On rate of convergence in non-central limit theorems
Vo Anh, Nikolai Leonenko, Andriy Olenko, Volodymyr Vaskovych
aa r X i v : . [ m a t h . P R ] M a r ON RATE OF CONVERGENCE IN NON-CENTRAL LIMITTHEOREMS
By Vo Anh , Nikolai Leonenko , AndriyOlenko and Volodymyr Vaskovych Queensland University of Technology, Cardiff Universityand La Trobe University
The main result of this paper is the rate of convergence to Hermite-type distributions in non-central limit theorems. To the best of ourknowledge, this is the first result in the literature on rates of conver-gence of functionals of random fields to Hermite-type distributionswith ranks greater than 2. The results were obtained under rathergeneral assumptions on the spectral densities of random fields. Theseassumptions are even weaker than in the known convergence resultsfor the case of Rosenblatt distributions. Additionally, L´evy concen-tration functions for Hermite-type distributions were investigated.
1. Introduction.
This research will focus on the rate of convergenceof local functionals of real-valued homogeneous random fields with long-range dependence. Non-linear integral functionals on bounded sets of R d arestudied. These functionals are important statistical tools in various fields ofapplication, for example, image analysis, cosmology, finance, and geology.It was shown in [10], [34] and [35] that these functionals can produce non-Gaussian limits and require normalizing coefficients different from those incentral limit theorems.Since many modern statistical models are now designed to deal with non-Gaussian data, non-central limit theory is gaining more and more popularity.Some novel results using different models and asymptotic distributions wereobtained during the past few years, see [1], [6], [22], [30], [34] and referencestherein. Despite such development of the asymptotic theory, only a few of Supported in part under Australian Research Council’s Discovery Projects fundingscheme (project number DP160101366) Supported in part by project MTM2012-32674 (co-funded with FEDER) of the DGI,MINECO, and under Cardiff Incoming Visiting Fellowship Scheme and International Col-laboration Seedcorn Fund Supported in part by the La Trobe University DRP Grant in Mathematical and Com-puting Sciences
MSC 2010 subject classifications:
Primary 60G60; secondary 60F05, 60G12
Keywords and phrases:
Rate of convergence, Non-central limit theorems, Random field,Long-range dependence, Hermite-type distribution ANH, LEONENKO, OLENKO AND VASKOVYCH the studies obtained the rate of convergence, especially in the non-centralcase.There are two popular approaches to investigate the rate of convergencein the literature: the direct probability approach [1], [17], and the Stein-Malliavin method introduced in [25].As the name suggests, the Stein-Malliavin method combines Malliavincalculus and Stein’s method. The main strength of this approach is thatit does not use any restrictions on the moments of order higher than four(see, for example, [25]) and even three in some cases (see [23]). For a moredetailed description of the method, the reader is referred to [25]. At thismoment, the Stein-Malliavin approach is well developed for stochastic pro-cesses. However, many problems concerning non-central limit theorems forrandom fields remain unsolved. The full list of the already solved problemscan be found in [37].One of the first papers which obtained the rate of convergence in thecentral limit theorem using the Stein-Malliavin approach was [25]. The caseof stochastic processes was considered. Further refinement of these resultscan be found in [26], where optimal Berry-Esseen bounds for the normalapproximation of functionals of Gaussian fields are shown. However, it isknown that numerous functionals do not converge to the Gaussian distri-bution. The conditions to obtain the Gaussian asymptotics can be found inso-called Breuer-Major theorems, see [2] and [11]. These results are based onthe method of cumulants and diagram formulae. Using the Stein-Malliavinapproach, [27] derived a version of a quantitative Breuer-Major theoremthat contains a stronger version of the results in [2] and [11]. The rate ofconvergence for Wasserstein topology was found and an upper bound for theKolmogorov distance was given as a relationship between the Kolmogorovand Wasserstein distances. In [16] the authors directly derived the upper-bound for the Kolmogorov distance in the same quantitative Breuer-Majortheorem as in [27] and showed that this bound is better than the knownbounds in the literature, since it converges to zero faster. The results de-scribed above are the most general results currently known concerning therate of convergence in the central limit theorem using the Stein-Malliavinapproach.Related to [27] is the work [32] where, using the same arguments, theauthor found the rate of convergence for the central limit theorem of sojourntimes of Gaussian fields. Similar results for the Kolmogorov distance wereobtained in [16].Concerning non-central limit theorems, only partial results have beenfound. It is known from [8],[11] and [34] that, depending on the value of the
ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION Hurst parameter, functionals of fractional Brownian motion can convergeeither to the standard Gaussian distribution or a Hermite-type distribution.This idea was used in [6] and [7] to obtain the first rates of convergencein non-central limit theorems using the Stein-Malliavin method. Similar tothe case of central limit theorems, these results were obtained for stochasticprocesses. In [7] fractional Brownian motion was considered, and rates ofconvergence for both Gaussian and Hermite-type asymptotic distributionswere given. Furthermore all the results of [7] were refined in [6] for the case ofthe fractional Brownian sheet as an initial random element. It makes [7] theonly known work that uses the Stein-Malliavin method to provide the rateof convergence of some local functionals of random fields with long-rangedependence.Separately stands [3]. This work followed a new approach based onlyon Stein’s method without Malliavin calculus. The authors worked withWasserstein-2 metrics and showed the rate of convergence of quadratic func-tionals of i.i.d. Gaussian variables. It is one of the convergence results whichcan’t be obtained using the regular Stein-Malliavin method [3]. However,we are not aware of extensions of these results to the multi-dimensional andnon-Gaussian cases.The classical probability approach employs direct probability methods tofind the rate of convergence. Its main advantage over the other methodsis that it directly uses the correlation functions and spectral densities ofthe involved random fields. Therefore, asymptotic results can be explicitlyobtained for wide classes of random fields using slowly varying functions.Using this approach, the first rate of convergence in the central limit the-orem for Gaussian fields was obtained in [17]. In the following years, someother results were obtained, but all of them studied the convergence to theGaussian distribution.As for convergence to non-Gaussian distributions, the only known resultusing the classical probability approach is [1]. For functionals of Hermiterank-2 polynomials of long-range dependent Gaussian fields, it investigatedthe rate of convergence in the Kolmogorov metric of these functionals tothe Rosenblatt-type distribution. In this paper, we generalize these resultsto some classes of Hermite-type distributions. It is worth mentioning thatour present results are obtained under more natural and much weaker as-sumptions on the spectral densities than those in [1]. These quite generalassumptions allow to consider various new asymptotic scenarios even for theRosenblatt-type case in [1].It’s also worth mentioning that in the known Stein-Malliavin results, therate of convergence was obtained only for a leading term or a fixed number
ANH, LEONENKO, OLENKO AND VASKOVYCH of chaoses in the Wiener chaos expansion. However, while other expansionterms in higher level Wiener chaoses do not change the asymptotic distri-bution, they can substantially contribute to the rate of convergence. Themethod proposed in this manuscript takes into account all terms in theWiener chaos expansion to derive rates of convergence.It is well known, see [8, 24, 33], that the probability distributions ofHermite-type random variables are absolutely continuous. In this paper weinvestigate some fine properties of these distributions required to derive ratesof convergence. Specifically, we discuss the cases of bounded probability den-sity functions of Hermite-type random variables. Using the method proposedin [28], we derive the anti-concentration inequality that can be applied toestimate the L´evy concentration function of Hermite-type random variables.The article is organized as follows. In Section 2 we recall some basicdefinitions and formulae of the spectral theory of random fields. The mainassumptions and auxiliary results are stated in Section 3. In Section 4 wediscuss some fine properties of Hermite-type distributions. Section 5 providesthe results concerning the rate of convergence. Discussions and conclusionsare presented in Section 6.
