National University of Kaohsiung Repository System:Item 310360000Q/10517

全文

(1)國立高雄大學統計學研究所碩士論文. Normal Approximation for Random Vectors 隨機向量之常態逼近. 研究生：莊婷婷撰指導教授：李育嘉博士. 中華民國九十八年七月.

(2) 謝辭在這兩年求學的過程中，我受到了大家很多的幫助以及鼓勵。剛上研究所時，抱持著對研究有很大的期待，但在這些歷程當中，一開始的熱情漸漸變成了徬徨無助，開始產生了無力感，但在論文完成時，那所產生的喜悅之情，是無法言喻的。這個過程是研究生必經的過程。首先，在這裡我要感謝我的指導教授. 李育嘉教授，在我做論. 文時，遇到了困難以及對論文有不解時，老師總是提供了我不一樣的想法，這對我來說是很大的衝擊，也為我提供了新的思考方向，並且不厭其煩的指導我、帶領著我，讓我了解到做研究的過程，體驗到做研究的樂趣；也感謝施信宏教授，給我一些指導，跟我們分享一些心得；感謝陳瑞彬教授在口試中給我的指導跟寶貴的建議；謝謝正忠學長給我的鼓勵及信心；謝謝廣杰學長在我對論文有問題時，給我一些建議以及幫助，幫我解決一些難題，這對我有很大的幫助。在這兩年裡，謝謝黃文璋教授在我修習隨機過程時，給予我一些指導；謝謝黃錦輝教授在課堂上，對於一些我們以前對於某些視為理所當然的觀念，給予我們一些新的概念；謝謝俞淑惠教授，在課堂上分享一些讀書的經歷，讓我有很大的收穫；謝謝蘭屏姐在我們遇到挫折時給我們打氣，當我們的輔導老師；也感謝一路陪我度過這兩年的同班同學（岱霙、于琇、怡如、立琳、怜竹、暉展、聲杰、耀升、信.

(3) 堡、恩豪）以及學長姐們（裕盛、昱頡、華芳、基祥、孝穎、顯忠）我這兩年的生活因為有大家在而變得更加的精彩，一起歡笑、一起打鬧，一起在學校熬夜完成作業、一起挑燈夜戰複習功課、一起出遊… 等等，種種的事情都會成為我美好的回憶；也感謝以前的大學同學為仁，在我有困難的時候給予我一些幫助，讓我可以解決問題。最後，我要感謝我的父母親，他們給予了我無條件的支持跟鼓勵，在我遇到瓶頸時，他們總是一直鼓勵著我，讓我可以度過那段時期，在我難過時，他們安慰著我，讓我有更多的力量來完成這件事情。這一篇論文總算告一段落，謝謝大家陪我度過這兩年的時光，因為有大家在，我這兩年變得十分精采也十分的快樂。婷婷 2009.7.29 .

(4) Normal Approximation for Random Vectors. by Ting-Ting Chaung Advisor Yuh-Jia Lee. Institute of Statistics, National University of Kaohsiung Kaohsiung, Taiwan 811 R.O.C. July 2009.

(5) Contents. Z` zZ`. ii iv. 1 Introduction. 1. 2 Stein’e lemma for the standard normal random vector. 3. 3 An alternative proof for key lemma. 4. 4 Normal Approximation for Random Vectors. 8. References. 14. i.

(6) ^'ðV¿ ¼0>0: A }ÿ »ñ{..Tàó. .ß: §§ »ñ{..Ù.@~X ` !y3Stein½(¡Z[3]XèÕÝStein]°&ÆÊStein equationÝ¨× ËP F 00 (w) − wF 0 (w) = e h,. Í eh := h − Eh(Z)v Z × ^óýãðV5µÍETÝStein identity-îW E[F 00 (W )] = E[W F 0 (W )],. ô¿àhP¼ÑýãðV5µÝ ^óîP F = F Ý &ÆôÞÍî e h. Z. Feh (w) =. ∞ 0. 1 √ 2π. Z. e h(e−t w +. √. 1 − e−2t y)e−y. 2 /2. dydt.. R. 9øî°Ý8F¬;OÝÄvôÞÍ.ÂnîÝµ9Î æ¼]°XP°

