Noncentral limit theorem for the generalized Rosenblatt process
aa r X i v : . [ m a t h . P R ] M a y NONCENTRAL LIMIT THEOREM FOR THE GENERALIZEDROSENBLATT PROCESS
DENIS BELL AND DAVID NUALART
Abstract.
We use techniques of Malliavin calculus to study the convergence in law of afamily of generalized Rosenblatt processes Z γ with kernels defined by parameters γ takingvalues in a tetrahedral region ∆ of R q . We prove that, as γ converges to a face of ∆, theprocess Z γ converges to a compound Gaussian distribution with random variance given bythe square of a Rosenblatt process of one lower rank. The convergence in law is shown tobe stable. This work generalizes a previous result of Bai and Taqqu, who proved the resultin the case q = 2 and without stability. This paper is dedicated to the memory of Salah Mohammed Introduction
Let W = { W x , x ∈ R } be a two-sided Brownian motion on the real line. The generalizedRosenblatt process is defined by(1.1) Z γ ( t ) = Z R q f γ ( x , . . . , x q ) dW x · · · dW x q , t ≥ , where q ≥ γ = ( γ , . . . , γ q ) and f γ ( x ) = A γ Z t ( s − x ) γ + ( s − x ) γ + · · · ( s − x q ) γ q + ds for x = ( x , . . . , x q ) ∈ R q . The constant A γ is a normalizing constant, chosen so that E [ Z γ ( t ) ] = t ¯ γ +1+ q , where ¯ γ = γ + · · · + γ q . For f γ to be in L ( R q ) it is necessary that theexponent γ live in the region∆ = { γ : − < γ i < − , ≤ i ≤ q, γ + · · · + γ q > − q + 12 } . For q = 1, this process reduces to the fractional Brownian motion B H with Hurst parameter H = γ + ∈ ( , q = 2, the process has been considered by Maejima and Tudorin [9], and it generalizes the classical Rossenblatt process ( q = 2, γ = γ ) introduced byTaqqu in [11].In a recent work by Bai and Taqqu [2], the authors study the convergence in law of thisprocess when q = 2 and the parameter γ = ( γ , γ ) converges to the boundary of the region∆. In particular, when γ → − and γ is fixed, the limit in distribution is ηB γ + ( t ), where η is a standard normal Gaussian variable independent of the fractional Brownian motion Mathematics Subject Classification.
Key words and phrases.
Multiple stochastic integrals, Rosenblatt process, Skorohod integral, central andnoncentral limit theorems.D. Nualart was supported by the NSF grant DMS 1512891. γ + . Two different proofs are given of this result, one based on the method of momentsand a second constructive proof based on a discretization argument.The goal of this paper is to derive this result as an application of a general theorem ofconvergence in law of multiple stochastic integrals to a mixture of Gaussian distributions (seeTheorem 3.2), which is of independent interest. This theorem is proved using a noncentrallimit theorem for Skorohod integrals derived by Nourdin and Nualart in [3]. This allows usto extend Bai and Taqqu’s result in two directions: We can deal with a general Rosenblattin the q th Wiener chaos, and we can show that the convergence is stable.On the other hand, using a version of the Fourth Moment Theorem of Nualart and Peccati[8], we show (see Theorem 4.5) that when γ + · · · + γ q → − q +12 and γ i > − ǫ , 1 ≤ i ≤ q ,for a fixed ǫ >
0, then the limit is a standard Brownian motion B ( t ). For q = 2 this wasalso proved in [2]. 2. Preliminaries
Multiple stochastic integrals.