2. Notations.
In what follows |·| and k·k denote the Lebesgue measureand the Euclidean distance in R d , respectively. We use the symbols C and δ to denote constants which are not important for our exposition. Moreover,the same symbol may be used for different constants appearing in the sameproof.We consider a measurable mean-square continuous zero-mean homoge-neous isotropic real-valued random field η ( x ) , x ∈ R d , defined on a proba-bility space (Ω , F , P ) , with the covariance functionB( r ) := Cov ( η ( x ) , η ( y )) = Z ∞ Y d ( rz ) dΦ( z ) , x, y ∈ R d , where r := k x − y k , Φ( · ) is the isotropic spectral measure, the function Y d ( · )is defined by Y d ( z ) := 2 ( d − / Γ (cid:18) d (cid:19) J ( d − / ( z ) z (2 − d ) / , z ≥ ,J ( d − / ( · ) being the Bessel function of the first kind of order ( d − / . Definition . The random field η ( x ) , x ∈ R d , as defined above is saidto possess an absolutely continuous spectrum if there exists a function f ( · ) ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION such thatΦ( z ) = 2 π d/ Γ − ( d/ Z z u d − f ( u ) d u, z ≥ , u d − f ( u ) ∈ L ( R + ) . The function f ( · ) is called the isotropic spectral density function of thefield η ( x ) . In this case, the field η ( x ) with an absolutely continuous spectrumhas the isonormal spectral representation η ( x ) = Z R d e i ( λ,x ) p f ( k λ k ) W (d λ ) , where W ( · ) is the complex Gaussian white noise random measure on R d . Consider a Jordan-measurable bounded set ∆ ⊂ R d such that | ∆ | > r ) , r > , be the homotheticimage of the set ∆ , with the centre of homothety at the origin and thecoefficient r > , that is | ∆( r ) | = r d | ∆ | . Consider the uniform distribution on ∆( r ) with the probability densityfunction (pdf) r − d | ∆ | − χ ∆( r ) ( x ) , x ∈ R d , where χ A ( · ) is the indicator func-tion of a set A. Definition . Let U and V be two random vectors which are indepen-dent and uniformly distributed inside the set ∆( r ) . We denote by ψ ∆( r ) ( z ) ,z ≥ , the pdf of the distance k U − V k between U and V. Note that ψ ∆( r ) ( z ) = 0 if z > diam { ∆( r ) } . Using the above notations,one can obtain the representation Z ∆( r ) Z ∆( r ) Υ ( k x − y k ) d x d y = | ∆ | r d E Υ ( k U − V k )(2.1) = | ∆ | r d Z diam { ∆( r ) } Υ ( z ) ψ ∆( r ) ( z ) d z, where Υ ( · ) is an integrable Borel function. Remark . If ∆( r ) is the ball v ( r ) := { x ∈ R d : k x k < r } , then ψ v ( r ) ( z ) = d r − d z d − I − ( z/ r ) (cid:18) d + 12 , (cid:19) , ≤ z ≤ r, where I µ ( p, q ) := Γ( p + q )Γ( p ) Γ( q ) Z µ u p − (1 − u ) q − d u, µ ∈ (0 , , p > , q > , is the incomplete beta function, see [15]. ANH, LEONENKO, OLENKO AND VASKOVYCH
Remark . Let H k ( u ), k ≥ u ∈ R , be the Hermite polynomials,see [30]. If ( ξ , . . . , ξ p ) is a 2 p -dimensional zero-mean Gaussian vector with E ξ j ξ k = , if k = j,r j , if k = j + p and 1 ≤ j ≤ p, , otherwise,then E p Y j =1 H k j ( ξ j ) H m j ( ξ j + p ) = p Y j =1 δ m j k j k j ! r k j j . The Hermite polynomials form a complete orthogonal system in theHilbert space L ( R , φ ( w ) dw ) = (cid:26) G : Z R G ( w ) φ ( w ) d w < ∞ (cid:27) , φ ( w ) := 1 √ π e − w . An arbitrary function G ( w ) ∈ L ( R , φ ( w ) dw ) admits the mean-squareconvergent expansion(2.2) G ( w ) = ∞ X j =0 C j H j ( w ) j ! , C j := Z R G ( w ) H j ( w ) φ ( w ) d w. By Parseval’s identity(2.3) ∞ X j =0 C j j ! = Z R G ( w ) φ ( w ) d w. Definition . [34] Let G ( w ) ∈ L ( R , φ ( w ) dw ) and assume there existsan integer κ ∈ N such that C j = 0, for all 0 ≤ j ≤ κ − , but C κ = 0 . Then κ is called the Hermite rank of G ( · ) and is denoted by H rank G. Definition . [4] A measurable function L : (0 , ∞ ) → (0 , ∞ ) is said tobe slowly varying at infinity if for all t > , lim r →∞ L ( rt ) L ( r ) = 1 . By the representation theorem [4, Theorem 1.3.1], there exists
C > r ≥ C the function L ( · ) can be written in the form L ( r ) = exp (cid:18) ζ ( r ) + Z rC ζ ( u ) u d u (cid:19) , ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION where ζ ( · ) and ζ ( · ) are such measurable and bounded functions that ζ ( r ) → ζ ( r ) → C ( | C | < ∞ ) , when r → ∞ . If L ( · ) varies slowly, then r a L ( r ) → ∞ , r − a L ( r ) → a > r → ∞ , see Proposition 1.3.6 [4]. Definition . [4] A measurable function g : (0 , ∞ ) → (0 , ∞ ) is saidto be regularly varying at infinity, denoted g ( · ) ∈ R τ , if there exists τ suchthat, for all t > , it holds thatlim r →∞ g ( rt ) g ( r ) = t τ . Definition . [4] Let g : (0 , ∞ ) → (0 , ∞ ) be a measurable function and g ( x ) → x →
0. Then a slowly varying function L ( · ) is said to be slowlyvarying with remainder of type 2, or that it belongs to the class SR2, if ∀ x > L ( rx ) L ( r ) − ∼ k ( x ) g ( r ) , r → ∞ , for some function k ( · ).If there exists x such that k ( x ) = 0 and k ( xµ ) = k ( µ ) for all µ , then g ( · ) ∈ R τ for some τ ≤ k ( x ) = ch τ ( x ), where(2.4) h τ ( x ) = ( ln( x ) , if τ = 0 , x τ − τ , if τ = 0 .
