 X By similar computation, a t β (2k 2, 2j 2)2 θ2(s) ds, n − − −→ π k j 1 Z0 ≤X− b t β (2k 2, 2j 1)2 1 θ2(s) ds n − − −→ π k j 1 Z0 ≤X− and b t β (2k 1, 2j 2)2 2 θ2(s) ds, n − − −→ π k j 1 Z0 ≤X− CLT FOR A STRATONOVICH INTEGRAL 43 where ∞ b = (√2m + 2 2√2m +1+ √2m)2, 1 − m=1 X ∞ b = (√2m 2√2m 1+ √2m 2)2. 2 − − − m=1 X We have proved the following result: Proposition 5.9. Under the above assumptions, ρ(r, t) satisfies condi- tion (v) of Section 4, where t 2 + 4a 2b1 2b2 2 η(t)= − − (u(q(s),s))− ds. π Z0 The coefficient 2 + 4a 2b1 2b2 is approximately 1.3437, while u(q(t),t) depends on f and α. − − 6. Proof of the technical lemmas. We begin with two technical lemmas. The first is a version of Corollary 4.2 with disjoint intervals. Lemma 6.1. For 0 t Proof. We may assume t0 =0 and t1 = t2. Observe that 2 2 2 2 ∂(2⊗j 1)/n ∂(2⊗j 2)/n, ∂(2⊗k 1)/n ∂(2⊗k 2)/n H⊗2 h − − − − − − i = β (2j 1, 2k 1)2 β (2j 1, 2k 2)2 β (2j 2, 2k 1)2 n − − − n − − − n − − + β (2j 2, 2k 2)2. n − − Therefore, it is enough to show that nt2 nt3 ⌊ ⌋ ⌊ ⌋ 2 ε (40) β (j, k) Cn− n ≤ j=0 k= nt2 +1 X ⌊X⌋ for some ε> 0. We can decompose the sum in (40) as nt3 nt3 nt2 1 nt3 ⌊ ⌋ ⌊ ⌋ ⌊ ⌋− ⌊ ⌋ β (0, k)2 + β ( nt , k)2 + β (j, k)2. n n ⌊ 2⌋ n k= nt2 +1 k= nt2 +1 j=1 k= nt2 +1 ⌊X⌋ ⌊X⌋ X ⌊X⌋ 44 D. HARNETT AND D. NUALART By condition (iv), for some γ> 0 we have nt3 nt3 ⌊ ⌋ ⌊ ⌋ 2 βn(0, k) sup βn(0, k) βn(0, k) ≤ 1 j nt3 | | | | k= nt2 +1 k= nt2 +1 ⌊X⌋ ≤ ≤⌊ ⌋ ⌊X⌋ nt3 ⌊ ⌋ 1 γ 1 γ Cn− (k 1)− + Cn− Cn− . ≤ − ≤ k= nt2 +2 ⌊X⌋ By condition (ii), for some 1 < α 3 , ≤ 2 nt3 nt3 ⌊ ⌋ ⌊ ⌋ 2 2 1 β ( nt , k) β ( nt , nt + 1) + Cn− β ( nt , k) n ⌊ 2⌋ ≤ n ⌊ 2⌋ ⌊ 2⌋ n ⌊ 2⌋ k= nt2 +1 k= nt2 +2 ⌊X⌋ ⌊X⌋ nt3 ⌊ ⌋ 1 1 α 1 Cn− + Cn− (k nt )− Cn− ≤ − ⌊ 2⌋ ≤ k= nt2 +1 ⌊X⌋ and again by condition (ii), for β = 3 α, 2 − nt2 1 nt3 ⌊ ⌋− ⌊ ⌋ 2 βn(j, k) j=1 k= nt2 +1 X ⌊X⌋ nt2 1 nt3 ⌊ ⌋− ⌊ ⌋ 1 α β 3/2 Cn− [(k nt )− j− + (k j)− ] ≤ − ⌊ 2⌋ − j=1 k= nt2 +1 X ⌊X⌋ nt3 nt2 nt2 ⌊ ⌋ ⌊ ⌋ ⌊ ⌋ 1 α β 1 1/2 Cn− k− j− + Cn− ( nt2 j)− ≤ ! ! ⌊ ⌋− Xk=1 Xj=1 Xj=1 β 1/2 Cn− + Cn− ; ≤ hence the sum is bounded by Cn ε for ε = min β,γ, 1 . − { 2 } Lemma 6.2. For 0 t T and integer j 1, ≤ ≤ ≥1/2 ε , ∂ H Cn− |h t j/ni |≤ for a positive constant C which depends on T . Proof. By conditions (i) and (ii), we have for j 1 and t> 0, ≥ nt 1 ⌊ ⌋− εt, ∂j/n H ∂k/n, ∂j/n H + εt ε nt , ∂j/n H |h i |≤ |h i | |h − ⌊ ⌋ i | k=0 (41) X ∞ 1/2 α 1/2 1/2 C n− ( j k − 1) + O(n− ) Cn− . ≤ | − | ∧ ≤ Xk=0 CLT FOR A STRATONOVICH INTEGRAL 45 6.1. Proof of Lemma 4.4. By the Lagrange theorem for the Taylor ex- pansion remainder, the terms R0(W2j/n),R1(W(2j 2)/n) can be expressed in integral form, − 1 W2j/n R (W )= (W u)2f (3)(u) du 0 2j/n 2 2j/n − ZW(2j−1)/n and W(2j−1)/n 1 2 (3) R1(W(2j 2)/n)= (W(2j 2)/n u) f (u) du. − −2 − − ZW(2j−2)/n After a change of variables, we obtain