We denote by W = { W ( x ) , x ∈ R } a two-sided Brownianmotion on the real line defined on some probability space (Ω , F , P ). Then we can define theWiener integral W ( h ) = R R h ( x ) dW x for any function h in the Hilbert space H := L ( R ),and { W ( h ) , h ∈ H} is an isonormal Gaussian process. We recall that this means that thisis a centered Gaussian family with covariance given by the scalar product in H : E [ W ( h ) W ( g )] = h h, g i H . For every integer q ≥
1, consider the tensor product H ⊗ q = L ( R q ) and the symmetric tensorproduct, denoted by H ⊙ q , formed by the symmetric functions in L ( R q ). For any symmetricfunction f ∈ H ⊙ q we denote by I q ( f ) the multiple Wiener-Itˆo stochastic integral of f withrespect to W , that can be defined as an iterated Itˆo integral: I q ( f ) = Z R q f ( x , . . . , x q ) dW x · · · dW x q . Then the following isometry formula holds: E [ I q ( f ) ] = q ! k f k L ( R q ) . If f ∈ L ( R q ), we put I q ( f ) = I q ( ˜ f ), where ˜ f denotes the symmetrization of f , that is,˜ f γ ( x , . . . , x q ) = 1 q ! X σ f ( x σ , . . . , x σ q ) , where σ runs over all the permutations of { , . . . , q } .Let m, q ≥ I ⊂ { , . . . , q } of cardinality r = 0 , . . . , q ∧ m , aone-to-one mapping ψ : I → { , . . . , m } , and two functions f ∈ L ( R q ) and g ∈ L ( R m ), wedenote by f ⊗ I,ψ g the element in L ( R q + m − r ) given by f ⊗ I,ψ g = Z R r f ( x , . . . , x q ) g ( y , . . . , y m ) r Y j =1 δ ( x i − y ψ ( i ) ) dx · · · dx r . That is, f ⊗ I,ψ g is the function in f, g ∈ L ( R q − r ) obtained by contracting the variables x i from f with the variables y ψ ( i ) from g . When q = m and I = { , . . . , q } , we simply write ⊗ ψ g . Then the following product formula for multiple stochastic integrals holds. For any f ∈ L ( R q ) and g ∈ L ( R m ),(2.1) I q ( f ) I m ( g ) = q ∧ m X r =0 X I,ψ I q + m − r ( f ⊗ I,ψ g ) , where the sum runs over all sets I ⊂ { , . . . , q } of cardinality r and one-to-one mappings ψ : I → { , . . . , m } . Notice that when f and g are symmetric, this reduces to the well-knownformula I q ( f ) I m ( g ) = q ∧ m X r =0 (cid:18) qr (cid:19)(cid:18) mr (cid:19) r ! I q + m − r ( f ⊗ r g ) , where f ⊗ r g is the contraction of r indices of f and g . On the other hand, for any function f ∈ H ⊗ q , which is not necessarily symmetric, we have E [ I q ( f ) ] = X ψ f ⊗ ψ f, where ψ runs over all bijections of { , . . . , q } .Let { F n } be a sequence of random variables, all defined on the probability space (Ω , F , P )and let F be a random variable defined on some extended probability space (Ω ′ , F ′ , P ′ ). Wesay that F n converges stably to F , iflim n →∞ E (cid:2) Ze iλF n (cid:3) = E ′ (cid:2) Ze iλ,F (cid:3) for every λ ∈ R and every bounded F –measurable random variable Z .2.0.2. Elements of Malliavin calculus.
We introduce some basic elements of the Malliavincalculus with respect to the two-sided Brownian motion W . We refer the reader to Nualart[7] for a more detailed presentation of these notions. Let S be the set of all smooth andcylindrical random variables of the form(2.2) F = g ( W ( h ) , . . . , W ( h n )) , where n ≥ g : R n → R is a infinitely differentiable function with compact support, and h i ∈ H . The Malliavin derivative of F with respect to X is the element of L (Ω; H ) definedas DF = n X i =1 ∂g∂x i ( W ( h ) , . . . , W ( h n )) h i . By iteration, one can define the q th derivative D q F for every q ≥
2, which is an element of L (Ω; H ⊙ q ).For q ≥ p ≥ D q,p denotes the closure of S with respect to the norm k · k D q,p ,defined by the relation k F k p D q,p = E [ | F | p ] + q X i =1 E (cid:0) k D i F k p H ⊗ i (cid:1) . If V is a real separable Hilbert space, we denote by D q,p ( V ) the corresponding Sobolev spaceof V -valued random variables.We denote by δ the adjoint of the operator D , also called the divergence operator. Theoperator δ is an extension of the Itˆo integral. It is also called the Skorohod integral because n the case of the Brownian motion it coincides with the anticipating stochastic integralintroduced by Skorohod in [10]. A random element u ∈ L (Ω; H ) belongs to the domain of δ , denoted Dom δ , if and only if it satisfies (cid:12)(cid:12) E (cid:0) h DF, u i H (cid:1)(cid:12)(cid:12) ≤ c u p E ( F )for any F ∈ D , , where c u is a constant depending only on u . If u ∈ Dom δ , then the randomvariable δ ( u ) is defined by the duality relationship(2.3) E ( F δ ( u )) = E (cid:0) h DF, u i H (cid:1) , which holds for every F ∈ D , . The operators D and δ satisfy the following commutationrelation:(2.4) D ( δ ( u )) = u + δ ( Du ) , for any u ∈ D , ( H ).3. Noncentral limit theorems for multiple stochastic integrals
The following result has been proved by Nourdin and Nualart in [3].