(7) §ÝI5Ü»¼1nîÝ ^'ÍStein equation¶W ∆F (w) − w · ∇F (w) = e h,. Stein identity¶W E[∆F (W )] = E[W · ∇F (W )]. £&ÆÞ-ÞÍîW Z. Feh (w) =. ∞ 0. µ. 1 √ 2π. ¶n Z. e h(e−t w + Rn. ii. √. 2 /2. 1 − e−2t y)e−|y|. dydt..

(8) 39S¡Z&ÆÞ¥±JÍxS§(¢[1]S§2.3)¬ÿÕWasserstein distanceõKolmogorov distanceÝî&£ n"CStein]°Stein ]Stein P ^'Wasserstein ?ûKolmogorov ?û. iii.

(9) Normal Approximation for Random Vectors Advisor: Dr. Yuh-Jia Lee Department of Applied Mathematics National University of Kaohsiung. Student: Ting-Ting Chaung Institute of Statistics National University of Kaohsiung. Abstract Unlike the Stein’s method introduced in his celebrated paper[3], we consider the following alternative Stein’s equation F 00 (w) − wF 0 (w) = e h,. (0.1). where e h := −Eh(Z) and Z is a standard normal distributed random variable. The corresponding Stein identity now becomes E[F 00 (W )] = E[W F 0 (W )], which also characterized the standard normal random variable as well. The solution of (0.1) F = Feh is given by Z ∞ Z √ 1 2 e √ h(e−t w + 1 − e−2t y)e−y /2 dydt. Feh (w) = 2π R 0. (0.2). The advantage of adapting equation (0.1) is not only that the solution (0.2) is itself very easy to handle but also that the solution is ready to be extended to vectorvalued random variable. For example, for random vectors taking values in Rn , the Stein equation (0.1) becomes ∆F (w) − w · ∇F (w) = e h, iv.

(10) the Stein identity becomes E[∆F (W )] = E[W · ∇F (W )], and the solution (0.2) now becomes ¶n Z Z ∞µ √ 1 2 e √ Feh (w) = h(e−t w + 1 − e−2t y)e−|y| /2 dydt. 2π 0 Rn In this paper we reprove the key lemma (Lemma 2.3 in [1]) and then obtain the estimation of upper bound for Wasserstein distance and Kolmogorov distance. Keyword: Stein’s method, Stein’s equation, Stein identity, random vectors, Wasserstein distance, Kolmogorov distance.. v.

(11) 1. Introduction. In his celebrated paper[3], Stein introduced a method for deriving explicit estimates of the accuracy of the approximation for one probability distribution by standard normal distribution. Stein first characterize the standard normal distribution by the following lemma, known as Stein’s Lemma. Lemma 1.1. [1] Let W be a real-valued random variable. Then W has a standard normal distribution if and only if Ef 0 (W ) = E{W f (W )}, for all f ∈ Cbd , where Cbd be the set of continuous and piecewise continuously differentiable functions, f : R → R with E|f 0 (Z)| < ∞. Let Z be the random variable which is N (0, 1). Then Stein estimated Eh(W ) − E(h(Z)) for various class of random variables W , where h is an absolute continuous function satisfying kh0 k := sup |h0 (x)| < ∞. x. The estimation is then shown to bound on the the accuracy of the standard normal approximation to W in terms of the Wasserstein distance dW , to be defined later. Unlike the Stein’s method, we consider the following alternative Stein’s equation F 00 (w) − wF 0 (w) = e h,. (1.3). where e h := h − Eh(Z) and Z is a random variable having the standard normal 0 distribution and the function F belongs to the space Cbd of all differentiable functions. f which is piecewise continuously differentiable with finite E|F 00 (Z)|. The alternative Stein equation comes naturally by putting f = F 0 in the Stein’s Lemma, then the corresponding Stein identity now becomes E[F 00 (W )] = E[W F 0 (W )], which also characterized the standard normal random variable as well. The solution of (1.3) F = Feh is given by Z ∞ Z √ 1 2 e √ h(e−t w + 1 − e−2t y)e−y /2 dydt. Feh (w) = 2π R 0 1. (1.4).

