KOLMOGOROV DISTANCE FOR MULTIVARIATE NORMAL APPROXIMATION. Yoon Tae Kim and Hyun Suk Park

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1 Korean J. Math. 3 (015, No. 1, pp KOLMOGOROV DISTANCE FOR MULTIVARIATE NORMAL APPROXIMATION Yoon Tae Kim and Hyun Suk Park Abstract. This paper concerns the rate of convergence in the multidimensional normal approximation of functional of Gaussian fields. The aim of the present work is to derive explicit upper bounds of the Kolmogorov distance for the rate of convergence instead of Wasserstein distance studied by Nourdin et al. [Ann. Inst. H. Poincaré(B Probab.Statist. 46(1 ( ]. 1. Introduction Let Z be a standard Gaussian random variable on a probability space (Ω, F, P. Suppose that {F n } is a sequence of real-valued random variables of an infinite-dimensional Gaussian field. In the paper [6] and [7], authors combine Stein s method and Malliavin calculus to derive explicit upper bounds for quantities of the type (1 E[h(F n ] E[h(Z], Received October 14, 014. Revised January 14, 015. Accepted January 19, Mathematics Subject Classification: 60H07. Key words and phrases: Malliavin calculus, Kolmogorov distance, Stein s method, multidimensional normal approximation, Wasserstein distance, fractional Brownian motion. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF funded by the Ministry of Education, Science and Technology (01R1A1A4A and NRF-013R1A1A Corresponding author. c The Kangwon-Kyungki Mathematical Society, 015. This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License ( -nc/3.0/ which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.

2 Y.T. Kim and H.S. Park where h is a suitable test function. In the paper [9], authors extend the results of [6]and [7] to the multidimensional normal approximation of functional of Gaussian fields in Wasserstein distance. For a given function f : R R, the Stein equation associated with f is defined by ( f(x E[f(Z] = h (x xh(x for all x R. A solution to the equation ( is a function h such that h is Lebesguealmost everywhere differentiable and there exists a version of h satisfying (. If h Lip(1, where Lip(1 is the collection of all functions with Lipschitz constant bounded by 1, then the equation ( has a soluiton h such that h 1 and h. Recall that Wasserstein distance between the laws of two real-valued random variables X and Y is defined by d W (X, Y = sup E[h(X] E[h(Y ]. h Lip(1 In the paper [9], authors obtain explicit upper bounds of d W in the case when Z is a d-dimensional Gaussian vector, F = (F (1,..., F (d of smooth functionals of Gaussian fields, and d W is Wasserstein distance probability law on R d. In this paper, we consider the case when the test function h is nonsmooth such as the indicator functions of Borel-measurable convex sets. The test function of the Kolmogorov distance is such a class. This distance is defined by d Kol (X, Y = sup E[h(X] E[h(Y ]. {h=1 (,z] :z R d } For the proof of quantitative Breuer-Major theorems in [8], the upper bound of the Kolmogorov distance is obtained by using the relation (see Theorem 3.1 in [3] (3 d Kol (X, Y d W (X, Y. In this paper, by using the smoothing inequality, we directly derive an explicit upper bound of the Kolmogorov distance for a sequence {F n = (F n (1,..., F n (d, n 1} As an application, we find an explicit upper bound of the Kolmogorov distance in the Breuer-Major central limit theorem for fractional Brownian motion. (For the Wasserstein distance, see Theorem 4.1 in [9]. We stress that our upper bound is more

3 Kolmogorov distance for Multivariate normal approximation 3 efficient than the upper bound obtained by the relationship (3 as our upper bound converges to zero more fast.. Preliminaries In this section, we recall some basic facts about Malliavin calculus for Gaussian processes. The reader is referred to [10] for a more detailed explanation. Suppose that H is a real separable Hilbert space with scalar product denoted by <, > H. Let B = {B(h, h H} be an isonormal Gaussian process, that is a centered Gaussian family of random variables such that E[B(hB(g] = h, g H. Let S be the class of smooth and cylindrical random variables F of the form (4 F = f(b(ϕ 1,, B(ϕ n, where n 1, f Cb (Rn and ϕ i H, i = 1,, n. The Malliavin derivative of F with respect to B is the element of L (Ω, H defined by n f (5 DF = (B(ϕ 1,, B(ϕ n ϕ i, x i i=1 We denote by D l,p the closure of its associated smooth random variable class with respect to the norm l F p l,p = E( F p + E( D k F p. H k We denote by δ the adjoint of the operator D, also called the divergence operator. The domain of δ, denoted by Dom(δ, is an element u L (Ω; H such that k=1 E(< DF, u > H C(E F 1/ for all F D l,. If u Dom(δ, then δ(u is the element of L (Ω defined by the duality relationship E[F δ(u] = E[ DF, u H ] for every F D 1,. Let F L (Ω be a square integrable random variable. The operator L is defined through the projection operator J n, n = 0, 1,..., as L = n=0 nj nf, and is called the infinitesimal generator of the Ornstein- Uhlhenbeck semigroup. The relationship between the operator D, δ,