3. Assumptions and auxiliary results.
In this section, we list themain assumptions and some auxiliary results from [20] which will be usedto obtain the rate of convergence in non-central limit theorems.
Assumption . Let η ( x ) , x ∈ R d , be a homogeneous isotropic Gaussianrandom field with E η ( x ) = 0 and a covariance function B ( x ) such that B (0) = 1 , B ( x ) = E η (0) η ( x ) = k x k − α L ( k x k ) , where L ( k·k ) is a function slowly varying at infinity. In this paper we restrict our consideration to α ∈ (0 , d/κ ) , where κ isthe Hermite rank in Definition 3. For such α the covariance function B ( x )satisfying Assumption 1 is not integrable, which corresponds to the case oflong-range dependence. ANH, LEONENKO, OLENKO AND VASKOVYCH
Let us denote K r := Z ∆( r ) G ( η ( x )) d x and K r,κ := C κ κ ! Z ∆( r ) H κ ( η ( x )) d x, where C κ is defined by (2.2). Theorem . [20] Suppose that η ( x ) , x ∈ R d , satisfies Assumption and H rank G = κ ∈ N . If at least one of the following random variables K r √ Var K r , K r p Var K r,κ and K r,κ p Var K r,κ , has a limit distribution, then the limit distributions of the other randomvariables also exist and they coincide when r → ∞ . Assumption . The random field η ( x ) , x ∈ R d , has the spectral density f ( k λ k ) = c ( d, α ) k λ k α − d L (cid:18) k λ k (cid:19) , where c ( d, α ) := Γ (cid:0) d − α (cid:1) α π d/ Γ (cid:0) α (cid:1) , and L ( k·k ) is a locally bounded function which is slowly varying at infinityand satisfies for sufficiently large r the condition (3.1) (cid:12)(cid:12)(cid:12)(cid:12) − L ( tr ) L ( r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C g ( r ) h τ ( t ) , t ≥ , where g ( · ) ∈ R τ , τ ≤ , such that g ( x ) → , x → ∞ , and h τ ( t ) is definedby (2.4). Remark . In applied statistical analysis of long-range dependent mod-els researchers often assume an equivalence of Assumptions 1 and 2. How-ever, this claim is not true in general, see [12, 19]. This is the main reasonof using both assumptions to formulate the most general result in Theo-rem 5. However, in various specific cases just one of the assumptions maybe sufficient. For example, if f ( · ) is decreasing in a neighbourhood of zeroand continuous for all λ = 0 , then by Tauberian Theorem 4 [19] both as-sumptions are simultaneously satisfied. A detailed discussion of relationsbetween Assumption 1 and 2 and various examples can be found in [19, 29].Some important models used in spatial data analysis and geostatistics that ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION simultaneously satisfy Assumptions 1 and 2 are Cauchy and Linnik’s fields,see [1]. Their covariance functions are of the form B ( x ) = (1 + k x k σ ) − θ ,σ ∈ (0 , , θ > . Exact expressions for their spectral densities in the formrequired by Assumption 2 are provided in Section 5 [1].The remarks below clarify condition (3.1) and compare it with the as-sumptions used in [1].
Remark . This assumption implies weaker restrictions on the spectraldensity than the ones used in [1]. Slowly varying functions in Assumption 2can tend to infinity or zero. This is an improvement over [1] where slowlyvarying functions were assumed to converge to a constant. For example, afunction that satisfies this assumption, but would not fit that of [1], is ln( · ). Remark . If we consider the equivalence in Definition 6 in the uniformsense, then all the functions in the class SR2 satisfy condition (3.1). If weconsider this equivalence in the non-uniform sense, then there are functionsfrom SR2 that do not satisfy (3.1). An example of such functions is ln ( · ). Remark . By Corollary 3.12.3 [4] for τ = 0 the slowly varying function L ( · ) in Assumption 2 can be represented as L ( x ) = C (cid:0) cτ − g ( x ) + o ( g ( x )) (cid:1) . As we can see L ( · ) converges to some constant as x goes to infinity. Thismakes the case τ = 0 particularly interesting as this is the only case when aslowly varying function with remainder can tend to infinity or zero. Lemma . If L satisfies (3.1) , then for any k ∈ N , δ > , and sufficientlylarge r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L k/ ( tr ) L k/ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C g ( r ) h τ ( t ) t δ , t ≥ . Proof.
