The Annals of Apple Probablty 1997, Vol. 7, No. 3, 82 814 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 1996. 1 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
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
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 = 1 2 3 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 2 + 2 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 2 + 2 y 2v t 1 y/v f n y t 2 y t 1
= # 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 2 + 2 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 (2.4 + 1 γ 1/ +1 + 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 + 1 + 1 Ɣ exp γ y +1 ( + 1 j y + 1 2 + 1 γ t 1 + t 2 2y 1 exp γ t +1 2 t 2 y t 1
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 2 1 2 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/ +1 2 + 1 λ I 2 = K γ 1/ +1 u + w 1 exp u + x +1 + x +1 w + x +1} u w x 2 + 1 K 1 + 1 ( Ɣ j j=1 ( j + 1 + 1 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: 1 2 3 4 K.37.213.195.27 σ 2 γ 1/ +1 /λ.342.439.515.579. 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.
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
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 2 +2 1 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 1 + 1 } v t 1 +t 2 +2 >k 1 t 2 + /v t 2 + 1 } 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 1 + 1 an z 2 + 1, respectvely, only f at least n 1 an m 1 more ponts of exst n z 1 + 1 t 1 t 1 + /v an z 2 + 1 t 2 t 2 + /v,
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 1 + 1 } ( t 2 + /v t 2 + 1 } 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 2 +2 1 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 1 2 1 + / 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 1 2 + 2v 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.
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 2 2 1 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 = 2 4 6. 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 2 2 1 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.
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 }
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 = 1 3 5 an J 2 = 2 4 6. Contonal on T j = t j j J 1 J 2, the forben regons ntersect only when v t j1 + t j2 + 2 2 > 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
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. 1 147 15. Braley, R. C. (1981. Central lmt theorems uner weak epenence. J. Multvarate Anal. 11 1 16. Chu, S. N. (1995. Lmt theorem for the tme of completon of Johnson Mehl tessellatons. Av. n Appl. Probab. 27 889 91. Chu, S. N. (1997. A central lmt theorem for lnear Kolmogorov s brth growth moels. Stochastc Process. Appl. 66 97 16. Herrnorf, N. (1984. A functonal central lmt theorem for weakly epenent sequences of ranom varables. Ann. Probab. 12 141 153. Holst, L., Qune, M. P. an Robnson, J. (1996. A general stochastc moel for nucleaton an lnear growth. Ann. Appl. Probab. 6 93 921. 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. 3 355 36 (n Russan. Mecke, J. (1967. Statonäre zufällge Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebete 9 36 58.
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