R0(W2j/n) 1 1 3 2 (3) = (W2j/n W(2j 1)/n) v f (vW(2j 1)/n + (1 v)W2j/n) dv 2 − − − − Z0 and 1 3 R1(W(2j 2)/n)= (W(2j 2)/n W(2j 1)/n) − 2 − − − 1 2 (3) v f (vW(2j 1)/n + (1 v)W(2j 2)/n) dv. × − − − Z0 Define 1 1 2 (3) G0(2j)= v f (vW(2j 1)/n + (1 v)W2j/n) dv 2 − − Z0 and 1 1 2 (3) G1(2j 2) = v f (vW(2j 1)/n + (1 v)W(2j 2)/n) dv. − 2 − − − Z0 We may assume r =0. Define ∆Wℓ/n = W(ℓ+1)/n Wℓ/n. We want to show that − nt/2 2 ⌊ ⌋ E 3 3 G0(2j)∆W(2j 1)/n + G1(2j 2)∆W(2j 2)/n { − − − } " j=1 ! # (42) X nt 3/2 C n− . ≤ 2 This part of the proof was inspired by a computation in [6]; see Lemma 4.2. 3 Consider the Hermite polynomial identity x = H3(x) + 3H1(x). We use the q q map δ (h )= H (W (h)) [see (2) in Section 2], for h H with h H = 1. For ⊗ q ∈ k k 46 D. HARNETT AND D. NUALART 1/4 each j, let wj := ∆Wj/n H, and note that condition (i) implies wj Cn− for all j. Then k k ≤ 3 ∆Wj/n ∆Wj/n ∆Wj/n = H + 3H w3 3 w 1 w j j j 3 ∂j/n⊗ ∂j/n = δ3 + 3δ w3 w j j so that 3 3 3 2 ∆Wj/n = δ (∂j/n⊗ ) + 3wj δ(∂j/n). It follows that we can write 3 3 G0(2j)∆W(2j 1)/n G1(2j 2)∆W(2j 2)/n − − − − 3 3 3 3 = G0(2j)δ (∂(2⊗j 1)/n) G1(2j 2)δ (∂(2⊗j 2)/n) − − − − 2 2 + 3w2jG0(2j)δ(∂(2j 1)/n) 3w2j 1G1(2j 2)δ(∂(2j 2)/n). − − − − − It is enough to verify the individual inequalities nt/2 2 ⌊ ⌋ E 3 3 nt 3/2 (43) G0(2j)δ (∂(2⊗j 1)/n) C n− , " − # ≤ 2 j=1 X nt/ 2 2 ⌊ ⌋ E 3 3 nt 3/2 (44) G1(2j 2)δ (∂(2⊗j 2)/n) C n− , " − − # ≤ 2 j=1 X nt/2 2 ⌊ ⌋ E 2 nt 3/2 (45) w2jG0(2j)δ(∂(2j 1)/n) C n− " − # ≤ 2 j=1 X and nt/2 2 ⌊ ⌋ E 2 nt 3/2 (46) w2j 1G1(2j 2)δ(∂(2j 2)/n) C n− . " − − − # ≤ 2 j=1 X We will show (43) and (45), with (44) and (46) being essentially similar. Proof of (43). Using (3) and the duality property, nt/2 2 ⌊ ⌋ E 3 3 G0(2j)δ (∂(2⊗j 1)/n) " − ! # Xj=1 nt/2 ⌊ ⌋ = E G0(2j)G0(2k) " j,kX=1 CLT FOR A STRATONOVICH INTEGRAL 47 3 6 2r 3 r 3 r r δ − (∂⊗ − ∂⊗ − ) ∂(2j 1)/n, ∂(2k 1)/n H × (2j 1)/n ⊗ (2k 1)/n h − − i r=0 − − !# X nt/2 3 ⌊ ⌋ r ∂(2j 1)/n, ∂(2k 1)/n H ≤ |h − − i | r=0 j,kX=1 X E 6 2r 3 r 3 r [ D − (G0(2j)G0(2k)), ∂(2⊗j−1)/n ∂(2⊗k−1)/n H⊗6−2r ]. × |h − ⊗ − i | For integers r 0, we have ≥ 1 r r 1 2 (3) D G0(2j)= D v f (vW(2j 1)/n + (1 v)W2j/n) dv 2 − − Z0 1 1 2 (3+r) (47) = v f (vW(2j 1)/n + (1 v)W2j/n) 2 − − Z0 r r (vε(2⊗j 1)/n + (1 v)ε2⊗j/n) dv. × − − By product rule and (47) we have E 6 2r 3 r 3 r [ D − (G0(2j)G0(2k)), ∂(2⊗j−1)/n ∂(2⊗k−1)/n H⊗6−2r ] |h − ⊗ − i | E (a) C sup f (vW(2j 1)/n + (1 v)W(2j 2)/n) ≤ 0 v,w 1| − − − a+b=6 2r ≤ ≤ X− h (b) f (wW(2k 1)/n + (1 w)W(2k 2)/n) × − − − | (48) 1 1 i a a (vε⊗ + (1 v)ε⊗ ) × |h (2j 1)/n − 2j/n Z0 Z0 − b b (wε(2⊗k 1)/n + (1 w)ε2⊗k/n), ⊗ − − 3 r 3 r ∂(2⊗j−1)/n ∂(2⊗k−1)/n H⊗6−2r dv dw. − ⊗ − i | Notice that by condition (0), E[sup f (3+r)(ξ) p] < , where the supremum is taken over the random variables | ξ = vW | + (1∞ v)W , 0 v 1, 0 { s1 − s2 ≤ ≤ ≤ s1,s2 T . From Lemma 6.2, for integers a and b with a + b = 6 2r, we have the≤ following} estimate: − 1 1 a a (vε⊗ + (1 v)ε⊗ ) |h (2j 1)/n − 2j/n Z0 Z0 − b b (wε⊗(2k 1)/n + (1 w)ε2⊗k/n), (49) ⊗ − − 3 r 3 r ∂(2⊗j−1)/n ∂(2⊗k−1)/n H⊗6−2r dv dw − ⊗ − i | (3 r) Cn− − . ≤ 48 D. HARNETT AND D. NUALART It follows that if r = 0, then by Lemma 4.1, equation (48) and equation (49), 6 nt/2 ⌊ ⌋ r C ∂(2j 1)/n, ∂(2k 1)/n H |h − − i | j,kX=1 E 6 r 3 r 3 r [ D − (G0(2j)G0(2k)), ∂(2⊗j−1)/n ∂(2⊗k−1)/n H⊗6−2r ] × |h − ⊗ − i | nt/2 ⌊ ⌋ r 3 r Cn − ∂(2j 1)/n, ∂(2k 1)/n H ≤ |h − − i | j,kX=1 nt r/2 3 C n − , ≤ 2 which satisfies (42) because r 3 3 . On the other hand, if r =0, then 2 − ≤− 2 nt/2 ⌊ ⌋ 3 nt 2 Cn− C n− , ≤ 2 j,kX=1 and we are done with (43). Proof of (45). Proceeding along the same lines as above, nt/2 2 ⌊ ⌋ E 2 w2jG0(2j)δ(∂(2j 1)/n) " − ! # Xj=1 nt/2 ⌊ ⌋ E 2 2 = w2jw2kG0(2j)G0(2k) " j,kX=1 2 δ (∂(2j 1)/n ∂(2k 1)/n)+ ∂(2j 1)/n, ∂(2k 1)/n H × { − ⊗ − h − − i }# nt/2 ⌊ ⌋ 1 E E 2 Cn− sup G0(ℓ) ∂(2j 1)/n, ∂(2k 1)/n H ≤ 0 ℓ nt/2 | | |h − − i | j,kX=1 h ≤ ≤⌊ ⌋ i nt/2 ⌊ ⌋ 1 a b + Cn− E E D G (2j)D G (2k), |h 0 0 j,kX=1 a+Xb=2 2 δ (∂(2j 1)/n ∂(2k 1)/n) H⊗2 . − ⊗ − i | By Lemma 4.1 we have an estimate for the second term, nt/2 ⌊ ⌋ 1 nt 3/2 Cn− ∂(2j 1)/n, ∂(2k 1)/n H C n− . |h − − i |≤ 2 j,kX=1 CLT FOR A STRATONOVICH INTEGRAL 49 Then the first term has the same estimate as (48) when r = 2, which proves (45) and the lemma. 6.2. Proof of Lemma 4.5. As in Lemma 4.4, we may assume r = 0. Start with Bn(t). Define nt/2 ⌊ ⌋ (3) 2 2 γn(t):= f (W(2j 1)/n) ε(2j 1)/n, ∂(2⊗j 1)/n ∂(2⊗j 2)/n H − h − − − − i Xj=1 nt/2 ⌊ ⌋ (3) = f (W(2j 1)/n) ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H − h − − − − i Xj=1 (∂(2j 1)/n ∂(2j 2)/n), × − − − so that Bn(t) = 2δ(γn(t)). By Lemma 2.1, we have 2 2 2 δ(γ (t)) 2 E γ (t) + E Dγ (t) ⊗2 . k n kL (Ω) ≤ k n kH k n kH We can write nt/2 ⌊ ⌋ 2 (3) (3) γn(t) H = f (W(2j 1)/n)f (W(2k 1)/n) k k − − j,kX=1 ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H × h − − − − i ε(2k 1)/n, ∂(2k 1)/n ∂(2k 2)/n H × h − − − − i ∂(2j 1)/n ∂(2j 2)/n, ∂(2k 1)/n ∂(2k 2)/n H × h − − − − − − i (3) 2 2 sup f (Ws) sup ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H ≤ 0 s t| | 1 j nt h − − − − i ≤ ≤ ≤ ≤⌊ ⌋ nt/2 ⌊ ⌋ ∂(2j 1)/n ∂(2j 2)/n, ∂(2k 1)/n ∂(2k 2)/n H . × |h − − − − − − i | j,kX=1 E (3) 2 Note that [sup0 s t f (Ws) ]= C by condition (0), and by Lemma 6.2, ≤ ≤ | | 1/2 εt, ∂(2j 1)/n ∂(2j 2)/n H C2n− for all j, t. By Corollary 4.2 we know |h − − − i |≤ nt/2 ⌊ ⌋ nt 1/2 ∂(2j 1)/n ∂(2j 2)/n, ∂(2k 1)/n ∂(2k 2)/n H C n− . |h − − − − − − i |≤ 2 j,kX=1 Hence, it follows that E γ (t) 2 C nt n 1n 1/2 C nt n 3/2. Next, k n kH ≤ ⌊ 2 ⌋ − − ≤ ⌊ 2 ⌋ − nt/2 ⌊ ⌋ (4) Dγn(t)= f (W(2j 1)/n) ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H − h − − − − i Xj=1 (ε(2j 1)/n (∂(2j 1)/n ∂(2j 2)/n)), × − ⊗ − − − 50 D. HARNETT AND D. NUALART and this implies 2 (4) 2 Dγn(t) H⊗2 sup f (Ws) k k ≤ 0 s t| | ≤ ≤ nt/2 ⌊ ⌋ ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H × h − − − − i j,k=1 X ε(2k 1)/n, ∂(2k 1)/n ∂(2k 2)/n H × h − − − − i ε(2j 1)/n (∂(2j 1)/n ∂(2j 2)/n), × |h − ⊗ − − − ε(2k 1)/n (∂(2k 1)/n ∂(2k 2)/n) H⊗2 − ⊗ − − − i | (4) 2 2 sup f (Wt) sup ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H ≤ 0 s t| | j h − − − − i ≤ ≤ sup εs, εr H × 0 s,r t|h i | ≤ ≤ nt/2 ⌊ ⌋ ∂(2j 1)/n ∂(2j 2)/n, ∂(2k 1)/n ∂(2k 2)/n H . × |h − − − − − − i | j,kX=1 E (4) By condition (0), [sup0 s t f (Ws) ] is bounded, and sup0 s,r t εr, εs H ≤ ≤ | | ≤ ≤ |h i | is bounded. Hence, it can be seen that E Dγ (t) 2 gives the same estimate k n kH⊗2 as γn(t). 2 2 For Cn(t), using condition (0) and the identity a b = (a b)(a + b), we can write − − 2 (4) 2 E[Cn(t) ] E sup f (Ws) ≤ 0 s t| | h ≤ ≤ i sup ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H × 1 j nt/2|h − − − − i | ≤ ≤ nt/2 2 ⌊ ⌋ ε(2j 1)/n, ∂(2j 1)/n + ∂(2j 2)/n H . × |h − − − i |! Xj=1 1/2 By Lemma 6.2, ε(2j 1)/n, ∂(2j 1)/n ∂(2j 2)/n H C2n− and by condi- tion (iv), |h − − − − i |≤ nt/2 nt/2 ⌊ ⌋ ⌊ ⌋ 1 1/2 1/2 1/2 ε(2j 1)/n, [(2j 2)/n,2j/n) H Cn− + Cn− (2j 2)− |h − − i |≤ − Xj=1 Xj=2 1/2 nt 1/2 C n− . ≤ 2 CLT FOR A STRATONOVICH INTEGRAL 51 E 2 nt 2 Hence it follows that [Cn(t) ] C 2 n− for some constant C, and the lemma is proved. ≤ ⌊ ⌋ 6.3. Proof of Lemma 4.9. For i = 1,...,d, set nt /2 ⌊ i ⌋ i 2 2 un = f ′′(W(2j 1)/n)(∂(2⊗j 1)/n ∂(2⊗j 2)/n) − − − − j= nt − /2 +1 ⌊ Xi 1 ⌋ i 2 i and recall that Fn = δ (un). As in Remark 3.3, we want to show: Condition (a). For each 1 i d, the following converge to zero in L1(Ω): i ≤ ≤ (a.1) un, h1 h2 H⊗2 for all h1, h2 H of the form ετ (see Remark 3.4). h i ⊗j i ∈ (a.2) un, DFn h H⊗2 for each 1 j d and h H. h i j ⊗ i k ≤ ≤ ∈ (a.3) u , DFn DF H⊗2 for each 1 j, k d. h n ⊗ n i ≤ ≤ Condition (b). i 2 j 1 (b.1) un, D Fn H⊗2 0 in L if i = j. h i 2 i i −→ 16 (b.2) u , D F H⊗2 converges in L to a random variable of the form h n ni ti j 2 F = c f ′′(Ws) η(ds). ∞ Zti−1 The proofs of (a.1) and (a.2) are essentially the same as those given in [5] (see Theorem 5.2), but the proof of (a.3) is new. 2 Proof of (a.1). We may assume i = 1. Let h1 h2 = εs ετ H⊗ for some values s,τ [0,t]. Then ⊗ ⊗ ∈ ∈ nt1/2 ⌊ ⌋ 1 un, h1 h2 H⊗2 = f ′′(W(2j 1)/n) ∂(2j 1)/n ∂(2j 2)/n, εs H h ⊗ i − h − − − i Xj=1 ∂(2j 1)/n ∂(2j 2)/n, ετ H; × h − − − i so that 1 u , h h ⊗2 |h n 1 ⊗ 2iH | sup f ′′(Ws) sup sup ∂(2j 1)/n ∂(2j 2)/n, εs H ≤ 0 s t| | 1 j nt1/2 0 s t1|h − − − i | ≤ ≤ ≤ ≤⌊ ⌋ ≤ ≤ nt1/2 ⌊ ⌋ ∂(2j 1)/n ∂(2j 2)/n, ετ H . × |h − − − i | Xj=1 It follows from condition (iii) that for fixed τ 0, ≥ nt1/2 ⌊ ⌋ ∂(2j 1)/n ∂(2j 2)/n, ετ H |h − − − i | Xj=1 52 D. HARNETT AND D. NUALART