Applying the mean value theorem to the function f ( u ) = u n ,n ∈ R , on A = [min(1 , u ) , max(1 , u )] we obtain the inequality1 − x n = nθ n − (1 − x ) ≤ n (1 − x ) max(1 , x n − ) , θ ∈ A . Now, using this inequality for x = L ( tr ) L ( r ) and n = k/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L k/ ( tr ) L k/ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n (cid:12)(cid:12)(cid:12)(cid:12) − L ( tr ) L ( r ) (cid:12)(cid:12)(cid:12)(cid:12) max , (cid:18) L ( tr ) L ( r ) (cid:19) k − ! . ANH, LEONENKO, OLENKO AND VASKOVYCH
By Theorem 1.5.6 [4] we know there exists c > δ > L ( tr ) L ( r ) ≤ C · t δ , t ≥ . Applying this result and condition (3.1) to (3.2) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L k/ ( tr ) L k/ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C g ( r ) h τ ( t ) max (cid:16) , t δ ( k − ) (cid:17) ≤ C g ( r ) h τ ( t ) t δ , t ≥ . Let us denote the Fourier transform of the indicator function of the set∆ by K ∆ ( x ) := Z ∆ e i ( x,u ) d u, x ∈ R d . Lemma . [20] If t , ..., t κ , κ ≥ , are positive constants such that itholds P κi =1 t i < d, then Z R dκ | K ∆ ( λ + · · · + λ κ ) | d λ . . . d λ κ k λ k d − t · · · k λ κ k d − t κ < ∞ . Theorem . [20] Let η ( x ) , x ∈ R d , be a homogeneous isotropic Gaussianrandom field with E η ( x ) = 0 . If Assumptions and hold, then for r → ∞ the finite-dimensional distributions of X r,κ := r ( κα ) / − d L − κ/ ( r ) Z ∆( r ) H κ ( η ( x )) d x converge weakly to the finite-dimensional distributions of X κ (∆) := c κ/ ( d, α ) Z ′ R dκ K ∆ ( λ + · · · + λ κ )(3.3) × W (d λ ) . . . W (d λ κ ) k λ k ( d − α ) / · · · k λ κ k ( d − α ) / , where R ′ R dκ denotes the multiple Wiener-Itˆo integral. Remark . If κ = 1 the limit X κ (∆) is Gaussian. However, for the case κ > X κ (∆) are almost unknown. It was shownthat the integrals in (3.3) posses absolutely continuous densities, see [8, 33]. ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION The article [1] proved that these densities are bounded if κ = 2 . Also, forthe Rosenblatt distribution, i.e. κ = 2 and a rectangular ∆, the densityand cumulative distribution functions of X κ (∆) were studied in [36]. Anapproach to investigate the boundedness of densities of multiple Wiener-Itˆointegrals was suggested in [8]. However, it is difficult to apply this approachto the case κ > n th degree forms. Definition . Let Y and Y be arbitrary random variables. The uni-form (Kolmogorov) metric for the distributions of Y and Y is defined bythe formula ρ ( Y , Y ) = sup z ∈ R | P ( Y ≤ z ) − P ( Y ≤ z ) | . The following result follows from Lemma 1.8 [31].
Lemma . If X, Y and Z are arbitrary random variables, then for any ε > ρ ( X + Y, Z ) ≤ ρ ( X, Z ) + ρ ( Z + ε, Z ) + P ( | Y | ≥ ε ) .
4. L´evy concentration functions for X k (∆). In this section, wewill investigate some fine properties of probability distributions of Hermite-type random variables. These results will be used to derive upper boundsof ρ ( X κ (∆) + ε, X κ (∆)) in the next section. The following function fromSection 1.5 [31] will be used in this section. Definition . The L´evy concentration function of a random variable X is defined by Q ( X, ε ) := sup z ∈ R P( z < X ≤ z + ε ) , ε ≥ . We will discuss three important cases, and show how to estimate the L´evyconcentration function in each of them.If X k (∆) has a bounded probability density function p X κ (∆) ( · ) , then itholds(4.1) Q ( X κ (∆) , ε ) ≤ ε sup z ∈ R p X κ (∆) ( z ) ≤ ε C. This inequality is probably the sharpest known estimator of the L´evyconcentration function of X k (∆). It is discussed in cases 1 and 2. ANH, LEONENKO, OLENKO AND VASKOVYCH
Case 1.
If the Hermite rank of G ( · ) is equal to κ = 2 we are dealingwith the so-called Rosenblatt-type random variable. It is known that theprobability density function of this variable is bounded, consult [1, 8, 9, 18,21] for proofs by different methods. Thus, one can use estimate (4.1). Case 2.
Some interesting results about boundedness of probability den-sity functions of Hermite-type random variables were obtained in [14] byMalliavin calculus. To present these results we provide some definitions fromMalliavin calculus.Let X = { X ( h ) , h ∈ L ( R d ) } be an isonormal Gaussian process defined ona complete probability space (Ω , F , P ). Let S denote the class of smooth ran-dom variables of the form F = f ( X ( h ) , . . . X ( h n )), n ∈ N , where h , . . . , h n are in L ( R d ), and f is a function, such that f itself and all its partialderivatives have at most polynomial growth.The Malliavin derivative DF of F = f ( X ( h ) , . . . X ( h n )) is the L ( R d )valued random variable given by DF = n X i =1 ∂f∂x i ( X ( h ) , . . . X ( h n )) h i . The derivative operator D is a closable operator on L (Ω) taking values in L (Ω; L ( R d )). By iteration one can define higher order derivatives D k F ∈ L (Ω; L ( R d ) ⊙ k ), where ⊙ denotes the symmetric tensor product. For anyinteger k ≥ p ≥ D k,p the closure of S with respectto the norm k · k k,p given by k F k pk,p = k X i =0 E (cid:16)(cid:13)(cid:13) D i F (cid:13)(cid:13) pL ( R d ) ⊗ i (cid:17) . Let’s denote by δ the adjoint operator of D from a domain in L (Ω; L ( R d ))to L (Ω). An element u ∈ L (Ω; L ( R d )) belongs to the domain of δ if andonly if for any F ∈ D , it holds E [ h DF, u i ] ≤ c u p E [ F ] , where c u is a constant depending only on u .The following theorem gives sufficient conditions to guarantee bounded-ness of Hermite-type densities. Theorem . [14] Let F ∈ D ,s such that E [ | F | q ] < ∞ and (4.2) E h k DF k − rL ( R d ) i < ∞ , ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION where q, r, s > satisfying q + r + s = 1 .Denote w = k DF k L ( R d ) and u = w − DF . Then u ∈ D ,q ′ with q ′ = qq − and F has a density given by p F ( x ) = E [ F >x δ ( u )] . Furthermore, p F ( x ) is bounded and p F ( x ) ≤ C q k w − k r k F k ,s min(1 , | x − k F k q ) , for any x ∈ R ,where C q is a constant depending only on q . Note, that the Hermite-type random variable X κ (∆) does belong to thespace D ,s , s >
1, and E [ | X κ (∆) | q ] < ∞ by the hypercontractivity property,see (2.11) in [14]. Thus, if the condition (4.2) holds, one can use (4.1). Case 3.
When there is no information about boundedness of the proba-bility density function, anti-concentration inequalities can be used to obtainestimates of the L´evy concentration function.Let us denote by I κ ( · ) a multiple Wiener-Itˆo stochastic integral of or-der dκ , i.e. I κ ( f ) = R ′ R dκ f ( λ , · · · , λ κ ) W (d λ ) . . . W (d λ κ ) , where f ( · ) ∈ L s ( R dκ ). Here L s ( R dκ ) denotes the space of symmetrical functions in L ( R dκ ).Note, that any F ∈ L (Ω) can be represented as F = E ( F ) + ∞ P q =1 I q ( f q ),where the functions f q are determined by F . The multiple Wiener-Itˆo inte-gral I q ( f q ) coincides with the orthogonal projection of F on the q -th Wienerchaos associated with X .The following lemma uses the approach suggested in [28]. Lemma . For any κ ∈ N , t ∈ R , and ˆ ε > it holds P ( | X κ (∆) − t | ≤ ˆ ε ) ≤ c κ ˆ ε /κ (cid:16) C k ˆ K ∆ k L ( R dκ ) + t (cid:17) /κ , where ˆ K ∆ ( x , . . . , x κ ) := K ∆ ( x + ··· + x κ ) k λ k ( d − α ) / ···k λ κ k ( d − α ) / and c κ is a constant thatdepends on κ . Proof.
