CENTRAL LIMIT THEORY FOR THE NUMBER OF SEEDS IN A GROWTH MODEL IN d WITH INHOMOGENEOUS POISSON ARRIVALS

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1 The Annals of Apple Probablty 1997, Vol. 7, No. 3, CENTRAL LIMIT THEORY FOR THE NUMBER OF SEEDS IN A GROWTH MODEL IN WITH INHOMOGENEOUS POISSON ARRIVALS By S. N. Chu 1 an M. P. Qune Hong Kong Baptst Unversty an Unversty of Syney A Posson pont process n -mensonal Euclean space an n tme s use to generate a brth growth moel: sees are born ranomly at locatons x n R at tmes t. Once a see s born, t begns to create a cell by growng raally n all rectons wth spee v >. Ponts of contane n such cells are scare, that s, thnne. We stuy the asymptotc strbuton of the number of sees n a regon, as the volume of the regon tens to nfnty. When = 1, we establsh contons uner whch the evoluton over tme of the number of sees n a regon s approxmate by a Wener process. When 1, we gve contons for asymptotc normalty. Rates of convergence are gven n all cases. 1. Introucton. Conser the followng spatal brth growth moel n R. Sees are born (or forme ranomly at locatons x at tme t, = 1 2 accorng to a spatal temporal pont process x t R. Once a see s born, t mmeately generates a cell by growng raally n all rectons wth a constant spee v >. The space occupe by cells s regare as covere. Cells an new sees contnue to grow an form, respectvely, only n uncovere space n R. The pont process s assume to be a Posson process wth ntensty measure l, where l s the Lebesgue measure n R, whle s an arbtrary locally fnte measure on such that > an (1.1 µ exp t } ω v t u u t < where ω = π /Ɣ 1 + /2 s the volume of a unt ball n R. It wll be shown n the next secton that µ s the ntensty of the sees forme n R. Throughout the paper we use t to enote t. Such a brth growth process was frst suggeste an stue by Kolmogorov (1937 n the case = 2 to moel crystal growth [see Chu (1995, 1996 for etals of subsequent evelopments]. Interestngly, specal cases of ths brth growth process when = 1 have foun applcatons n several fferent bologcal contexts [see Holst, Qune an Robnson (1996 an the references theren]. Receve October Research partally supporte by an Australan Research Councl Insttutonal Grant. AMS 1991 subject classfcatons. Prmary 6G55, 6D5, 6F5; seconary 6G6, 6F17. Key wors an phrases. Brth growth, nhomogeneous Posson process, R, central lmt theorem, Brownan moton, rate of convergence. 82

2 CLT FOR A GROWTH MODEL IN R 83 Denote by the spatal temporal pont process of the sees forme, whch s a epenently thnne verson of the Posson process. For ease of presentaton, we conser an as both ranom sets of ponts n R an ranom measures efne on the Borel σ-algebra of R. Denote by ξ z the ranom varables z + 1, where z Z an z + 1 = z + x x 1. Then ξ z z Z s a real-value statonary ranom fel. It s statonary because s spatally homogeneous, an so s. For z 1 an z 2 n Z, let z 1 z 2 = max 1 z 1 z 2, where z, 1, are the components of z. For Ɣ Z, enote by # Ɣ the number of elements n Ɣ an by Ɣ the set z Ɣ there exsts z Ɣ such that z z = 1. Let Ɣ n Z be a fxe sequence of fnte subsets of Z satsfyng the regularty conton that lm n # Ɣ n /# Ɣ n =. It mples that the sequence Ɣ n oes not ncrease n only one recton, except n the case = 1. Defne S n to be z Ɣ n ξ z µ for each n N. Let S =. Qune an Robnson (199 establshe asymptotc normalty for the number of sees n the case = 1 wth a homogeneous arrval rate. Ther metho was extene to cover more general arrval regmes by Chu (1996. Holst, Qune an Robnson (1996 prove results smlar to Chu s by conserng an assocate Markov process. In ths paper we use a completely fferent metho, base on mxng propertes, to establsh asymptotc normalty n an arbtrary menson 1 for a very general class of. In partcular, when = 1, we prove the functonal central lmt theorem for S n ; that s, after sutable normalzaton an lnear nterpolaton, S n behaves asymptotcally lke a Brownan moton. Rates of convergence are also scusse. 2. Moments. Let t enote the ranom regon n R whch s covere just before tme t by the -generate brth growth process. For each pont x t n, x t t = x t \ x t t = x t because the frst two events mply that at tme t the poston x has not yet been covere by the -generate brth growth process, an consequently a see s forme at x t. Therefore, we have [ ] E ξ z = E 1 x t t x t z+ 1 where 1 enotes the ncator functon. By Mecke [(1967, Satz 3.1] or Møller [(1992, equaton (3.1], E ξ z = E [ 1 x t x t t ] l x t z+ 1 = E [ 1 x t t ] l x t 1