nt1/2 ⌊ ⌋ E = [Wτ (W2j/n 2W(2j 1)/n + W(2j 2)/n)] | − − − | j=1 (50) X nt1/2 ⌊ ⌋ 1/2 1/2 3/2 3/2 Cn− + Cn− ((2j 2)− + τ 2j − 1) ≤ − | − | ∧ Xj=2 1/2 Cn− , ≤ and Lemma 6.2 implies 1/2 sup sup ∂(2j 1)/n ∂(2j 2)/n, εs H Cn− 1 j nt1/2 0 s t|h − − − i |≤ ≤ ≤⌊ ⌋ ≤ ≤ so that 1 1 E( u , h h ⊗2 ) Ct n− 0. |h n 1 ⊗ 2iH | ≤ 1 −→ Proof of (a.2). As in (a.1), assume i = 1. Using the same technique as in j j (a.1), we can write DFn h as DFn ετ for some τ [0, T ]. By Lemma 2.1, j 2 j 2 ⊗j j ⊗ ∈ DFn = Dδ (un)= δ (Dun)+ δ(un), which gives 1 j 1 2 j 1 j u , DF ε ⊗2 = u , δ (Du ) ε ⊗2 + u , δ(u ) ε ⊗2 . h n n ⊗ τ iH h n n ⊗ τ iH h n n ⊗ τ iH For the first term, 1 2 j E u , δ (Du ) ε ⊗2 |h n n ⊗ τ iH | nt1/2 ⌊ ⌋ E 2 j = f ′′(W(2ℓ 1)/n) ∂(2ℓ 1)/n ∂(2ℓ 2)/n, δ (Dun) H | − h − − − i Xℓ=1 ∂(2ℓ 1)/n ∂(2ℓ 2)/n, ετ H × h − − − i | E E 2 j 2 sup f ′′(Ws) sup ∂ℓ/n, δ (Dun) H ≤ 0 s t1| | 1 ℓ nt1/2 |h i | h ≤ ≤ i h ≤ ≤⌊ ⌋ i nt1/2 ⌊ ⌋ ∂(2ℓ 1)/n ∂(2ℓ 2)/n, ετ H . × |h − − − i | Xℓ=1 1/2 By (50), the sum has estimate Cn− , and for the second term we can take 2 j 2 j ∂ℓ/n, δ (Dun) H sup ∂ℓ/n H δ (Dun) H. |h i |≤ ℓ k k k k 1/4 2 j It follows from condition (i) that ∂ℓ/n H Cn− . This leaves the δ (Dun) H term. By the Meyer inequality fork a processk ≤ taking values in H, k k 2 j 2 j 2 2 j 2 3 j 2 (51) E[ δ (Du ) ] c E Du ⊗3 + c E D u ⊗4 + c E D u ⊗5 , k n kH ≤ 1 k nkH 2 k nkH 3 k nkH CLT FOR A STRATONOVICH INTEGRAL 53 so that by Lemma 4.7, E[ δ2(Du) 2 ] C, and we have k kH ≤ 1 2 j 3/4 E u , δ (Du ) ε ⊗2 Cn− . |h n n ⊗ τ iH |≤ Then similarly, 1 j u , δ(u ) ε ⊗2 |h n n ⊗ tiH | j 2 sup f ′′(Ws) sup ∂ℓ/n, δ(un) H ∂(2ℓ 1)/n ∂(2ℓ 2)/n, εt H . ≤ 0 s t1| | ℓ |h i | |h − − − i | ≤ ≤ Xℓ Similar to the above case, for each 1 ℓ nt1 , ≤ ≤ ⌊ 2 ⌋ j j E[ ∂ , δ(u ) ] E[ ∂ H δ(u ) ] |h ℓ/n n iH| ≤ k ℓ/nk k n kH 1/4 j j Cn− (E u ⊗2 + E Du ⊗3 ) ≤ k nkH k nkH 1/4 Cn− , ≤ and hence with (50) we have 1 j 3/4 E[ u , δ(u ) ε ⊗2 ] Cn− . |h n n ⊗ τ iH | ≤ i j k Proof of (a.3). For this term we consider the product un, DFn DFn H⊗2 . Lemma 6.1 shows that scalar products of this kind are smallh in absolute⊗ valuei when the time intervals are disjoint, and therefore it is enough to consider 1 1 1 the worst case u , DF DF H⊗2 , and assume t = t. We have h n n ⊗ n i 1 1 1 1 E[ u , DF DF ⊗2 ] |h n n ⊗ n iH | nt/2 ⌊ ⌋ E 2 2 1 1 [ f ′′(W(2ℓ 1)/n)(∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n), DFn DFn H⊗2 ] ≤ | h − − − − ⊗ i | Xℓ=1 nt/2 ⌊ ⌋ E 1 2 1 2 C [ ∂(2ℓ 1)/n, DFn H ∂(2ℓ 2)/n, DFn H ] ≤ |h − i − h − i | Xℓ=1 nt/2 ⌊ ⌋ E 1 1 1 C [ ∂(2ℓ 1)/n ∂(2ℓ 2)/n, DFn H [(2ℓ 2)/n,2ℓ/n], DFn H ]. ≤ |h − − − i ||h − i | Xℓ=1 1 2 1 1 Using the decomposition DFn = δ (Dun)+ δ(un), the above summand ex- pands into four terms: 2 1 1 2 1 (1) ∂(2ℓ 1)/n ∂(2ℓ 2)/n, δ (Dun) H [(2ℓ 2)/n,2ℓ/n], δ (Dun) H ; |h − − − i ||h − i | 