Theorem 3.1.
Consider a sequence of Skorohod integrals of the form F n = δ ( u n ) , where u n ∈ D , ( H ) . Suppose that the sequence { F n , n ≥ } is bounded in L (Ω) and the followingconditions hold: (i) h u n , h i H converges to zero in L (Ω) for all elements h ∈ H , where H is a densesubset of H . (ii) h u n , DF n i H converges in L (Ω) to a nonnegative random variable S .Then F n converges stably to a random variable with conditional Gaussian law N (0 , S ) given W . On the other hand, from Proposition 3.1 of the paper by Nourdin, Nualart and Peccati[4], it follows that for any test function ϕ ∈ C , we have(3.1) | E [ ϕ ( F n )] − E [ ϕ ( Sη )] | ≤ k ϕ ′′ k ∞ E [ |h u n , DF n i H − S | ] + 13 k ϕ ′′′ k ∞ E [ |h u n , DS i H | ] , assuming S ∈ D , , and where η is a N (0 , W . This provides a rate of convergence in the previous theorem. Moreover, the in order toshow the convergence in law F n ⇒ Sη , it suffices to check the following two conditions:(i) h u n , DF n i H → S in L (Ω) as n tends to infinity, and(ii) h u n , DS i H → L (Ω) as n tends to infinity.Applying Theorem 3.1, we derive the following noncentral limit theorem for a sequence ofmultiple stochastic integrals of order q . Theorem 3.2.
Fix q ≥ . Let F n be given by F n = Z R q f n ( x , x , . . . , x q ) dW x dW x · · · dW x q , where f n ∈ L ( R q ) Assume that: i) For all elements h ∈ H , where H is a dense subset of H , we have Z R h ( ξ ) f n ( ξ, · ) dξ converges to zero in H ⊗ ( q − . (ii) For any subset I ⊂ { , . . . , q } of cardinality r = 1 , . . . , q and any one-to-one mapping ψ : I → { , . . . , q } such that ∈ I and ψ (1) = 1 , f n ⊗ I,ψ f n converges to zero in H ⊗ (2 q − r ) . (iii) There exists an element g ∈ L ( R q − ) with variables g ( x , . . . , x q ) , such that for anysubset I ⊂ { , . . . , q } of cardinality r = 0 , . . . , q − and any one-to-one mapping ψ : I → { , . . . , q } lim n →∞ Z R [ f n ( ξ, · ) ⊗ I,ψ f n ( ξ, · )] dξ = g ⊗ I,ψ g, where the convergence holds in H ⊗ (2 q − r − .Then F n converges stably to a random variable with conditional Gaussian law N (0 , S ) given W , where S = ( I q − ( g )) . Proof.