Let { e i } i ∈ N be an orthogonal basis of L ( R d ). Then, ˆ K ∆ ∈ L ( R dκ ) can be represented asˆ K ∆ = X ( i ,...,i κ ) ∈ N κ c i ,...,i κ e i ⊗ · · · ⊗ e i κ . For each n ∈ N , setˆ K n ∆ = X ( i ,...,i κ ) ∈{ ,...,n } κ c i ,...,i κ e i ⊗ · · · ⊗ e i κ . ANH, LEONENKO, OLENKO AND VASKOVYCH
Note, that both ˆ K ∆ and ˆ K n ∆ belong to the space L s ( R dκ ).By (3.3) it follows that X κ (∆) = c κ/ ( d, α ) I κ ( ˆ K ∆ ). Let us denote X nκ (∆) := c κ/ ( d, α ) I κ ( ˆ K n ∆ ).As n → ∞ , ˆ K n ∆ → ˆ K ∆ in L ( R dκ ). Thus, X nκ (∆) → X κ (∆) in L (Ω , F , P ).Hence, there exists a strictly increasing sequence n j for which X n j κ (∆) → X κ (∆) almost surely as j → ∞ .It also follows that X nκ (∆) = c κ/ ( d, α ) I κ X ( i ,...,i κ ) ∈{ ,...,n } κ c i ,...,i κ e i ⊗ · · · ⊗ e i κ = c κ/ ( d, α ) κ X m =1 n X ≤ i ′ < ··· (cid:18) | X nκ (∆) − t | ≤ ˆ ε (cid:16) E ( X nκ (∆) − t ) (cid:17) (cid:19) ≤ ˆ c κ ˆ ε /κ . Analogously to [28], using Fatou’s lemma we getP (cid:18) | X κ (∆) − t | ≤ ˆ ε (cid:16) E ( X κ (∆) − t ) (cid:17) (cid:19) ≤ ˆ c κ /κ ˆ ε /κ = c κ ˆ ε /κ . ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION It is known, see (1.3) and (1.5) in [13], that E X κ (∆) = 0 and E ( X κ (∆)) = C k ˆ K ∆ k L ( R dκ ) . Thus, the above inequality can be rewritten asP ( | X κ (∆) − t | ≤ ˆ ε ) ≤ c κ ˆ ε /κ (cid:16) E ( X κ (∆) − t ) (cid:17) κ = c κ ˆ ε /κ (cid:16) C k ˆ K ∆ k L ( R dκ ) + t (cid:17) /κ . The following theorem combines all three cases above and provides anupper-bound estimator of the L´evy concentration function.
Theorem . For any κ ∈ N and an arbitrary positive ε it holds Q ( X κ (∆) , ε ) ≤ Cε a , where the constant a depends on the cases discussed above. Proof.
For cases 1 and 2 it is an immediate corollary of (4.1) and theboundedness of p X κ (∆) ( · ).For case 3, applying Lemma 4 with t = z + ε and ˆ ε = ε we get Q ( X κ (∆) , ε ) = sup z ∈ R P (cid:16)(cid:12)(cid:12)(cid:12) X κ (∆) − ( z + ε (cid:12)(cid:12)(cid:12) ≤ ε (cid:17) ≤ sup z ∈ R c κ (cid:0) ε (cid:1) /κ (cid:16) C k ˆ K ∆ k L ( R dκ ) + (cid:0) z + ε (cid:1) (cid:17) κ ≤ c κ ε /κ (cid:16) C k ˆ K ∆ k L ( R dκ ) (cid:17) κ = Cε /κ . Remark . Notice, that by Definitions 7 and 8 Q ( X κ (∆) , ε ) = sup z ∈ R (P( X κ (∆) ≤ z + ε ) − P( X κ (∆) ≤ z ))= sup z ∈ R | P( X κ (∆) ≤ z ) − P( X κ (∆) + ε ≤ z ) | = ρ ( X κ (∆) + ε, X κ (∆)) .
5. Rate of convergence.
In this section we consider the case of Hermi-te-type limit distributions in Theorem 2. The main result describes the rateof convergence of K r to X κ (∆) when r → ∞ . To prove it we use sometechniques and facts from [5, 20, 18]. ANH, LEONENKO, OLENKO AND VASKOVYCH
Theorem . Let Assumptions and hold and H rank G = κ ∈ N .If τ ∈ (cid:0) − d − κα , (cid:1) then for any κ < a a min (cid:16) α ( d − κα ) d − ( κ − α , κ (cid:17) ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) = o ( r − κ ) , r → ∞ , where a is a constant from Theorem , C κ is defined by (2.2) , and κ := min − τ, d − α + · · · + d − κα + d +1 − κα ! . If τ = 0 then ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) = g ( r ) , r → ∞ . Remark . This theorem generalises the result for the Rosenblatt-typecase ( κ = 2) in [1] to Hermite-type asymptotics ( κ > Proof.