(12) The advantage of adapting equation (1.3) is not only that the solution (1.4) is itself very easy to handle but also that the solution is ready to be extended to vectorvalued random variables. For example, for random vectors taking values in Rn , the Stein equation (1.3) becomes ∆F (w) − w · ∇F (w) = e h, the Stein identity becomes E[∆F (W )] = E[W · ∇F (W )], and the solution (1.4) now becomes ¶n Z Z ∞µ √ 1 2 e √ Feh (w) = h(e−t w + 1 − e−2t y)e−|y| /2 dydt. 2π 0 Rn We then use the estimation supw |∆Feh (w) − w · ∇Feh (w)| to bound on the accuracy of standard norm approximation to W in terms of Wasserstein distance and Kolmogorov distance defined below. Definition 1.2. [1][Kolmogorov distance] If P and Q are two distribution measures and the test functions are the indicators of all half-lines, then we define the Kolmogorov distance to be Z Z dK (P, Q) := sup | hdP − hdQ| = sup |P (−∞, z] − Q(−∞, z]|, z∈R. h∈H. where H = {1(−∞,z] ; z ∈ R}. Definition 1.3. [1][Wasserstein distance] If P and Q are two distribution measures and the test functions consist of all uniformly Lipschitz functions h with constant bounded by 1, then we define the Wasserstein distance to be Z Z dW (P, Q) := sup | hdP − hdQ|, h∈H. where H = {h : R → R; kh0 k ≤ 1} =: Lip(1). In this paper we will reprove the following key lemmas (with some minor modifications). 2.

(13) Lemma 1.4. [1][key lemma I] For any absolutely continuous function h : R → R, the solution fh of the Stein’s equation satisfies. kfh k ≤ min(. p π/2kh(·) − Eh(Z)k, 2kh0 k);. kfh0 k ≤ min(2kh(·) − Eh(Z)k, 4kh0 k), and kfh00 k ≤ 2kh0 k. Next, we modify the lemma using Z ∞ Z ∞ √ y2 1 √ Feh (w) = (e h(e−t w + 1 − e−2t y))e− 2 dydt. 2π −∞ 0 Lemma 1.5. [1][key lemma II] Assume that there exists a δ such that, for any uniformly Lipschitz function h, |Eh(W ) − Eh(Z)| ≤ δkh0 k.. (1.5). dW (L(W ), N (0, 1)) := sup |Eh(W ) − Eh(Z)| ≤ δ;. (1.6). Then. h∈Lip(1). dK (L(W ), N (0, 1)) := sup |P(W ≤ z) − Φ(z)| ≤ 2δ 1/2 .. (1.7). z∈R. The paper is arranged in such a way that, in section 2, we first show that our alternative Stein’s identity also characterizes the normal distribution. In section 2, we give an easy proof for the key lemmas and in section 3, we prove the key lemmas for random vectors.. 2. Stein’e lemma for the standard normal random vector. Lemma 2.1. Let W = (W1 , W2 , · · · , Wn ) be a real-valued random vectors. Then W is distributed from N (0, I) if and only if E [∆F (W ] = E {W · ∇F (W )} , 3.