We can write F n = δ ( u n ) where u n ( ξ ) = I q − ( f n ( ξ, · )). Then, we claim that F n and u n satisfy the conditions of Theorem 3.1. Notice tat u n ∈ D , ( H ) because u n is a multiplestochastic integral. To show condition (i) of Theorem 3.1, fix h ∈ H . Then, E [ h u n , h i H ] = E "(cid:12)(cid:12)(cid:12)(cid:12) I q − (cid:18)Z R h ( ξ ) f n ( ξ, · ) dξ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q ! (cid:13)(cid:13)(cid:13)(cid:13)Z R h ( ξ ) f n ( ξ, · ) dξ (cid:13)(cid:13)(cid:13)(cid:13) H ⊗ ( q − , which converges to zero by condition (i).It remains to check condition (ii) of Theorem 3.1. Let us first compute the inner product h u n , DF n i H . Recall that F n = δ ( u n ), where u n ( ξ ) = I q − ( f n ( ξ, · )). Using the commutationrelation (2.4), we can write D ξ F n = u n ( ξ ) + δ ( D ξ u n )= u n ( ξ ) + q X i =2 Z R q − f n ( x , x , . . . , x i − , ξ, x i +1 , · · · , x q − ) dW x · · · dW x q − =: u n ( ξ ) + q X i =2 G n,i ( ξ ) . We claim that h u n , G n,i i H converges to zero in L (Ω). Indeed, h u n , G n,i i H = Z R (cid:18)Z R q − f n ( ξ, x , . . . , x q − ) dW x · · · dW x q − (cid:19) × (cid:18)Z R q − f n ( x , x , . . . , x i − , ξ, x i +1 , · · · , x q − ) dW x · · · dW x q − (cid:19) dξ s a consequence, using the product formula for multiple stochastic integrals (see (2.1)), wecan write h u n , G n,i i H = q X r =1 X I,ψ I q − r − ( f n ⊗ I,ψ f n ) , where the sum is over all sets I ⊂ { , . . . , q } of cardinality r and one-to-one mappings ψ : I → { , . . . , q } such that i ∈ I and ψ (1) = i . Because i = 1, by condition (i) this termconverges to zero in L (Ω). Finally, taking into account that h u n , DF n i H = k u n k H + q X i =2 h u n , G n,i i H , it suffices to consider the convergence of k u n k H . For this term, we have k u n k H = Z R I q − ( f n ( ξ, · )) dξ = q − X r =0 X I,ψ I q − r − (cid:18)Z R ( f n ( ξ, · ) ⊗ I,ψ f n ( ξ, · )) dξ (cid:19) , where the sum is over all sets I ⊂ { , . . . , q } of cardinality r and one-to-one mappings ψ : I → { , . . . , q } . By our hypothesis (ii), this sum converges in L (Ω) to q − X r =0 X I,ψ I q − r − ( g ⊗ I,ψ g ) = ( I q − ( g )) , where the sum runs over all sets I ⊂ { , . . . , q } of cardinality r = 0 , . . . , q − ψ : I → { , . . . , q } . This completes the proof. (cid:3) It will have been noted that the proof of Theorem 3.2 depends crucially upon expressing F n as the Skorohod integral of a multiple Wiener integral of rank q −
1, i.e. choosing akernel f n such that u n ( ξ ) = I q − ( f n ( ξ )). Obviously, the choice of f n is not unique, e.g. onecould equally well choose f n ( ξ ) = f n ( x , x , . . . , , x i − , ξ, x i +1 , . . . , x q − ), for any 2 ≤ i ≤ q .However, any such choice will lead to the term (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z R q − f n ( x , x , . . . , , x i − , ξ, x i +1 , . . . , x q − ) dW x · · · dW x q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H . in the computation of h u n , DF n i H . This term evidently does not converge in L (Ω) sinceit can be shown that its L norm converges to a non-zero limit, while the integrand f n ( x , x , . . . , , x i − , ξ, x i +1 , . . . , x q − ) converges pointwise to zero outside of the diagonal in R q − . Thus the choice i = 1 is the only one that will work in the argument.The case where the limit is Gaussian is not included in Theorem 3.2. We state thisconvergence in the next theorem, whose proof would be similar to that of Theorem 3.2.Notice that Theorem 4.4 below is just the Fourth Moment Theorem proved by Nualart andPeccati in [8] (see the reference [5] for extensions and applications of this result). In theversion below of the Fourth Moment Theorem we do not require the kernels to be symmetricand we add condition (i) which ensures the stability of the convergence. heorem 3.3. Fix q ≥ . Let F n be given by F n = Z R q f n ( x , x , . . . , x q ) dW x dW x · · · dW x q , where f n ∈ L ( R q ) . Suppose that condition (i) in Theorem 3.1 holds, and moreover, (ii) For any subset I ⊂ { , . . . , q } of cardinality r = 1 , . . . , q − and any one-to-onemapping ψ : I → { , . . . , q } , we have f n ⊗ I,ψ f n → in H ⊗ (2 q − r ) . (iii) lim n →∞ E [ F n ] = σ .Then, F n converges stably to a random variable with Gaussian law N (0 , σ ) , independent of W . Generalized Rosenblatt process
We are interested in the asymptotic behavior of the generalized Rosenblatt process Z γ ( t )defined in (1.1), when the parameter γ converges to the boundary of the region ∆. Considerfirst the case when one of the parameters (for simplicity we choose the first one) convergesto − .We will make use of the following technical lemmas. The first lemma was proved by Baiand Taqqu in [1, Lemma 3.2]. Lemma 4.1.