3 84 S. N. CHIU AND M. P. QUINE Note that each x t oes not belong to t f an only f (2.1 y u y x v t u u t = where s the Euclean stance. Thus, t } E ξ z = exp ω v t u u t = µ where µ has been assume to be fnte n conton (1.1. By observng that x 1 t 1 x 1 t 1 x 2 t 2 t x 1 t 1 t an usng Møller [(1992, equaton (3.1], we obtan (2.2 E ξ z ξ z 1 ξ z j µ j+1 < for j = Thus, E ξz j < for each postve nteger j. Let Ɣ n + 1 = z + 1 z Ɣ n. Usng Møller [(1992, equaton (3.1] agan, we have [ ( ] E ξ z ξ z 1 z Ɣ n z Ɣ n (2.3 = E = x t Ɣ n + 1 =1 2 x 1 x 2 1 x 1 t 1 t 1 Ɣn+ 1 1 x 2 t 2 t 2 exp t 1 t 2 x 1 x 2 >v t 2 t 1 x 2 Ɣ n + 1 ( } v t1 + t exp 2 x 1 x 2 l x 2v 2 t 2 l x 1 t 1 where t = t ω v t u u an x y = max x y. Suppose X 1 an X 2 are two nepenent unformly strbute ponts n Ɣ n + 1. Denote by f n the ensty of Y X 1 X 2 an let r n = sup y f n y >. From (2.3, we have E S n S n 1 + # Ɣ n 2 µ 2 # Ɣ n µ ( } = # Ɣ n 2 v t1 + t exp t 1 t y 2v = # Ɣ n 2 y>v t 1 t 2 rn t1 +y/v f n y y t 2 t 1 ( } v t1 + t exp t 1 t y 2v t 1 y/v f n y t 2 y t 1

4 = # Ɣ n 2 CLT FOR A GROWTH MODEL IN R 85 rn exp t 1 f n y exp t 2 t 2 y/v t 1 exp t 2 t 2 t1 ( +y/v v t1 + t + exp t y y/v t 1 t 1 y/v 2v = # Ɣ n 2 µ 2 # Ɣ n 2 exp t 1 exp t 2 + # Ɣ n 2 rn exp t 1 f n y t1 +y/v exp t 2 + t 1 y/v v t1 +t 2 r n ( v t1 + t 2 y 2v } } t 2 y t 1 f n y y t 2 t 1 } t 2 y t 1 where x y = mn x y. The ensty f n epens on the shape of Ɣ n + 1 but σ 2 lm n var S n /# Ɣ n oes not. We can erve σ 2 by evaluatng the above ntegrals wth Ɣ n + 1 an # Ɣ n replace by a ball of large raus R an volume ω R, respectvely. The ensty of the stance between two nepenent unformly strbute ponts n ths ball s f y = R y 1 B +1 /2 1/2 1 y 2 / 4R 2 where B a b s the strbuton functon of the beta strbuton wth parameters a an b [Kenall an Moran (1963, equaton (2.122]. Therefore σ 2 = µ + exp t 1 t1 exp t 1 exp y ω v t 1 + t 2 exp t 2 t 2 t 1 y 2 ω v t 1 + t 2 2y 1 exp t 2 t 2 u t 1 In partcular, f t = λ t, where < λ <, then wrtng γ = λω v / + 1, ( λ 1 µ = Ɣ ( γ 1/ σ 2 = µ I 1 + I 2 where λ I 1 = + 1 γ 1/ +1 I 2 = ( Ɣ j j= t1 λ exp γ t +1 1 ( j Ɣ exp γ y +1 ( + 1 j y γ t 1 + t 2 2y 1 exp γ t +1 2 t 2 y t 1