(14) 0 0 for all F ∈ Cbd (Rn ), where Cbd (Rn ) is the set of all differentiable functions on Rn. with finite E||∆F (Z)|| and, for each i, ∂F/∂wi ∈ Cbd . 0n Proof. Necessity. If W has standard normal distribution, then for F ∈ Cbd. E{W · ∇F (W )} =. n X. E(Wi. i=1. =. n Z X. ∂ F (W )) ∂xi. Wi n. =. i=1 R n Z X. Rn. i=1. ∂ F (W )dW ∂xi. ∂2 F (W )dW ∂x2i. = E∆F (W ). Sufficiency. For arbitrary ξ ∈ Rn , let F (W ) = eihW,ξi and let g(t) = E[F (tW )]. Then we obtain tg 0 (t) = E(W · ∇F (tW )) = E∆F (tW ) = E{W · ∇F (tW )} = −|ξ|2 t2 g(t) It follows that g 0 (t) = −g(t)t|ξ|2 . Since g(0) = 1, we must have g(t) = e−t. 2 |ξ|2 /2. 2 /2. so that F (ξ) = e−|ξ|. . Therefore W. is N (0, I).. 3. An alternative proof for key lemma. 0 Assume that Z distributed from the standard normal distribution. Let Cbd be the. set of that F 0 is continuous and piecewise continuously differentiable functions and E|F 00 (Z)| < ∞. Let Z Feh (w) =. 0. Z Feh0 (w). ∞. ∞. = 0. 1 √ 2π. 1 √ 2π. Z. Z. ∞. (e h(e−t w +. √. y2. 1 − e−2t y))e− 2 dydt,. (3.8). −∞. ∞. (e h0 (e−t w +. −∞. 4. √. y2. 1 − e−2t y)e−t )e− 2 dydt.. (3.9).

(15) We replace the Stein equation by the following equation, F 00 (w) − wF 0 (w) = h(w) − Eh(Z) = e h.. (3.10). Therefore, we revised the key lemma as follows. Lemma 3.1. For any absolutely continuous function h : R → R, the solution Fh0 given as (3.9) satisfies. √ kFeh0 k ≤ min(. 2π e khk∞ , ke h0 k∞ ), 2. 2 kFeh00 k ≤ √ ke h0 k∞ , 2π. (3.11) (3.12). and 4 kFeh000 k ≤ ( √ + 2)ke h0 k. (3.13) 2π Proof. First, we verify (3.11) by using the solution of (3.10) which is Z ∞ Z ∞ √ y2 1 0 √ Feh (w) = (e h0 (e−t w + 1 − e−2t y)e−t )e− 2 dydt, 2π −∞ 0 then Z Z ∞ √ y2 e−t ∂ e −t 1 0 √ √ Feh (w) = [h(e w + 1 − e−2t y)]e− 2 dydt 2π 1 − e−2t R ∂y Z ∞ Z0 ∞ √ x2 e−t 1 √ √ [e h(e−t w + 1 − e−2t x) − e h(e−t w)]xe− 2 dxdt. = 2π 1 − e−2t −∞ 0 And we estimate the bound of Feh0 (w) in the following: Z ∞ Z ∞ √ x2 e−t 1 0 √ √ |Feh (w)| = | [e h(e−t w + 1 − e−2t x) − e h(e−t w)]xe− 2 dxdt| 2π 1 − e−2t −∞ Z ∞ Z0 ∞ √ x2 1 e−t √ √ ≤ | ke h0 k∞ 1 − e−2t x2 e− 2 dxdt| 2π 1 − e−2t −∞ 0 = ke h0 k∞ . Z ∞ Z ∞ √ x2 1 e−t 0 √ √ [e h(e−t w + 1 − e−2t x) − e h(e−t w)]xe− 2 dxdt| |Feh (w)| = | 2π 1 − e−2t −∞ Z0 ∞ Z ∞ √ x2 1 e−t e √ √ = | h(e−t w + 1 − e−2t x)xe− 2 dxdt| 2π 1 − e−2t −∞ Z0 ∞ Z ∞ x2 1 e−t √ √ ≤ | ke hk∞ |x|e− 2 dxdt 2π 1 − e−2t −∞ 0 Z ∞ e−t 2 √ √ dt| = ke hk∞ | 1 − e−2t 2π 0 √ 2π e khk∞ . = 2 5.