Suppose − < γ , γ < − / and s , s > . Then Z ∞−∞ ( s − x ) γ + ( s − x ) γ + dx = ( s − s ) γ + γ + B (1 + γ , − − γ − γ )+( s − s ) γ + γ + B (1 + γ , − − γ − γ ) . The second lemma concerns the asymptotic behavior of the Beta function (see [2, Lemma3.8]).
Lemma 4.2. As α → , we have αB ( α, β ) → , uniformly in β ∈ [ b , b ] , where < b < b < ∞ . In the next lemma, we compute the explicit value of the constant A γ . Lemma 4.3.
The constant A γ is given by A γ = (2 | γ | + q + 1)(2 | γ | + q + 2)2 P σ Q qj =1 B ( γ j + 1 , − γ j − γ σ j − , where the sum runs over all permutations σ of { , . . . , n } and we recall that | γ | = P qj =1 γ j .Proof. By a scaling argument, we can take t = 1. We have A γ = q ! k ˜ f γ k − L ( R q ) , here ˜ f γ denotes the symmetrization of f γ . Then˜ f γ ( x , . . . , x q ) = 1( q !) X σ,τ Z [0 , ( s − x σ ) γ + · · · ( s − x σ q ) γ q + × ( s − x τ ) γ + · · · ( s − x τ q ) γ q + ds ds . As a consequence, k ˜ f γ k L ( R q ) = 1 q ! X σ Z R q Z [0 , ( s − x ) γ + · · · ( s − x q ) γ q + × ( s − x σ ) γ + · · · ( s − x σ q ) γ q + ds ds dx · · · dx q = 1 q ! X σ Z [0 , q Y j =1 Z R ( s − x ) γ j + ( s − x ) γ σj + dx ! ds ds . By Lemma 4.1, we have Z R ( s − x ) γ j + ( s − x ) γ σj + dx = ( s − s ) γ j + γ σj +1+ B ( γ j + 1 , − γ j − γ σ j − s − s ) γ j + γ σj +1+ B ( γ σ j + 1 , − γ j − γ σ j − . Substituting this formula in the above expression for k ˜ f γ k L ( R q ) , we obtain k ˜ f γ k L ( R q ) = 1 q ! X σ Z [0 , " ( s − s ) | γ | + q + q Y j =1 B ( γ j + 1 , − γ j − γ σ j − s − s ) | γ | + q + q Y j =1 B ( γ σ j + 1 , − γ j − γ σ j − ds ds = 2 P σ Q qj =1 B ( γ j + 1 , − γ j − γ σ j − q !(2 | γ | + q + 1)(2 | γ | + q + 2) , which completes the proof of the lemma. (cid:3) The following is the main result of this paper.
Theorem 4.4. As γ converges to − , the random variable Z γ ( t ) converges stably to a ran-dom variable whose distribution given W is Gaussian with zero mean and variance Z γ ,...,γ q ( t ) .Proof. To simplify, by a scaling argument we can assume that t = 1. The asymptotic behaviorof the constant A γ , when γ → − is obtained from Lemma 4.3, taking into account theasymptotic behavior of the Beta function given by Lemma 4.2:(4.1) lim γ →− A γ − − γ = (2 P qj =2 γ j + q )(2 P qj =2 γ j + q + 1)2 P σ Q qj =2 B ( γ j + 1 , − γ j − γ σ j −
1) = A γ ,...,γ q , where in the denominator of the second expression, σ runs over all permutations of { , . . . , q } .The proof will be done in three steps. tep 1. Let us show condition (i) of Theorem 3.2. We can take h = [ a,b ] . Then, Z ba f γ ( ξ, · ) dξ = A γ Z (cid:18)Z ba ( s − ξ ) γ + dξ (cid:19) ( s − x ) γ + · · · ( s − x q ) γ q + ds. The term R ba ( s − ξ ) γ + dξ is uniformly bounded as γ → − and A γ converges to zero. Therefore,the above expression converges to zero in H ⊗ ( q − . Step 2.