Since H rank G = κ, it follows that K r can be represented in thespace of squared-integrable random variables L (Ω) as K r = K r,κ + S r := C κ κ ! Z ∆( r ) H κ ( η ( x )) d x + X j ≥ κ +1 C j j ! Z ∆( r ) H j ( η ( x )) d x, where C j are coefficients of the Hermite series (2.2) of the function G ( · ) . Notice that E K r,κ = E S r = E X κ (∆) = 0 , and X r,κ = κ ! K r,κ C κ r d − κα L κ ( r ) . It follows from Assumption 1 that | L ( u ) /u α | = | B ( u ) | ≤ B (0) = 1 . Thus,by the proof of Theorem 4 [20],
Var S r ≤ | ∆ | r d − ( κ +1) α X j ≥ κ +1 C j j ! Z diam { ∆ } z − ( κ +1) α L κ +1 ( rz ) ψ ∆ ( z ) dz ≤ | ∆ | r d − κα L κ ( r ) X j ≥ κ +1 C j j ! Z diam { ∆ } z − κα L κ ( rz ) L κ ( r ) L ( rz )( rz ) α ψ ∆ ( z ) d z. ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION We represent the integral in (5) as the sum of two integrals I and I withthe ranges of integration [0 , r − β ] and ( r − β , diam { ∆ } ] respectively, where β ∈ (0 , . It follows from Assumption 1 that | L ( u ) /u α | = | B ( u ) | ≤ B (0) = 1 andwe can estimate the first integral as I ≤ Z r − β z − κα L κ ( rz ) L κ ( r ) ψ ∆ ( z ) d z ≤ sup ≤ s ≤ r s δ/κ L ( s ) r δ/κ L ( r ) ! κ × Z r − β z − δ z − κα ψ ∆ ( z ) d z, where δ is an arbitrary number in (0 , min( α, d − κα )) . By Assumption 1 the function L ( · ) is locally bounded. By Theorem 1.5.3in [4], there exists r > C > r ≥ r sup ≤ s ≤ r s δ/ L ( s ) r δ/ L ( r ) ≤ C. Using (2.1) we obtain Z r − β z − δ z − κα ψ ∆ ( z ) d z ≤ C | ∆ | Z r − β τ d − κα − − δ d τ = C r − β ( d − κα − δ ) ( d − κα − δ ) | ∆ | . Applying Theorem 1.5.3 [4] we get I ≤ sup r − β ≤ s ≤ r · diam { ∆ } s δ L κ ( s ) r δ L κ ( r ) · sup r − β ≤ s ≤ r · diam { ∆ } L ( s ) s α Z diam { ∆ } z − ( δ + κα ) ψ ∆ ( z ) d z ≤ C · o ( r − ( α − δ )(1 − β ) ) , when r is sufficiently large.Notice that by (2.3) X j ≥ κ +1 C j j ! ≤ Z R G ( w ) φ ( w ) d w < + ∞ . Hence, for sufficiently large r Var S r ≤ C r d − κα L κ ( r ) (cid:16) r − β ( d − κα − δ ) + o (cid:16) r − ( α − δ )(1 − β ) (cid:17)(cid:17) . ANH, LEONENKO, OLENKO AND VASKOVYCH
Choosing β = αd − ( κ − α to minimize the upper bound we get Var S r ≤ Cr d − κα L κ ( r ) r − α ( d − κα ) d − ( κ − α + δ . It follows from Theorem 4 that ρ ( X κ (∆) + ε, X κ (∆)) ≤ Cε a . Applying Chebyshev’s inequality and Lemma 3 to X = X r,κ , Y = κ ! S r C κ r d − κα L κ ( r ) , and Z = X κ (∆) , we get ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) = ρ (cid:18) X r,κ + κ ! S r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) ≤ ρ ( X r,κ , X κ (∆)) + C (cid:18) ε a + ε − r − α ( d − κα ) d − ( κ − α + δ (cid:19) , for a sufficiently large r. Choosing ε := r − α ( d − κα )(2+ a )( d − ( κ − α ) to minimize the second term we obtain(5.1) ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) ≤ ρ ( X r,κ , X κ (∆)) + C r − aα ( d − κα )(2+ a )( d − ( κ − α ) + δ . Applying Lemma 3 once again to X = X κ (∆) , Y = X r,κ − X κ (∆) , and Z = X κ (∆) we obtain ρ ( X r,κ , X κ (∆)) ≤ ε a C + P {| X r,κ − X κ (∆) | ≥ ε }≤ ε a C + ε − Var ( X r,κ − X κ (∆)) . (5.2)Now we show how to estimate Var ( X r,κ − X κ (∆)) . By the self-similarity of Gaussian white noise and formula (2.1) [10] X r,κ D = c κ ( d, α ) Z ′ R κd K ∆ ( λ + · · · + λ κ ) Q r ( λ , . . . , λ κ ) × W (d λ ) . . . W (d λ κ ) k λ k ( d − κα ) / . . . k λ κ k ( d − κα ) / , where Q r ( λ , . . . , λ κ ) := r κ ( α − d ) L − κ ( r ) c − κ ( d, α ) " κ Y i =1 k λ i k d − α f (cid:18) k λ i k r (cid:19) / . ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION Notice that X κ (∆) = c κ ( d, α ) Z ′ R κd K ∆ ( λ + · · · + λ κ ) W ( dλ ) . . . W ( dλ κ ) k λ k ( d − α ) / . . . k λ κ k ( d − α ) / . By the isometry property of multiple stochastic integrals R r := E | X r,κ − X κ (∆) | c κ ( d, α )= Z R κd | K ∆ ( λ + · · · + λ κ ) | ( Q r ( λ , . . . , λ κ ) − k λ k d − α . . . k λ κ k d − α d λ . . . d λ κ . Let us rewrite the integral R r as the sum of two integrals I and I withthe integration regions A ( r ) := { ( λ , . . . , λ κ ) ∈ R κd : max i =1 ,κ ( || λ i || ) ≤ r γ } and R κd \ A ( r ) respectively, where γ ∈ (0 , . Our intention is to use the monotoneequivalence property of regularly varying functions in the regions A ( r ) . First we consider the case of ( λ , . . . λ κ ) ∈ A ( r ) . By Assumption 2 andthe inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)vuut κ Y i =1 x i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ X i =1 (cid:12)(cid:12)(cid:12) x κ i − (cid:12)(cid:12)(cid:12) we obtain | Q r ( λ , . . . , λ ) − | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)vuuut κ Y j =1 L (cid:16) r k λ j k (cid:17) L ( r ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ κ X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L κ (cid:16) r k λ j k (cid:17) L κ ( r ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By Lemma 1, if || λ j || ∈ (1 , r γ ) , j = 1 , κ, then for arbitrary δ > r we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L κ (cid:16) r k λ j k (cid:17) L κ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = L κ (cid:16) r k λ j k (cid:17) L κ ( r ) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L κ ( r ) L κ (cid:16) r k λ j k (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C L κ (cid:16) r k λ j k (cid:17) L κ ( r ) g (cid:18) r k λ j k (cid:19) × k λ j k δ h τ ( k λ j k ) = C k λ j k δ h τ ( k λ j k ) g ( r ) g (cid:16) r k λ j k (cid:17) g ( r ) L (cid:16) r k λ j k (cid:17) L ( r ) κ . ANH, LEONENKO, OLENKO AND VASKOVYCH