5 86 S. N. CHIU AND M. P. QUINE When = 1, we can obtan an analytc soluton by means of the transformaton u = t 2 y / 2, w = t 2 + y / 2 an a seres expanson, gvng I 2 = πλ = log 2 v λ exp λvt j λvt 2 Ɣ j/2 1 j/2 t j! 1 j=1 For 2 we can wrte I 2 n a form sutable for numercal ntegraton as follows. Put u = γ 1/ +1 t 1 y, w = γ 1/ +1 t 2 y an x = γ 1/ +1 y. Then where K = an (2.4 gves σ 2 = λ γ 1/ λ I 2 = K γ 1/ +1 u + w 1 exp u + x +1 + x +1 w + x +1} u w x K ( Ɣ j j=1 ( j Ɣ ( + 1 j + 1 By means of substtutons lke α = u + x +1, K can be reuce to an ntegral of the varable x alone, the ntegral contanng strbuton functons of gamma varables. In ths form the ntegral can be realy evaluate usng an S-Plus program. The numercal values to three ecmal places for = 1, 2, 3 an 4 are as follows: K σ 2 γ 1/ +1 /λ Hereafter we conser only the class of wth σ 2 >. 3. Mxng coeffcents. Denote by P the probablty space nuce by ξ z z Z. For Ɣ 1 Ɣ 2 Z, let Ɣ 1 Ɣ 2 = nf z 1 z 2 z Ɣ = 1 2. Defne the mxng coeffcents to be } α a b k sup P A 1 A 2 P A 1 P A 2 A σ ξ z z Ɣ # Ɣ 1 a # Ɣ 2 b Ɣ 1 Ɣ 2 k } where k N, a b N an σ ξ z z Ɣ s the σ-algebra generate by ξ z z Ɣ. We mpose the followng conton on to govern how fast t goes to nfnty.

6 CLT FOR A GROWTH MODEL IN R 87 Conton 3.1. There exsts a constant M < such that t } t + c t + 1 s + c s + 1 exp ω v t u u M for all s t <, where c = /v. In ths secton we erve an upper boun only for α 1 1 k. Conser ξ z1 an ξ z2 such that ξ z1 ξ z2 k. For each A σ ξ z, there exsts an nex set J of nonnegatve ntegers such that A = j J A j where A j = ξ z = j an = 1 or 2. Let P ( A n ( n ( m 2 P A 1 P A 2 = β n m k Then, for any A σ ξ z, = 1 an 2, (3.1 Note that P A 1 A 2 P A 1 P A 2 P ( A ( 2 P A 1 ( = P A n 1 n 1 P ( A m 2 P ( A m 2 n= m= P ( n 1 A n β n m k 2 Hence we obtan β m k n=1 β n m k, β n k m=1 β n m k an β k n=1 m=1 β n m k. Consequently, t suffces to conser only n=1 m=1 β n m k because (3.2 n= β n m k 4 m= n=1 m=1 β n m k Let T = nf t x t z + 1 an let X be the poston of the see corresponng to the brth-tme T, for = 1 an 2. Because s a Posson process whch s spatally homogeneous, T 1 an T 2 are nepenent whenever z 1 z 2. They have the same strbuton functon F whch s gven by (3.3 F t = 1 exp t for t an zero otherwse. The ranom postons X are unformly strbute n z + 1. Recall that for each x t, x t f an only f (2.1 hols. That means for each x t there s a forben regon R x t n whch no ponts of exst. For = 1 an 2, R x t s a trangle an a cone n R, respectvely. For x j t j j = 1 n, the forben regon s just the unon n j=1 R x j t j. Snce s a Posson process, for n 1 an m 1, P ( A n 2 T = t = 1 2 P ( A n 1 T ( m 1 = t 1 P A 2 T 2 = t 2