Now we show condition (ii) of Theorem 3.2. Fix a subset I ⊂ { , . . . , q } of cardinality r = 1 , . . . , q and any one-to-one mapping ψ : I → { , . . . , q } such that 1 ∈ I and ψ (1) = 1.Set J = ψ ( I ). Let us compute( f n ⊗ I,ψ f n )( x, y ) = A γ Z R r Z [0 , Y i ∈ I ( s − ξ i ) γ i + Y j ∈ I c ( s − x j ) γ j + × Y i ∈ I ( s − ξ i ) γ ψ ( i ) + Y k ∈ J c ( s − y k ) γ k + ds ds dξ, where x = ( x j ) j ∈ I c , y = ( y k ) k ∈ J c , ξ = ( ξ i ) i ∈ I and dξ = Q i ∈ I dξ i . Using Lemma 4.1, we obtain( f n ⊗ I,ψ f n )( x, y ) = A γ Z [0 , (cid:16) Y i ∈ I ( s − s ) γ i + γ ψ ( i ) +1+ B ( γ i + 1 , − γ i − γ ψ ( i ) − Y i ∈ I ( s − s ) γ i + γ ψ ( i ) +1+ B ( γ ψ ( i ) + 1 , − γ i − γ ψ ( i ) − (cid:17) × Y j ∈ I c ( s − x j ) γ j + Y k ∈ J c ( s − y k ) γ k + ds ds , Set Φ ( s , s ) = Y i ∈ I ( s − s ) γ i + γ ψ ( i ) +1+ B ( γ i + 1 , − γ i − γ ψ ( i ) − Y i ∈ I ( s − s ) γ i + γ ψ ( i ) +1+ B ( γ ψ ( i ) + 1 , − γ i − γ ψ ( i ) − . With this notation, we can write k f n ⊗ I,ψ f n k L ( R q − r ) = A γ Z [0 , Φ ( s , s )Φ ( s , s )Φ ( s , s )Φ ( s , s ) ds ds ds ds , where Φ ( s , s ) = Y j ∈ I c | s − s | γ j +1 B ( γ j + 1 , − γ j − ( s , s ) = Y k ∈ J c | s − s | γ k +1 B ( γ k + 1 , − γ k − . We know that A γ ( − − γ ) − converges to a finite limit. On the other hand, Φ convergesto a finite limit because 1 ∈ I but ψ (1) = 1. Also, Φ converges to a finite sum because1 I c and Φ diverges as ( − − γ ) − as γ → − . Therefore,lim γ →− ( − − γ ) Z [0 , Φ ( s , s )Φ ( s , s )Φ ( s , s )Φ ( s , s ) ds ds ds ds = 0 tep 3. It remains to show condition (iii). Define g ( x , . . . , x q ) = A γ ,...,γ q Z ( s − x ) γ + · · · ( s − x q ) γ q + ds = f γ ,...,γ q ( x , . . . , x q ) . Fix r = 0 , . . . , q −
1, a set I ⊂ { , . . . , q } of cardinality r and a one-to-one mapping ψ := I → { , . . . , q } . Set J = ψ ( I ). We also write ¯ I = I ∪ { } and ¯ ψ is the extension of ψ to ¯ I such that ¯ ψ (1) = 1. We claim that f γ ⊗ ¯1 , ¯ ψ f γ converges in L ( R q − r − ) to g ⊗ I,ψ g . We have( f γ ⊗ ¯ I, ¯ ψ f γ )( x, y ) = A γ Z R r Z [0 , ( s − ξ ) γ + Y i ∈ I ( s − ξ i ) γ i + Y j ∈ I c ,j =1 ( s − x j ) γ j + × ( s − ξ ) γ + Y i ∈ I ( s − ξ i ) γ ψ ( i ) + Y k ∈ J c ,k =1 ( s − y k ) γ k + ds ds dξ dξ, (4.2)where x = ( x j ) j ∈ I c , y = ( y k ) k ∈ J c and ξ = ( ξ i ) i ∈ I . By Lemma 4.1, we have Z R ( s − ξ ) γ + ( s − ξ ) γ dξ = | s − s | γ +1 B ( γ + 1 , − γ − . Therefore,( f γ ⊗ ¯ I, ¯ ψ f γ )( x, y ) = A γ Z [0 , | s − s | γ +1 B ( γ + 1 , − γ − × (cid:16) Y i ∈ I ( s − s ) γ i + γ ψ ( i ) + B ( γ i + 1 , − γ i − γ ψ ( i ) − Y i ∈ I ( s − s ) γ i + γ ψ ( i ) + B ( γ ψ ( i ) + 1 , − γ i − γ ψ ( i ) − (cid:17) × Y j ∈ I