For any positive β and β , applying Theorem 1.5.6 [4] to g ( · ) and L ( · )and using the fact that h τ (cid:0) t (cid:1) = − t τ h ( t ) we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L κ (cid:16) r k λ j k (cid:17) L κ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k λ j k δ + κβ + β k λ j k − τ h τ ( k λ j k ) g ( r )(5.3) = C k λ j k δ h τ (cid:18) k λ j k (cid:19) g ( r ) . By Lemma 1 for || λ j || ≤ , j = 1 , κ , and arbitrary δ > , we obtain(5.4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − L κ (cid:16) r k λ j k (cid:17) L κ ( r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k λ j k − δ h τ (cid:18) k λ j k (cid:19) g ( r ) . Hence, by (5.3) and (5.4) | Q r ( λ , . . . λ κ ) − | ≤ k κ X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L κ (cid:16) r k λ j k (cid:17) L κ ( r ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C κ X j =1 h τ (cid:18) k λ j k (cid:19) g ( r ) max (cid:16) k λ j k − δ , k λ j k δ (cid:17) , for ( λ , . . . λ κ ) ∈ A ( r ).Notice, that in the case τ = 0 for any δ > C > h ( x ) = ln( x ) < Cx δ , x ≥
1, and h ( x ) = ln( x ) < Cx − δ , x <
1. Hence, byLemma 2 for − τ ≤ d − κα we get Z A ( r ) ∩ [0 , κd h τ (cid:16) k λ j k (cid:17) max (cid:16) k λ j k − δ , k λ j k δ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) K ∆ (cid:18) κ P i =1 λ i (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) d λ . . . d λ κ k λ k d − α . . . k λ κ k d − α < ∞ . Therefore, we obtain for sufficiently large rI ≤ C g ( r ) κ X j =1 Z A ( r ) ∩ R κd h τ (cid:16) k λ j k (cid:17) · max (cid:16) k λ j k − δ , k λ j k δ (cid:17) k λ k d − α . . . k λ κ k d − α ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION ×| K ∆ ( λ + . . . λ κ ) | d λ . . . d λ κ ≤ C g ( r ) Z A ( r ) ∩ R κd h τ (cid:16) k λ k (cid:17) k λ k d − α . . . k λ κ k d − α (5.5) × max (cid:16) k λ k − δ , k λ k δ (cid:17) | K ∆ ( λ + . . . λ κ ) | d λ . . . d λ κ ≤ C g ( r ) . It follows from Assumption 2 and the specification of the estimate (23)in the proof of Theorem 5 [20] that for each positive δ there exists r > r ≥ r , ( λ , . . . , λ κ ) ∈ B (1 ,µ ,...,µ κ ) = { ( λ , . . . , λ κ ) ∈ R κd : || λ j || ≤ , if µ j = − , and || λ j || > , if µ j = 1 , j = 1 , k } , and µ j ∈ {− , } , it holds | K ∆ ( λ + · · · + λ κ ) | ( Q r ( λ , . . . λ κ ) − k λ k d − α . . . k λ κ k d − α ≤ C | K ∆ ( λ + · · · + λ κ ) | k λ k d − α . . . k λ κ k d − α + C | K ∆ ( λ + · · · + λ κ ) | k λ k d − α − δ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ . Since the integrands are non-negative, we can estimate I as it is shownbelow I ≤ κ Z R ( κ − d Z || λ || >r γ | K ∆ ( λ + · · · + λ κ ) | ( Q r ( λ , . . . , λ κ ) − d λ . . . d λ κ k λ k d − α . . . k λ κ k d − α ≤ C Z R ( κ − d Z || λ || >r γ | K ∆ ( λ + · · · + λ ) | d λ . . . d λ κ k λ k d − α . . . k λ κ k d − α + C X µi ∈{ , , − } i ∈ ,κ Z R ( κ − d Z || λ || >r γ | K ∆ ( λ + · · · + λ κ ) | d λ . . . d λ κ k λ k d − α − δ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ ≤ C max µi ∈{ , , − } i ∈ ,κ Z R ( κ − d Z || λ || >r γ | K ∆ ( λ + · · · + λ κ ) | (5.6) × d λ . . . d λ κ k λ k d − α − δ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ . Replacing λ + λ by u we obtain I ≤ C max µi ∈{ , , − } i ∈ ,κ Z R ( κ − d Z || λ || >r γ | K ∆ ( u + λ + · · · + λ κ ) | k λ k d − α − δ k u − λ k d − α − µ δ ANH, LEONENKO, OLENKO AND VASKOVYCH × d λ d u d λ . . . d λ κ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ ≤ C max µi ∈{ , , − } i ∈ ,κ Z R ( κ − d k u k d − α − ( µ +1) δ × | K ∆ ( u + λ + · · · + λ κ ) | k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ Z k λ k > rγ k u k d λ d u d λ . . . d λ κ k λ k d − α − δ (cid:13)(cid:13)(cid:13) u k u k − λ (cid:13)(cid:13)(cid:13) d − α − µ δ . Taking into account that for δ ∈ (0 , min( α, d/κ − α ))sup u ∈ R d \{ } Z R d d λ k λ k d − α − δ (cid:13)(cid:13)(cid:13) u k u k − λ (cid:13)(cid:13)(cid:13) d − α − µ δ ≤ C, we obtain I ≤ C max µi ∈{ , , − } i ∈ ,κ Z R ( κ − d max µ ∈{ , , − } Z || u ||≤ r γ | K ∆ ( u + λ + · · · + λ κ ) | k u k d − α − ( µ +1) δ × d λ . . . d λ κ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ Z || λ || >r γ − γ d λ d u k λ k d − α − δ (cid:13)(cid:13)(cid:13) u k u k − λ (cid:13)(cid:13)(cid:13) d − α − µ δ + max µ i ∈{ , , − } Z || u || >r γ | K ∆ ( u + λ + · · · + λ κ ) | d u d λ . . . d λ κ k u k d − α − ( µ +1) δ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ , where γ ∈ (0 , γ ) . By Lemma 2, there exists r > r ≥ r the first summandis bounded by C max µ ∈{ , , − } Z || u ||≤ r γ | K ∆ ( u + λ + · · · + λ κ ) | d u d λ . . . d λ κ k u k d − α − ( µ +1) δ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ × Z || λ || >r γ − γ d λ k λ k d − α − δ − µ δ ≤ Cr − ( γ − γ )( d − α − δ ) . Therefore, for sufficiently large r,I ≤ Cr − ( γ − γ )( d − α − δ ) + C max µi ∈{ , , − } i ∈ ,κ Z R ( κ − d Z || u || >r γ | K ∆ ( u + λ + · · · + λ κ ) | d u d λ . . . d λ κ k u k d − α − δ k λ k d − α − µ δ . . . k λ κ k d − α − µ κ δ . ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION Notice that the second summand here coincides with (5.6) if κ is replacedby κ −