(16) Moreover, we verify (3.12) by the following equation Z ∞ Z √ y2 1 00 √ Feh (w) = (e h00 (e−t w + 1 − e−2t y)e−2t )e− 2 dydt 2π R Z0 ∞ Z ∞ √ y2 1 e−2t ∂ e −t −2t y)]ye− 2 dydt. √ = [ h(e w + 1 − e 2π 1 − e−2t −∞ ∂y 0 Then Z. |Feh00 (w)|. = = ≤ = =. Z ∞ √ y2 1 e−2t ∂ e −t −2t y)]ye− 2 dydt| √ | [ h(e w + 1 − e 2π 1 − e−2t −∞ ∂y 0 Z ∞ Z ∞ √ x2 1 e−2t √ | (e h(e−t w + 1 − e−2t x) − e h(e−t w))(x2 − 1)e− 2 dxdt| −2t 2π 1 − e 0 −∞ Z ∞ Z ∞ −2t √ x2 1 e 0 −2t |x2 − 1|e− 2 dxdt| √ | kh k 1 − e ∞ 2π 1 − e−2t −∞ 0 Z ∞ Z ∞ x2 1 e−2t 0 √ √ kh k∞ | |x3 − x|e− 2 dxdt| 2π 1 − e−2t −∞ 0 2 √ kh0 k∞ . 2π ∞. Finally, we verify (3.13) by using the differentiation of (3.10) which is F 000 (w) = wF 00 (w) + F 0 (w) + e h0 . Then we use the above equation to find the bound of F 000 (w). Before calculating, we. 6.

(17) may assume that |h(t)| ≤ kh0 k∞ |t| by h(0) = 0.. = = − ≤ + ≤ + =. = =. |wF 00 (w)| Z ∞ Z ∞ √ 1 e−2t 2 e √ | w h(e−t w + 1 − e−2t y)(y 2 − 1)e−y /2 dydt| −2t 2π 1 − e Z0 ∞ Z ∞−∞ √ √ 1 e−t 2 −t −t −2t y)e √ | (e w + 1 − e h(e w + 1 − e−2t y)(y 2 − 1)e−y /2 dydt −2t 2π 1 − e 0 Z ∞ Z −∞ ∞ √ −t √ 1 e 2 −t −2t ye √ 1 − e h(e w + 1 − e−2t y)(y 2 − 1)e−y /2 dydt| −2t 2π 1 − e Z0 ∞ Z−∞∞ −t √ √ 1 e 2 −t −t −2t y)e √ | (e w + 1 − e h(e w + 1 − e−2t y)(y 2 − 1)e−y /2 dydt| −2t 2π 1 − e Z0 ∞ Z−∞ ∞ √ √ 1 e−t 2 √ | 1 − e−2t y(e h(e−t w + 1 − e−2t y) − e h(e−t w))(y 2 − 1)e−y /2 dydt| −2t 2π 1 − e 0 −∞ Z ∞ Z ∞ −t √ 1 e 2 0 −t −2t y)2 (y 2 − 1)e−y /2 dydt| √ ke h k∞ | (e w + 1 − e 2π 1 − e−2t −∞ Z0 ∞ Z ∞ y2 1 √ e−t |y 4 − y 2 |e− 2 dydt ke h0 k∞ | 2π Z0 ∞ −∞ e−t 1 √ ke h0 k∞ | 2π 1 − e−2t 0 Z ∞ √ 2 (e−2t w2 + 2e−t 1 − e−2t wy + (1 − e−2t )y 2 )(y 2 − 1)e−y /2 dydt| + 2|e h0 k −∞ Z Z ∞ ∞ e−t 1 2 0 e √ kh k∞ | (1 − e−2t )|y 4 − y 2 |kh0 ke−y /2 dydt + 2||e h0 || −2t 1 − e 2π 0 −∞ 4 e0 √ kh k∞ . 2π. 4 |Feh000 (w)| = |wF 00 (w) + F 0 (w) + e h0 | ≤ |wF 00 (w)| + |F 0 (w)| + |e h0 | ≤ ( √ + 2)||h0 ||. 2π. We compare the conclusions of Lemma 1.4 and Lemma 3.1. We find that √ 2π e kFeh0 k ≤ min( khk∞ , ke h0 k ∞ ) 2 is better than. p kfh k ≤ min( π/2kh(·) − Eh(Z)k, 2kh0 k); 2 kFeh00 k ≤ √ ke h0 k ∞ 2π. is better than kfh0 k ≤ min(2kh(·) − Eh(Z)k, 4kh0 k). 7.