7 88 S. N. CHIU AND M. P. QUINE only f contonal on T = t = 1 2 the forben regons for A n 1 an A m 2 have a nonempty ntersecton. Ths can happen only f v t 1 +t 2 +2 > k 1. Hence, (3.4 β n m k v t 1 +t 2 +2 >k 1 P ( A n 2 T = t = 1 2 F t 1 F t 2 P ( A n v t 1 +t T 1 = t 1 >k 1 P ( A m 2 T 2 = t 2 F t1 F t 2 Conser P A n T = t, = 1 an 2. Contonal on X T = x t, = 1 or 2, there are n sees forme n z + 1 only f x t t an at least n 1 more ponts of exst n z + 1 after t but before the cell generate by the see at x t covers z + 1, whch wll occur before t + /v. Thus, P ( A n T = t t } exp ω v t u u j n 1 for = 1 an 2. Hence (3.5 n=1 m=1 t + /v t j exp t + /v t j! v t 1 +t 2 +2 >k 1 P ( A n 1 T 1 = t 1 P ( A m 2 T 2 = t 2 F t1 F t 2 t1 + /v t } v t 1 +t 2 +2 >k 1 t 2 + /v t } t1 exp ω v t 1 u u t2 } ω v t 2 u u F t 1 F t 2 Smlarly, conser P A n 2 T = t = 1 2. Contonal on X T = x t = 1 2, there are n an m sees forme n z an z 2 + 1, respectvely, only f at least n 1 an m 1 more ponts of exst n z t 1 t 1 + /v an z t 2 t 2 + /v,

8 CLT FOR A GROWTH MODEL IN R 89 respectvely, an R x 1 t 1 R x 2 t 2 =. The probablty of the latter s at most exp t max ω v t max u u where t max = max t 1 t 2. Therefore, (3.6 n=1 m=1 = v t 1 +t 2 +2 >k 1 v t 1 +t 2 +2 >k 1 P ( A n 2 T = t = 1 2 F t 1 F t 2 ( t1 + /v t } ( t 2 + /v t } tmax } exp ω v t max u u F t 1 F t 2 v t 1 +t 2 +2 >k 1 I t 1 t 2 F t 1 F t 2 say. Uner Conton 3.1, there exsts a constant M such that I t 1 t 2 M for all t 1 t 2. From (3.4, (3.5 an (3.6, we have (3.7 n=1 m=1 β n m k 2 I t v t 1 +t t 2 F t 1 F t 2 >k 1 4M v t 1 +t 2 +2 >k 1 t 1 t 2 4M F t k / 2v F t 1 F t 2 where x + = max x. Thus, by the statonarty of ξ z z Z, (3.1, (3.2, (3.3 an (3.7, (3.8 ( α 1 1 k 16M exp ( k v whch tens to zero as k tens to nfnty. } exp = α k 4. Central lmt theorem. We prove the central lmt theorem for S n n an arbtrary menson 1 n ths secton. Lemma 4.1 [Bolthausen (1982]. Suppose that ξ z z Z s statonary. If k=1 k 1 α a b k < for a + b 4, α 1 k = o k, an E ξ z 2+δ < an k=1 k 1 α 1 1 k δ/ 2+δ < for some δ >, then z cov ξ z ξ z < an f σ 2 = z cov ξ z ξ z >, then the strbuton of S n / # Ɣ n σ 2 converges weakly to the stanar normal strbuton as n. In orer to use ths lemma to show the asymptotc normalty of S n, we have to know upper bouns of α 1 k an α a b k for a + b 4.

9 81 S. N. CHIU AND M. P. QUINE Lemma 4.2. Uner Conton 3.1, for all k a b N, α a b k abα k Proof. Conser Ɣ = z j j J for = 1 an 2 such that Ɣ 1 Ɣ 2 k, where J 1 = 2j 1 j = 1 a, J 2 = 2j j = 1 b, a b N, a a, b b an all z j are stnct. Let A n = ξ z = n, where n s a nonnegatve nteger an = 1 an 2. For each A σ ξ z z Ɣ, A = n= A n B n for some B n σ ξ z z Ɣ \ z. Let P ( A n 2 B n 1 B m ( n 2 P A 1 B n ( m 1 P A 2 B m 2 = γ n m k Then, n vew of (3.2, (3.7 an (3.8, t suffces to show that γ n m k a b β n m k n=1 m=1 n=1 m=1 Let T j = nf t x t z j + 1 where j J 1 J 2. Smlar to the argument n Secton 3, for n 1 an m 1, P A n 1 A m 2 B n 1 B m 2 T j = t j j J 1 J 2 P A n 1 B n 1 T j = t j j J 1 P A m 2 B m 2 T j = t j j J 2 only f the forben regons ntersect, that s, f v t j1 +t j2 +2 > k 1 for some j 1 J 1 an j 2 J 2. Ths par j 1 j 2 can be any one of the a b elements n the set j 1 j 2 j J = 1 2. Snce P A n 2 B n 1 B m 2 T j = t j j J 1 J 2 P A n 2 T j = t j j J 1 J 2 an P A n B n T j = t j j J P A n T j = t j j J for = 1 an 2, from (3.5, (3.6 an (3.7, the result follows. Lemma 4.3. Uner Conton 3.1, for all k N, α 1 k h 1 α h h=k Proof. We use the same argument an notaton as n the proof of Lemma 4.2 except that b =. Now J 1 = 1 an J 2 = Let J h 2 = j z 1 z j = h for all ntegers h k. Then the number of elements n J h 2 s 2h + 1 2h 1, whch s less than h 1. The forben regons ntersect only when v t 1 +t j +2 > h 1 for some t j J h j an h k. Therefore, from (3.5, (3.6 an (3.7, γ n m k h=k 22 1 h 1 β n m h, an the result follows. Remark. Lemmas 4.2 an 4.3 are qute smlar to Braley (1981, Lemma 8. However, n our context Braley s lemma s not applcable because hs conton, that the σ-algebras σ ξ zj j J h 2 be nepenent, s not fulflle.