2 1 1 1 (2) ∂(2ℓ 1)/n ∂(2ℓ 2)/n, δ (Dun) H [(2ℓ 2)/n,2ℓ/n], δ(un) H ; |h − − − i ||h − i | 1 1 2 1 (3) ∂(2ℓ 1)/n ∂(2ℓ 2)/n, δ(un) H [(2ℓ 2)/n,2ℓ/n], δ (Dun) H ; |h − − − i ||h − i | 1 1 1 (4) ∂(2ℓ 1)/n ∂(2ℓ 2)/n, δ(un) H [(2ℓ 2)/n,2ℓ/n], δ(un) H . |h − − − i ||h − i | 54 D. HARNETT AND D. NUALART We will show computations for the terms (1) and (4) only, with the others being similar. For (1) we have nt/2 ⌊ ⌋ E 2 1 1 2 1 C [ ∂(2ℓ 1)/n ∂(2ℓ 2)/n, δ (Dun) H [(2ℓ 2)/n,2ℓ/n], δ (Dun) H ] |h − − − i ||h − i | Xℓ=1 nt/2 ⌊ ⌋ E = C ∂(2ℓ 1)/n ∂(2ℓ 2)/n, ′ |h − − − ℓ,m,mX=1 2 (3) 2 2 δ (f (W(2m 1)/n)ε(2m 1)/n(∂(2⊗m 1)/n ∂(2⊗m 2)/n)) H − − − − − i | 1 [(2ℓ 2)/n,2ℓ/n], × |h − 2 (3) 2 2 δ (f (W(2m′ 1)/n)ε(2m′ 1)/n(∂(2⊗m′ 1)/n ∂(2⊗m′ 2)/n)) H − − − − − i | E 2 (3) 2 2 2 C sup ( [ δ (f (W(2k 1)/n)(∂(2⊗k 1)/n ∂(2⊗k 2)/n)) H⊗2 ]) ≤ 1 k nt/2 k − − − − k ≤ ≤⌊ ⌋ nt/2 ⌊ ⌋ ∂(2ℓ 1)/n ∂(2ℓ 2)/n, ε(2m 1)/n H × ′ |h − − − − i | ℓ,m,mX=1 1 [(2ℓ 2)/n,2ℓ/n], ε(2m′ 1)/n H . × |h − − i | 1/2 By Lemmas 2.1 and 4.7, the Skorohod integral term is bounded by Cn− , and we use conditions (iii) and (iv) for the scalar products to obtain an estimate of the form nt/2 ⌊ ⌋ 2 3/2 3/2 Cn− ((2m 1)− + 2ℓ 2m − ) ′ − | − | ℓ,m,mX=1 1/2 1/2 ((2ℓ 2)− + 2ℓ 2m′ − ) × − | − | 1/2 Cn− . ≤ For term (4), we have by a computation similar to the proof of Lemma 4.7, E (3) 1/4 [ δ(f (W(2k 1)/n)(∂(2k 1)/n ∂(2k 2)/n)) H] Cn− , k − − − − k ≤ and by conditions (i) and (ii) we have nt/2 ⌊ ⌋ 3/2 Cn− ∂(2ℓ 1)/n ∂(2ℓ 2)/n, ∂(2m 1)/n ∂(2m 2)/n H ′ |h − − − − − − i | ℓ,m,mX=1 1 [(2ℓ 2)/n,2ℓ/n], ∂(2m′ 1)/n ∂(2m′ 2)/n H × |h − − − − i | CLT FOR A STRATONOVICH INTEGRAL 55 nt/2 ⌊ ⌋ 3/2 α α Cn− ( 2ℓ 2m − )( 2ℓ 2m′ − ) ≤ ′ | − | | − | ℓ,m,mX=1 1/2 Cn− . ≤ 2 Proof of (b.1). By Lemma 2.1, we can expand D Fn as follows: i 2 j i 2 2 j i j un, D Fn H⊗2 = un, δ (D un) H⊗2 + 4 un, δ(Dun) H⊗2 (52) h i h i h i i j + 2 u , u ⊗2 . h n niH i Without loss of generality, we may assume that un is defined on [0,t1], j and Fn is defined on [t1,t2] for t1 nt1/2 ⌊ ⌋ i 2 2 un = f ′′(W(2ℓ 1)/n)(∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n) − − − − Xℓ=1 and nt2/2 ⌊ ⌋ j 2 2 un = f ′′(W(2m 1)/n)(∂(2⊗m 1)/n ∂(2⊗m 2)/n). − − − − m= nt1/2 +1 ⌊X ⌋ First term. i 2 2 j E u , δ (D u ) ⊗2 |h n n iH | nt1/2 ⌊ ⌋ E 2 2 = f ′′(W(2ℓ 1)/n)(∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n), * − − − − ℓ=1 X nt2/2 ⌊ ⌋ 2 (4) 2 δ f (W(2m 1)/n)ε(2⊗m 1)/n − − m= nt1/2 +1 ⌊X ⌋ 2 2 (∂⊗ ∂⊗ ) ⊗ (2m 1)/n − (2m 2)/n − − !