(18) But kfh00 k ≤ 2kh0 k is better than 4 kFeh000 k ≤ ( √ + 2)ke h0 k. 2π Theorem 3.2. For any n, let ξ1 , ξ2 , · · · , ξn be independent random variable satisfyP ing Eξi = 0 and E|ξi |3 < ∞, and such that ni=1 Eξi2 = 1. Then n X ¢ ¡ 00 4 0 0 e E|ξi |3 . |E Feh (W ) − W Feh (W ) | ≤ (( √ + 2)kh k∞ ) 2π i=1. (3.14). Proof. The proof is along the line with Lemma 3.2 in [1]. Theorem 2.2 can be applied to Lindeberg central limit theorem when n large enough .. 4. Normal Approximation for Random Vectors. For the random vectors on Rn , let Z Z ∞ √ |~ y |2 1 e Feh (w) ~ = h(e−t w ~ + 1 − e−2t ~y )e− 2 d~y dt n (2π) 2 Rn 0. (4.15). Then Feh satifies ~ =e 4F (w) ~ − w∇F ~ (w) ~ = h(w) ~ − Eh(Z) h.. (4.16). Lemma 4.1. For function h : Rn → R which is absolute continuous in such a way that for each i, the function wi → h(. . . , wi , . . . ), regarded as a function of wi is absolutely continuous, the solution Fh given in (4.15) satisfying √ π |h∇Feh (w), ~ ki| ≤ min( ke hk∞ |k|, nk∇e hk∞ |k|), 2. (4.17). √ π k∇Feh (w)k ~ ≤ min( ke hk∞ , nk∇e hk∞ ). 2. (4.18). |hD2 Feh (w)k ~ 1 , k2 i| ≤ k∆Feh (w)k ~ ≤. √ √. 2nk∇e hk∞ |k1 ||k2 |,. (4.19). 2nk∇e hk∞ .. (4.20). 8.

(19) Proof. h∇Feh (w), ~ ki Z ∞Z √ = e−t hDy e h(e−t w ~ + 1 − e−2t y), kiµ(dy)dt n Z0 ∞ ZR √ e−t e √ = h(e−t w ~ + 1 − e−2t y)hy, kiµ(dy)dt. 1 − e−2t 0 Rn |h∇Feh (w), ~ ki| Z ∞Z √ e−t e √ = | h(e−t w ~ + 1 − e−2t y)hy, kiµ(dy)dt| n 1 − e−2t Z0 ∞ ZR e−t √ ≤ | ke hk∞ hy, kiµ(dy)dt| −2t n 1 − e 0 R π e ≤ khk∞ |k|. 2 Moreover, we have Z k∇Feh (w)k ~ ≤. = = ≤ ≤ ≤. −∞. |h∇Feh (w), ~ ki|µ(dk) ≤. π e khk∞ . 2. |h∇Feh (w), ~ ki| Z ∞Z √ e−t e √ | h(e−t w ~ + 1 − e−2t y)hy, kiµ(dy)dt| n 1 − e−2t Z0 ∞ ZR √ e−t √ | (e h(e−t w ~ + 1 − e−2t y) − e h(e−t w))hy, ~ kiµ(dy)dt| −2t 1−e 0 Rn Z ∞Z √ e−t √ h(e−t w)|hy, ~ kiµ(dy)dt |e h(e−t w ~ + 1 − e−2t y) − e −2t 1 − e 0 Rn Z ∞Z e−t k∇e hk∞ |y||hy, ki|µ(dy)dt 0 Rn Z e k∇hk∞ |y||hy, ki|µ(dy) Rn. ½Z ≤ k∇e hk∞ =. ∞. √. ¾1/2 |k| |y| µ(dy) 2. Rn. nk∇e hk∞ |k|.. Consequently, Z k∇Feh (w)k ~ =. ∞ −∞. |h∇Feh (w), ~ ki|µ(dk) ≤. 9. √. nk∇e hk∞ ..