10 CLT FOR A GROWTH MODEL IN R 811 Now we mpose one more conton on. Conton 4.1. For suffcently large k N, for some τ. h 1 α h = o k τ h=k From (2.2 an Lemmas 4.2 an 4.3, f Conton 4.1 hols, whch mples that α k = o k 2+1 τ, then all the requrements of Lemma 4.1 are met when (1 τ an δ = 5 f 2 or (2 τ = ε for some ε > an δ > 2/ε f = 1. Thus, the followng central lmt theorem s obtane. Theorem 4.1. Uner Contons 3.1 an 4.1 where τ f 2 or τ > f = 1, the strbuton of S n / # Ɣ n σ 2 converges weakly to the stanar normal strbuton as n. Contons 3.1 an 4.1 are fulflle (for any τ when, for example, t Kt j for some postve K an 1 j <. If <, then Conton 3.1 hols, but Conton 4.1 requres a fast convergence of t. Conser, for example, t = λɣ α 1 t yα 1 e y y for some postve fnte α an λ so that = λ. Then there exsts a t o such that exp t exp λ = exp λ exp λ t 1 2 exp λ λ t for t > t o = O ( t α 1 exp t Thus, by (3.8, ths satsfes Contons 3.1 an 4.1 for any τ. 5. Functonal central lmt theorem. In partcular, we conser = 1 n ths secton, an so σ 2 = z cov ξ ξ z. For each n N, for ease of presentaton we assume # Ɣ n = n an efne W n t ω = S nt ω / σ 2 n for t 1 an ω where x s the greatest nteger not exceeng x. The functon ω W n ω s a measurable mappng from nto D, where D s the space of functons on 1 that are rght contnuous an have left-han lmts, an enotes the Borel σ-algebra nuce by the Skorokho topology [see, e.g., Bllngsley (1968]. Let α k sup n Z P A 1 A 2 P A 1 P A 2 A 1 σ ξ z z n for k N. Note that α k α k for all k. A 2 σ ξ z z n + k }

11 812 S. N. CHIU AND M. P. QUINE Lemma 5.1 [Herrnorf (1984, Corollary 1]. If there exsts some δ > such that k=1 α k δ/ 2+δ < an E ξ z 2+δ < for all z Z, an var S n /n σ 2, where < σ 2 <, then W n converges n strbuton to the stanar Wener measure on D as n. In vew of ths lemma, we shoul fn an upper boun for α k. Lemma 5.2. Uner Conton 3.1, for each k N, α k r + 1 α k + r = r= r=k α h Proof. We use agan the same argument an notaton as n the proof of Lemma 4.2 except that Ɣ 1 an Ɣ 2 have to be n the form z Z z n an z Z z n + k, respectvely, for some n Z. Now J 1 = an J 2 = Contonal on T j = t j j J 1 J 2, the forben regons ntersect only when v t j1 + t j > k + r 1 where z j1 z j2 = k + r for some r N an j J, = 1 an 2. For each such r, the number of elements n the set j 1 j 2 z j1 z j2 = k + r j J = 1 2 s at most r + 1. The statement s now obvous. If Contons 3.1 an 4.1 hol for τ = 1 + ε for some ε >, then by Lemma 4.1, var S n /n σ 2 <. Moreover, by Lemma 5.2, α k = r=k o r 2 ε = o k 1 ε/2 Thus, the requrements of Lemma 5.1 are met whenever δ > 4/ε. Hence, we have prove the functonal central lmt theorem for S n n one menson. Theorem 5.1. For = 1, uner Contons 3.1 an 4.1 where τ > 1, W n converges n strbuton to the stanar Wener measure on D as n. 6. Rates of convergence. In ths secton we assume that h=r (6.1 or (6.2 t Kt j for some postve K an 1 j <, t t = λ y α 1 e y y for some postve fnte α an λ. Ɣ α Ether (6.1 or (6.2 mples that α k = O e ρk for some postve fnte ρ. Thus, by Lemma 5.2, when = 1, α k = O e ρk. Denote by G n the strbuton functon of S n / # Ɣ n σ 2 an by G the stanar normal strbuton. Theorem 6.1. If (6.1 or (6.2 hols, then for 1, (6.3 sup G n x G x x R = O ( # Ɣ n 1/2 log # Ɣ n