1. Thus, we can repeat the above procedure κ − I ≤ Cr − ( γ − γ )( d − α − δ ) + · · · + Cr − ( γ κ − − γ κ − )( d − κα − κδ ) (5.7) + C Z k u k >r γκ − | K ∆ ( u ) | d u k u k d − κα − κδ , where γ > γ > γ > · · · > γ κ − . By the spherical L -average decay rate of the Fourier transform [5] for δ < d + 1 − κα and sufficiently large r we get the following estimate of theintegral in (5.7) Z k u k >r γκ − | K ∆ ( u ) | d u k u k d − κα − κδ ≤ C Z z>r γκ − Z S d − | K ∆ ( zω ) | z − κα − κδ d ω d z ≤ C Z z>r γκ − d zz d +2 − κα − κδ = C r − γ κ − ( d +1 − κα − κδ ) (5.8) = C r − ( γ κ − − γ κ − )( d +1 − κα − κδ ) , where S d − := { x ∈ R d : k x k = 1 } is a sphere of radius 1 in R d and γ κ − = 0 . Now let’s consider the case τ <
0. In this case by Theorem 1.5.6 [4] forany δ > g ( r ) as follows(5.9) g ( r ) ≤ C r τ + δ . Combining estimates (5.1), (5.2), (5.5), (5.7), (5.8),(5.9) and choosing ε := r − β , we obtain ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) ≤ C (cid:18) r − aα ( d − κα )(2+ a )( d − ( κ − α ) + δ + r − aβ + r τ +2 δ +2 β + r − ( γ − γ )( d − α − δ )+2 β + · · · + r − ( γ κ − − γ κ − )( d − κα − κδ )+2 β + r − ( γ κ − − γ κ − )( d +1 − κα − κδ )+2 β (cid:17) . Therefore, for any ˜ κ ∈ (0 , aa κ ) one can choose a sufficiently small δ > ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) ≤ Cr δ (cid:18) r − aα ( d − κα )(2+ a )( d − ( κ − α ) + r − a ˜ κ a (cid:19) , ANH, LEONENKO, OLENKO AND VASKOVYCH where κ := sup >γ>γ > ··· >γκ − β> min ( aβ, − τ − β, ( γ − γ )( d − α ) − β, . . . , ( γ κ − − γ κ − )( d − κα ) − β, ( γ κ − − γ κ − ) ( d + 1 − κα ) − β ) . Lemma . Let Γ = { γ = ( γ , . . . , γ n +1 ) | b = γ > γ > · · · > γ n +1 = 0 } and x = ( x , . . . , x n ) ∈ R n +1+ be some fixed vector.The function G ( γ ) = min i ( γ i − γ i +1 ) x i reaches its maximum at ¯ γ = (¯ γ , . . . , ¯ γ n +1 ) ∈ Γ such that for any ≤ i ≤ n it holds (5.11) ( ¯ γ i − ¯ γ i +1 ) x i = (¯ γ i +1 − ¯ γ i +2 ) x i +1 . Proof.
Let us show that any deviation of γ from ¯ γ leads to a smallerresult. Consider a vector ˆ γ such that for some i ∈ , n and some ε > γ i − ˆ γ i +1 = ¯ γ i − ¯ γ i +1 + ε. Since n P i =0 ˆ γ i − ˆ γ i +1 = ˆ γ − ˆ γ n +1 = b we can conclude that there exist some j = i, j ∈ , n, and ε > γ j − ˆ γ j +1 = ¯ γ j − ¯ γ j +1 − ε .Obviously, in this case G (ˆ γ ) ≤ ( ˆ γ j − ˆ γ j +1 ) x j = ( ¯ γ j − ¯ γ j +1 − ε ) x j = ( ¯ γ j − ¯ γ j +1 ) x j − ε x j Since ε > x j > G (ˆ γ ) ≤ ( ¯ γ j − ¯ γ j +1 ) x j − ε x j < ( ¯ γ j − ¯ γ j +1 ) x j = G (¯ γ ) . So it’s clearly seen that any deviation from ¯ γ will yield a smaller result.Note, that for fixed γ ∈ (0 ,
1) by Lemma 5sup γ>γ > ··· >γ κ − =0 min (( γ − γ )( d − α ) , . . . , ( γ κ − − γ κ − )( d − κα ) , ( γ κ − − γ κ − ) ( d + 1 − κα )) = γ d − α + · · · + d − κα + d +1 − κα and sup γ ∈ (0 , γ d − α + · · · + d − κα + d +1 − κα = 1 d − α + · · · + d − κα + d +1 − κα . ATE OF CONVERGENCE TO HERMITE-TYPE DISTRIBUTION Note that κ = sup β> min ( aβ, κ − β ) = a κ a . Finally, from (5.10) for ˜ κ < κ the first statement of the theorem follows.Now let’s consider the case τ = 0. In this case by Theorem 1.5.6 [4] forany s > r (5.12) g ( r ) > r − s . Combining estimates (5.1), (5.2), (5.5), (5.7), (5.8), replacing all powers of r for g ( r ) using (5.12), and choosing ε := g β ( r ) , β ∈ (0 ,
1) we obtain ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) ≤ C (cid:16) g ( r ) + g β ( r ) + g − β (cid:17) . Since sup β ∈ (0 , min(2 , β, − β ) = , it follows that ρ (cid:18) κ ! K r C κ r d − κα L κ ( r ) , X κ (∆) (cid:19) ≤ Cg ( r ) . This proves the second statement of the theorem.
Remark . The upper bound on the rate of convergence in Theorem 5is given by explicit formulae that are easy to evaluate and analyse. Forexample, for fixed values of α and κ it is simple to see that the upper boundfor κ approaches a a min ( α, − τ ) , when d → + ∞ . For fixed values of d and κ the upper bound for κ is of the order of magnitude of O ( d − κα ),when α → d/κ. This result is expected as the value α = d/κ correspondsto the boundary where a phase transition between short- and long-rangedependence occurs.
6. Conclusion.
The rate of convergence to Hermite-type limit distri-butions in non-central limit theorems was investigated. The results wereobtained under rather general assumptions on the spectral densities of theconsidered random fields, that weaken the assumptions used in [1]. Similarto [1], the direct probabilistic approach was used, which has, in our view, anindependent interest as an alternative to the methods in [6, 25, 26]. Addi-tionally, some fine properties of the probability distributions of Hermite-typerandom variables were investigated. Some special cases when their proba-bility density functions are bounded were discussed. New anti-concentrationinequalities were derived for L´evy concentration functions. ANH, LEONENKO, OLENKO AND VASKOVYCH
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Bernoulli https://sites.google.com/site/malliavinstein/home . Retrieved on16 February 2016. Vo AnhSchool of Mathematical SciencesQueensland University of TechnologyBrisbane, Queensland, 4001AustraliaE-mail: [email protected]
N. LeonenkoSchool of MathematicsCardiff UniversitySenghennydd Road, Cardiff CF24 4AGUnited KingdomE-mail:
LeonenkoN@cardiff.ac.uk ANH, LEONENKO, OLENKO AND VASKOVYCH