c ,j =1 ( s − x j ) γ j + Y k ∈ J c ,k =1 ( s − y k ) γ k + ds ds , It suffices to show that the following quantities converge to k g ⊗ I,ψ g k L ( R q − r − ) as γ → − :(4.3) k f γ ⊗ ¯ I, ¯ ψ f γ k L ( R q − r − ) , and(4.4) h f γ ⊗ ¯ I, ¯ ψ f γ , g ⊗ I,ψ g i L ( R q − r − ) . We will consider only the convergence of (4.3), and that of (4.4) is proved in the same way.As before, setΦ ( s , s ) = Y i ∈ I ( s − s ) γ i + γ ψ ( i ) +1+ B ( γ i + 1 , − γ i − γ ψ ( i ) − Y i ∈ I ( s − s ) γ i + γ ψ ( i ) +1+ B ( γ ψ ( i ) + 1 , − γ i − γ ψ ( i ) − . ith this notation, we can write k f n ⊗ ¯ I, ¯ ψ f n k L ( R q − r − ) = A γ B ( γ + 1 , − γ − Z [0 , | s − s | γ +1 | s − s | γ +1 × Φ ( s , s )Φ ( s , s )Φ ( s , s )Φ ( s , s ) ds ds ds ds , where Φ ( s , s ) = Y j ∈ I c ,j =1 | s − s | γ j +1 B ( γ j + 1 , − γ j − ( s , s ) = Y k ∈ J c ,k =1 | s − s | γ k +1 B ( γ k + 1 , − γ k − . As γ → − , by the monotone convergence theorem, we obtainlim γ →− = A γ ,...,γ q Z [0 , Φ ( s , s )Φ ( s , s )Φ ( s , s )Φ ( s , s ) ds ds ds ds = k g ⊗ I,ψ g k L ( R q − r − ) . This completes the proof. (cid:3)
When γ converges to the boundary of ∆ defined by ¯ γ = q +12 , we obtain the followingresult, that generalizes Theorem 2.1 in [2]. Theorem 4.5.
Suppose that γ + · · · + γ q → − q +12 with γ i > − ǫ , ≤ i ≤ q , forarbitrarily fixed ǫ > . Then, Z γ ( t ) converges in law to B ( t ) , where B is a Brownian motionindependent of W .Proof. Condition (i) of Theorem 3.1 is easy to show. Then, it suffices to check condition (ii)of Theorem 4.4. Fix a subset I ⊂ { , . . . , q } of cardinality r = 1 , . . . , q − ψ : I → { , . . . , q } . We have k f n ⊗ I,ψ f n k H ⊗ (2 q − r ) = A γ Z [0 , Φ ( s , s )Φ ( s , s )Φ ( s , s )Φ ( s , s ) ds ds ds ds , where Φ ( s , s ) = ( s − s ) P i ∈ I ( γ i + γ ψ ( i ) )+ r + Y i ∈ I B ( γ i + 1 , − γ i − γ ψ ( i ) − s − s ) P i ∈ I ( γ i + γ ψ ( i ) )+ r + Y i ∈ I B ( γ ψ ( i ) + 1 , − γ i − γ ψ ( i ) − , Φ ( s , s ) = | s − s | P j ∈ Ic γ j + q − r Y j ∈ I c B ( γ j + 1 , − γ j − ( s , s ) = | s − s | P k ∈ Jc γ k + q − r Y k ∈ J c B ( γ k + 1 , − γ k − . All the products of Beta functions are uniformly bounded by our hypothesis γ i > − ǫ ,1 ≤ i ≤ q . Therefore, k f n ⊗ I,ψ f n k H ⊗ (2 q − r ) ≤ CA γ Z [0 , | s − s | α | s − s | α | s − s | α | s − s | α ds ds ds ds , here α = X i ∈ I ( γ i + γ ψ ( i ) ) + r,α = 2 X j ∈ I c γ j + q − r,α = 2 X k ∈ J c γ k + q − r. We have 2 α + α + α = 2 q X j =1 +2 q > − . Therefore, from Lemma 3.3 in [2], the above integral has a finite limit as γ + · · · + γ q → − q +12 ,and because A γ converges to zero as γ + · · · + γ q → − q +12 , this proves condition (ii). (cid:3) Remark 1.