(20) = = = ≤ ≤ ≤. |hD2 Feh (w)k ~ 1 , k2 i| Z ∞Z √ e−2t √ | hDy2e h(e−t w ~ + 1 − e−2t y)k1 , k2 iµ(dy)dt| n 1 − e−2t Z0 ∞ ZR √ e−2t e −t | h(e w ~ + 1 − e−2t y)(hy, k1 ihy, k2 i − hk1 , k2 i)µ(dy)dt| −2t 1 − e n Z0 ∞ ZR √ e−2t e −t | h(e w ~ + 1 − e−2t y) − e h(e−t w))(hy, ~ k1 ihy, k2 i − hk1 , k2 i)µ(dy)dt| −2t 1 − e n 0 R Z ∞Z e−2t √ k∇e hk∞ |y||hy, k1 ihy, k2 i − hk1 , k2 i|µ(dy)dt −2t n 1 − e 0 R Z ∞ Z Z e−2t 2 1/2 e √ dt)k∇hk∞ ( |y| µ(dy)) ( |hy, k1 ihy, k2 i − hk1 , k2 i|2 µ(dy))1/2 ( −2t n n 1−e R R √0 e 2nk∇hk∞ |k1 ||k2 |.. It follows that Z k∆Feh (w)k ~ =. ∞ −∞. |hD2 Feh (w)k ~ 1 , k2 i|µ(dk) ≤. √. 2nk∇e hk∞ .. Theorem 4.2. Assume that there exists a δ such that, for any uniformly Lipschitz function h, ~ ) − Eh(Z)| ~ ≤ δ||∇h||∞ . |Eh(W. (4.21). Then Wasserstein distance and Kolmogorov distance for random vectors are ~ ), N (0, I)) := sup |Eh(W ~ ) − Eh(Z)| ~ ≤ δ, dW (L(W. (4.22). h∈Lip(1). and ~ ≤ ~z) − Φ(~z)| ≤ δ + ( √α )n . dK (PW , PN ) := sup |P(W α 2π ~ z ∈Rn. (4.23). Proof. The equation (4.22) comes from the definition of dW . For the equation (4.23), we may assume that ||h0α || = α1 . There exists a function   1 w j ≤ zj , hα (w1 , w2 , · · · , wn ) =  0 wj > zj + α.. 10.

(21) Then ~ ≤ ~z) − Φ(~z) P(W ≤ Eh(W ) − Eh(Z) + Eh(Z) − Φ(z) δ ≤ + P{z ≤ Z ≤ z + α} α α δ ≤ + ( √ )n . α 2π. 11.

(22) References [1] A D Barbour, Louis H Y Chen: An Introuction to Stein’s Method, Singapore University Press and World Scientific, 2004. [2] Kuo, H.-H.: Gaussian Measures in Banach Spaces, Springer-Verlag Berlin. Heidelberg. New York, 1975. [3] C. Stein: A bound for the error in normal approximation to the distribution of a sum of dependent variables. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 583-602. Univ. California Press, Berkelewy, CA. [4] Charles Stein: Approximate Computation of Expectations, Institute of Mathematical Statistics, Lecture Notes-monograph Series, Shanti S. Gupta, Series Editor, Volume 7, 1986. [5] Larry Goldstein, Gesine Reinert: Stein’s Method and The Zero Bias Transformation with Application to Simple Random Sampling, The Annals of Applied Probability , Vol. 7, No. 4, 935-952, 1997.. 12.

(23)