4 4 Y.T. Kim and H.S. Park and L is given as follows: δdf = LF, that is, for F L (Ω the statement F Dom(L is equivalent to F Dom(δD (i.e. F D 1, and DF Dom(δ, and in this case δdf = LF. We also define the operator L 1, which is the pseudo-inverse of L, as L 1 F = n=1 1 J n n(f. Note that L 1 is an operator with values in D, and LL 1 F = F E[F ] for all F L (Ω. 3. Main results In this section, we derive an explicit upper bound of the Kolmogorov distance for normal approximation. We begin by the following simple lemma. Lemma 3.1. Let ( 1 f(t = a log + b 1 e t for t > 0, 1 e t where a and b are positive constants such that a < b. Then the minimum with respect to t is attained for t = 1 ( a (1 log, b and inf f(t = a(log(b log(a + a, t>0 Proof. The solution t of the equation f (t = 0 is given by t = 1 ( a (1 log. b It is clear that inf t>0 f(t = f(t = a(log(b log(a + a. We define the following smoothing of h by T t h for small t > 0: [ T t h(x = E h(e t x + ] 1 e t Z, where Z N (0, I. We use the following differential equation in [5] or ( in the book [1], (6 T t h(x Φh = Ψ t (x x Ψ t (x,

5 Kolmogorov distance for Multivariate normal approximation 5 where Ψ t (x = t { h(e s x + } 1 e s yφ(ydy ds. R d Let C be the class of all Borel convex sets in R d and h = h hdφ. R d In [1], the bound for the error arising from this smoothing is given by (7 sup E[ h(f n ] sup E[T t h(fn ] + b 1 e t e t, where b is a positive constant being independent of n. We first estimate, for small t > 0, (8 E[T t h(fn ] = E[ Ψ t (F n F n Ψ t (F n ], Obviously, for i, j = 1,..., d, ( x i x Ψ e s t(x = j t 1 e s { h(e s x + } 1 e s y R y d i y φ(ydy ds. j From the estimate φ(y R d y i y dy 1 for i, j = 1,..., d we have j (9 sup x i x Ψ t(x j x R d t ( e s 1 ds = log 1 e s 1 e t. Theorem 3.. Let Σ be a d d be a symmetric positive-definite matrix. Suppose that {F n = (F n (1,..., F n (d, n 1} is a sequence of R d - valued centered square integrable random variables such that F n (i D 1, for every i = 1,..., d and n 1. For n 1 such that 1A n < b, we have that sup E[h(F n ] E[h(Σ 1/ Z] (10 ( 1A n log(b log( 1A n + 1A n,

6 6 Y.T. Kim and H.S. Park where b is a positive constant, the norm 1 denotes the subordinate matrix norm for a matrix based on l 1 vector norms, and (11 A n = [ ( E Σ lv DL 1 F n (l, DF n (v H ]. Proof. Since C is invariant nonsingular, linear transformation, we have sup = sup E[h(F n ] E[h(Σ 1/ Z] E[h( F n ] E[h(Z]. Using the smoothing inequality (7 and (11 yields (1 sup sup By (8 and (9, we estimate = = (13 E[T t h( F n ] [ E i,j=1 i,l=1 [ E i,j=1 i,j=1 E[ h( F n ] x i x j Ψ t( F n δ i,j il j,v=1 jv E E[T t h( F n ] + c 1 e t e t. [ x i x j Ψ t( F n x i x j Ψ t( F n ] ] x i x Ψ t( F j n DL 1 F n (l, DF n (v H il jv il jv [ E r=1 Σ 1/ rl Σ 1/ rv DL 1 F (l n ] ], DF n (v H

7 (14 (15 Kolmogorov distance for Multivariate normal approximation 7 [ = E x i x Ψ t( F j n il jv i,j=1 ( ] Σ lv DL 1 F n (l, DF n (v H ( 1 log 1 e t i,j=1 [ Σlv E DL 1 F n (l ( 1 ( log sup 1 e t 1 l d il jv, DF (v n H ] E [ ( Σ lv DL 1 F (l n il i=1, DF (v n H ]. Since we take n 1 such that 1A n < b, it follows, from (7 and Lemma 3.1 together with (15, that (16 sup E[T t h( F n ] ( 1A n log(b log( 1A n + 1A n. Remark 3.3. By the Cauchy-Schwartz inequality, the right-hand side in (14 can be estimated as (17 ( 1 ( log 1 e t ij E [ j=1 i=1 ] ( Σlv DL 1 F n (l, DF n (v H