Functional versions of theorems Theorem 4.4 and Theorem 4.5 in the space C ([0 , T ]) can be proved by the same arguments as in [2]. In fact, using the self-similarityand stationary-increment property of the process Z γ , together with the hypercontractiveinequality for multiple stochastic integrals, we can show that E ( | Z γ ( t ) − Z γ ( s ) | p ) ≤ c p | t − s | pH , for any p ≥
2, where H = γ + · · · + γ q + q ≥ . This leads to the tightness property andthe convergence of the finite dimensional distributions is also easy to obtain. Remark 2.
We can derive the rate of convergence in Theorem 4.4 using the inequality (3.1).More precisely, it is not difficult to show thatsup ϕ ∈C , k ϕ ′′′ k ∞ ≤ , k ϕ ′′ k ∞ ≤ | E [ ϕ ( F n )] − E [ ϕ ( Sη )] | ≤ C p − − γ . The same rate was obtained when q = 2 for the Wasserstein distance in [2, Theorem 5.3],using properties of the second order chaos. Concerning Theorem 4.5, using Stein’s methodand the optimal rate of convergence in the Fourth Moment Theorem derived by Nourdinand Peccati in [6], we can obtain the following rate of convergence for the total variationdistance, as in [2, Theorem 5.1]: c (¯ γ + ( q + 1) / ≤ d T V ( Z γ , η ) ≤ c (¯ γ + ( q + 1) / , where η is a N (0 ,
1) random variable. In this inequaliy γ satisfies γ i > − ǫ , 1 ≤ i ≤ q and the distance of γ to the boundary { ¯ γ + q +12 = 0 } is less than ǫ , for some ǫ >
0. To showthese inequalities we need to estimate E [ Z γ ] using again the product formula for multiplestochastic integrals. We omit the details of this proof. References [1] S. Bai and M. Taqqu: Structure of the third moment of the generalized Rosenblatt distribution.
Stoch.Proc. Appl. no. 4, 1710-1739, 2014.[2] S. Bai and M. Taqqu: Behavior of the generalized Rosenblatt process at extreme critical exponentvalues.
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Ann.Probab. no 1, 1-41, 2016.[5] I. Nourdin and G. Peccati. Normal Approximations with Malliavin Calculus. From Stein’s Method toUniversality . Cambridge University Press. 2012.[6] I. Nourdin and G. Peccati: The optimal fourth moment theorem.
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Ann.Probab. , no. 1, 177-193, 2005.[9] M. Maejima and C. A. Tudor: Selfsimilar processes with stationary increments in the second Wienerchaos. Probability and Mathematical Statistics , no. 1, 167-186, 2012.[10] A. V. Skorohod: On a generalization of a stochastic integral. Theory Probab. Appl. , 219-233, 1975.[11] M. Taqqu: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab.Theory Rel. Fields no. 4, 287-302, 1975. David Nualart: Department of Mathematics, University of Kansas, Lawrence, KS 66045,USA
E-mail address : [email protected] Denis Bell: Department of Mathematics, University of North Florida, Jacksonville, FL32224, USA
E-mail address : [email protected]@unf.edu