+ ⊗2 H E sup f ′′(Ws) ≤ 0 s t| | h ≤ ≤ i 2 2 2 E ε⊗ , ∂⊗ ∂⊗ ⊗2 × |h (2m 1)/n (2ℓ 1)/n − (2ℓ 2)/niH | m − − − Xℓ X 2 (4) 2 2 δ (f (W(2m 1)/n)(∂⊗ ∂⊗ )) × | − (2m 1)/n − (2m 2)/n | − − E sup f ′′(Ws) ≤ 0 s t| | h ≤ ≤ i 56 D. HARNETT AND D. NUALART nt2/2 nt2/2 ⌊ ⌋ ⌊ ⌋ 2 2 sup δ (g4) L2(Ω) [ ε(2m 1)/n, ∂(2ℓ 1)/n H × m k k h − − i Xℓ=1 mX=1 2 ε(2m 1)/n, ∂(2ℓ 2)/n H]. − h − − i 2 First we need an estimate for the δ (g4) term, where in the notation of Lemma 4.7, (4) 2 2 g4 := f (W(2m 1)/n)(∂(2⊗m 1)/n ∂(2⊗m 2)/n). − − − − 2 E E E 2 By Lemma 2.1, δ (g4) L2(Ω) c1 g4 H⊗2 +c2 Dg4 H⊗3 +c3 D g4 H⊗4 , 2 k k 1/≤2 k k nt1 k knt2 k k and so δ (g ) 2 Cn for each nt2/2 ⌊ ⌋ 2 2 [ ε(2m 1)/n, ∂(2ℓ 1)/n H ε(2m 1)/n, ∂(2ℓ 2)/n H ] |h − − i − h − − i | ℓ,mX=1 1 sup ε(2m 1)/n, [(2ℓ 2)/n,2ℓ/n] H ≤ ℓ,m |h − − i | nt2/2 ⌊ ⌋ ε(2m 1)/n, ∂(2ℓ 1)/n ∂(2ℓ 2)/n H × |h − − − − i | ℓ,mX=1 nt2/2 ⌊ ⌋ 1/2 1/2 3/2 3/2 Cn− C n− [( ℓ m − + (ℓ 1)− ) 1] ≤ 2 | − | − ∧ ℓ,mX=1 nt2/2 ⌊ ⌋ 1 ∞ 3/2 Cn− p− C. ≤ ≤ p=1 Xℓ=1 X This provides an upper bound for the double sum, and hence the first 1/2 term of (52) is O(n− ). Note that in the above estimate the double sum is nt2 taken over 1 ℓ,m 2 . It follows that this estimate also holds for the ≤ ≤ ⌊i ⌋2 2 i 1/2 case i = j, that is, E u , δ (D u ) H⊗2 Cn . |h n n i |≤ − Second term. Using t1 CLT FOR A STRATONOVICH INTEGRAL 57 nt2/2 ⌊ ⌋ (3) 2 2 δ f (W(2k 1)/n)ε(2k 1)/n (∂⊗ ∂⊗ ) − − ⊗ (2k 1)/n − (2k 2)/n − − !+ ⊗2 k= nt1/2 H ⌊X ⌋ E = f ′′(W(2j 1)/n) ε(2k 1)/n (∂(2k 1)/n ∂(2k 2)/n), − h − ⊗ − − − j k X X 2 (∂(2j 1)/n ∂(2j 2)/n)⊗ H⊗2 − − − i (3) δ(f (W(2k 1)/n)(∂(2k 1)/n ∂(2k 2)/n)) × | − − − − | E C sup f ′′(Ws) sup εs, ∂j/n H sup δ(g3) L2(Ω) ≤ 0 s t| | s,j |h i | k k k h ≤ ≤ i nt2 nt2 ⌊ ⌋ ⌊ ⌋ ∂ , ∂ H , × |h j/n k/ni | Xj=0 Xk=0 (3) where in this case, g3 corresponds to the term including f (Wt). It fol- 1/2 lows from Lemma 6.2 that sup εs, ∂k/n H Cn− ; and the double sum is 1/2 |h i |≤ bounded by Cn by Corollary 4.2. This leaves an estimate for δ(g ) 2 . k 3 kL (Ω) By Lemma 2.1, δ(g ) 2 c g H + c Dg H⊗2 . For this case, k 3 kL (Ω) ≤ 1k 3k 2k 3k 2 E (3) 2 2 1/2 g3 H sup f (Ws) ∂(2k 1)/n ∂(2k 2)/n H Cn− , k k ≤ 0 s t| | k − − − k ≤ h ≤ ≤ i 1/4 hence g H Cn . Similarly, k 3k ≤ − E (4) Dg3 H⊗2 sup f (Ws) sup εs H ∂(2k 1)/n ∂(2k 2)/n H k k ≤ 0 s t| | 0 s t k k k − − − k h ≤ ≤ i ≤ ≤ 1/4 Cn− , ≤ 1/4 hence the second term is O(n− ). As in the first term, the double sum i i estimate shows that this result also holds for u , δ(DF ) H⊗2 . h n n i Third term. We can write nt1/2 nt2/2 ⌊ ⌋ ⌊ ⌋ i j 2 2 2 un, un H⊗2 sup f ′′(Ws) ∂(2⊗ℓ 1)/n ∂(2⊗ℓ 2)/n, |h i |≤ 0 s t| | |h − − − ≤ ≤ ℓ=1 m= nt1/2 +1 X ⌊X ⌋ 2 2 ∂(2⊗m 1)/n ∂(2⊗m 2)/n H⊗2 , − − − i | i j ε and it follows from Lemma 6.1 that E u , un H⊗2 Cn , for some ε> 0. |h n i |≤ − 58 D. HARNETT AND D. NUALART Proof of (b.2). As in case (b.1), this has the expansion (52). From remarks in the proof of (b.1), the first two terms have the same estimate as the i = j i i 6 case, hence only the term u , u H⊗2 is significant. h n ni Third term. Assume for the summation terms that the indices run over nti−1 + 1 j, k nti . We have ⌊ 2 ⌋ ≤ ≤ ⌊ 2 ⌋ i i un, un H⊗2 = f ′′(W(2j 1)/n)f ′′(W(2k 1)/n) h i − − Xj,k 2 2 2 2 ∂(2⊗j 1)/n ∂(2⊗j 2)/n, ∂(2⊗k 1)/n ∂(2⊗k 2)/n H⊗2 . × h − − − − − − i Expanding the product, observe that 2 2 2 2 ∂(2⊗j 1)/n ∂(2⊗j 2)/n, ∂(2⊗k 1)/n ∂(2⊗k 2)/n H⊗2 h − − − − − − i = β (2j 1, 2k 1)2 β (2j 1, 2k 2)2 n − − − n − − β (2j 2, 2k 1)2 + β (2j 2, 2k 2)2, − n − − n − − where βn(ℓ,m) is as defined for condition (v). For each n, define discrete measures on 1, 2,... 2 by { }⊗ ∞ µ+ := β (2j 1, 2k 1)2 + β (2j 2, 2k 2)2δ ; n n − − n − − jk j,kX=1 ∞ 2 2 µ− := β (2j 1, 2k 2) + β (2j 2, 2k 1) δ , n n − − n − − jk j,kX=1 where in this case δjk denotes the Kronecker delta. In the following, we show + only ηn , with ηn− being similar. It follows from condition (v) that for each t> 0, + 2 nt nt µ ([0,t] ) := lim µn , n 2 2 →∞ nt/2 ⌊ ⌋ 2 2 = lim βn(2j 1, 2k 1) + βn(2j 2, 2k 2) n − − − − j,kX=1 = η+(t). Moreover, if 0 REFERENCES [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. MR0233396 [2] Burdzy, K. and Swanson, J. (2010). A change of variable formula with Itˆocorrec- tion term. Ann. Probab. 38 1817–1869. MR2722787 [3] Houdre,´ C. and Villa, J. (2003). An example of infinite dimensional quasi-helix. In Stochastic Models (Mexico City, 2002). Contemporary Mathematics 336 195–201. Amer. Math. Soc., Providence, RI. MR2037165 60 D. HARNETT AND D. NUALART [4] Lei, P. and Nualart, D. (2009). A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett. 79 619–624. MR2499385 [5] Nourdin, I. and Nualart, D. (2010). Central limit theorems for multiple Skorokhod integrals. J. Theoret. Probab. 23 39–64. MR2591903 [6] Nourdin, I. and Reveillac,´ A. (2009). Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H = 1/4. Ann. Probab. 37 2200–2230. MR2573556 [7] Nourdin, I., Reveillac,´ A. and Swanson, J. (2010). The weak Stratonovich in- tegral with respect to fractional Brownian motion with Hurst parameter 1/6. Electron. J. Probab. 15 2117–2162. MR2745728 [8] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin. MR2200233 [9] Swanson, J. (2007). Weak convergence of the scaled median of independent Brow- nian motions. Probab. Theory Related Fields 138 269–304. MR2288071 [10] Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35 2122–2159. MR2353385 [11] Swanson, J. (2011). Fluctuations of the empirical quantiles of independent Brownian motions. Stochastic Process. Appl. 121 479–514. MR2763093 Department of Mathematics University of Kansas 405 Snow Hall Lawrence, Kansas 66045-2142 USA E-mail: [email protected] 0 and 2s r, t T with t r 2s, ≤ ≤ | − |≥ E[(Wt Wt s)(Wr Wr s)] | − − − − | 2 α β 2 3/2 C1s t r − (t r s)− + s t r − ≤ | − | ∧ − 3 | − | 3 for positive constants α, β, γ, such that 1 < α 2 and α + β = 2 . (iii) For 0 1 and β = 1 2H. Next,− consider the distance term ψ(t r, s). Without loss of generality, assume r