8 8 Y.T. Kim and H.S. Park By a similar estimate as for (15, we have, from (17, that sup E[T t h( F n ] (18 ( ab n log(b log(ab n where a = ( d d j=1 i=1 Σ 1/ ij and B n = E [ + ab n, ( Σlv DL 1 F n (l, DF n (v H ]. 4. Applications In this section, we use our main results in order to obtain an explicit upper bound of the Kolmogorov distance instead of the Wasserstein distance used for Theorem 4.1 in the paper [9] corresponding to Lemma 4.1 below. We recall that a fractional Brownian motion B H = {B H t, t 0}, with Hurst parameter H, is a centered Gaussian process with covariance R(s, t = E[B H t B H s ] = 1 (th + s H t s H. Fix an integer q. We assume that H < 1 1. Let us set q S n (t = 1 [nt] 1 σ H q (Bk+1 H Bk H, for t 0, n k=0 where H q is the qth Hermite polynomial function and σ = q! r Z ρ (r, ρ(r = 1 ( r + 1 H + r 1 H r H. In the paper [] or [4], authors prove that as n, S n f.d.d B H, where the notation f.d.d denotes convergence in the sense of finite-dimensional distributions. In the paper [9], authors obtain the multidimensional bound for the Wasserstein distance proved for {S n (t, t 0}.

9 Kolmogorov distance for Multivariate normal approximation 9 Lemma 4.1. For any fixed d 1 and 0 = t 1 < < t d, there exists a constant c, depending only on d, H and (t 0, t 1,..., t d such that for every n 1 d W (F n, Z c n 1/ for H (0, 1] n H 1 for H ( 1, q 3 n qh q+ 1 where Z N d (0, I d and F n = (F n (1,..., F n (d, F (i n = S n(t i S n (t i 1 ti t i 1. q ], for H ( q 3, q 1 ] q q If we use the relation (3, then the upper bound of the Kolmogorov distance equals n 1/4 for H (0, 1] (19 d Kol (F n, Z c n H 1 for H ( 1, q 3 ] q. n qh q+1 4 for H ( q 3, q 1 ] q q Theorem 4.. Let F n be a sequence given in Lemma 4.1. Then for sufficiently large n 1, we have log(nn 1/ for H (0, 1] (0 d Kol (F n, Z c log(nn H 1 for H ( 1, q 3 ] q. log(nn qh q+ 1 for H ( q 3, q 1 ] q q Proof. By ignoring terms in the upper bound (10 of Theorem 3. being of lower order than log(a n A n, we have, taking C = (, z], that (1 d Kol (F n, Z c log(a n A n. By the estimate (1 and Lemma 4.1, we get the results. Remark 4.3. we can see that the upper bound (0 obtained by using Theorem 3. is more efficient than that in (19 obtained by using Lemma 4.1 in the sense that the latter one converges to zero more slowly as n tends to infinity. Acknowledgements We would like to thank the editor and the anonymous referees for their comments which considerably improve the paper.

10 10 Y.T. Kim and H.S. Park References [1] Bhattacharya, R. N. and Ranga Rao. R (1986. Normal Approximation and Asymptotic Expansion. Krieger, Melbourne, FL. [] Breuer, P and Major, P. (1983. Central limit theorems for nonlinear functionals of Gaussian fieds, J. Multivariate Analysis, 13 (3, [3] Chen, L and Shao, Q.-M. (005. Stein s method for normal approximation, in: An Introduction to Stein s method, in Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 4, Singapore Univ. Press, Singapore, [4] Giraitis, L and Surgailis, D. (1985. CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 70 (3, [5] Götze, F.(009. On the rate of convergence in the multivariate CLT, Ann. Probab., 19, [6] Nourdin, I. and Peccati, G. (009. Stein s method on Wiener Chaos, Probab.Theory Related Fields, 145, [7] Nourdin, I. and Peccati, G.(009. Stein s method and exact Berry-Esseen asymptotics for functionals of Gaussian fields, Annals of Probab., 37 (6, [8] Nourdin, I. and Peccati, G and Podolskij, M. (011. Quantitative Breuer-Major theorems, Stochasic rocesses and their Applicatrions, 11, [9] Nourdin, I. and Peccati, G and Reveillac, A. (010. Multivariate normal approximation using Stein s method and Malliavin calculus, Ann. Inst. H. Poincaré(B Probab.Statist., 46 (1, [10] D. Nualart (006, Malliavin calculus and related topics, nd Ed. Springer. Yoon Tae Kim Department of Finance and Information Statistics Hallym University Chunchon 00-70, Korea ytkim@hallym.ac.kr Hyun Suk Park Department of Finance and Information Statistics Hallym University Chunchon 00-70, Korea hspark@hallym.ac.kr