12 CLT FOR A GROWTH MODEL IN R 813 Furthermore, when = 1, ( log 3 # Ɣ G n x G x = O n (6.4 # Ɣn 1 + x 4 for each x R. Proof. For 2, (6.3 follows from (2.2, Lemma 4.2 an Takahata (1983, Theorem 1, whereas for = 1, (6.3 an (6.4 follow from (2.2 an Tkhomrov (198, Theorem 4. In orer to obtan a rate of convergence for the functonal central lmt theorem, we nee to conser a smoothe verson of W n. For each n N we assume # Ɣ n = n an efne W n t ω = S nt ω σ2 n nt nt ( + S nt +1 ω S nt ω σ2 n for t 1 an ω. That means W n s the ranom polygonal lne wth noes at j/n S j / σ 2 n, j = n. Thus, W n belongs not only to D but also to C, the space of boune, contnuous, real-value functons efne on 1. Let P n an W be the strbutons of W n an the stanar Wener process on D. Denote by L the Lévy Prokhorov stance between two probablty measures efne on the Borel σ-algebra of the metrc space C wth the supnorm. The followng theorem follows from (2.2 an Utev (1985, Corollary 7.2. Theorem 6.2. where ε >. If (6.1 or (6.2 hols, then L P n W = O ( n 1/4+ε REFERENCES Bllngsley, P. (1968. Convergence of Probablty Measures. Wley, New York. Bolthausen, E. (1982. On the central lmt theorem for statonary mxng ranom fels. Ann. Probab Braley, R. C. (1981. Central lmt theorems uner weak epenence. J. Multvarate Anal Chu, S. N. (1995. Lmt theorem for the tme of completon of Johnson Mehl tessellatons. Av. n Appl. Probab Chu, S. N. (1997. A central lmt theorem for lnear Kolmogorov s brth growth moels. Stochastc Process. Appl Herrnorf, N. (1984. A functonal central lmt theorem for weakly epenent sequences of ranom varables. Ann. Probab Holst, L., Qune, M. P. an Robnson, J. (1996. A general stochastc moel for nucleaton an lnear growth. Ann. Appl. Probab Kenall, M. G. an Moran, P. A. P. (1963. Geometrcal Probablty. Grffn, Lonon. Kolmogorov, A. N. (1937. On statstcal theory of metal crystallsaton. Izvesta Acaemy of Scence, USSR, Ser. Math (n Russan. Mecke, J. (1967. Statonäre zufällge Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebete

13 814 S. N. CHIU AND M. P. QUINE Møller, J. (1992. Ranom Johnson Mehl tessellatons. Av. n Appl. Probab Qune, M. P. an Robnson, J. (199. A lnear ranom growth moel. J. Appl. Probab Takahata, H. (1983. On the rates n the central lmt theorem for weakly epenent ranom fels. Z. Wahrsch. Verw. Gebete Tkhomrov, A. N. (198. On the convergence rate n the central lmt theorem for weakly epenent ranom varables. Theory Probab. Appl Utev, S. S. (1985. Inequaltes for sums of weakly epenent ranom varables an estmates of the convergence rate n the nvarance prncple. In Lmt Theorems for Sums of Ranom Varables (A. A. Borovkov, e Optmzaton Software, Inc., New York. Department of Mathematcs Hong Kong Baptst Unversty Kowloon Tong Hong Kong E-mal: snchu@math.hkbu.eu.hk School of Mathematcs an Statstcs Unversty of Syney N. S. W. 26 Australa E-mal: malcolmq@maths.usy.eu.au