and the initial value R 0 = 0, 0 = fall equivalence classes ae singletons fig; i = 1; : : : ; ng: (3) Since the tansition pobability p := P (R = j R?1

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1 A CLASSIFICATION OF COALESCENT PROCESSES FOR HAPLOID ECHANGE- ABLE POPULATION MODELS Matin Mohle, Johannes Gutenbeg-Univesitat, Mainz and Seik Sagitov 1, Chalmes and Gotebogs Univesities, Gotebog Abstact We conside a class of haploid population models with non-ovelapping geneations and xed population size N assuming that the family sizes within a geneation ae exchangeable andom vaiables. A weak convegence citeion is established fo a popely scaled ancestal pocess as N! 1. It esults in a full classication of the coalescent geneatos in the case of exchangeable epoduction. In geneal the coalescent pocess allows fo simultaneous multiple meges of ancestal lines. 1 Intoduction Conside the class of haploid population models with non-ovelapping geneations and xed population size N 2 IN := f1; 2; : : :g intoduced by Cannings (1974, 1975). Each model in this class is chaacteized by an exchangeable joint distibution of family sizes 1 ; : : : ; N, (1) whee i denotes the numbe of osping of the i-th individual. Recall that accoding to (1) the distibution of the andom vecto ( i1 ; : : : ; ik ) with paiwise distinct indices depends only upon k and not upon the paticula set of indices. As the population size is xed the condition N = N (2) has to be satised. We ae inteested in the asymptotics of the genealogical stuctue in such a population in the spiit of Kingman (1982a,b,c). Fix n N and sample n individuals at andom fom the 0-th geneation. Let R denote the equivalence elation which contains the pai (i; j) i the i-th and the j-th individual of this sample have a common ancesto in the -th geneation backwads in time, 2 IN 0 := f0; 1; 2; : : :g. The pocess (R ) 2IN0 is a time homogeneous Makov chain with the state space E n = the set of all equivalence elations on f1; : : : ; ng 1 Suppoted by the Bank of Sweden Tecentenay Foundation poject \Dependence and Inteaction in Stochastic Population Dynamics" AMS 1991 subject classications. Pimay 92D25, 60J70; Seconday 92D15, 60F17. Key wods and phases. Ancestal pocesses, coalescent, exchangeability, geneato, neutality, population genetics, weak convegence. 1

2 and the initial value R 0 = 0, 0 = fall equivalence classes ae singletons fig; i = 1; : : : ; ng: (3) Since the tansition pobability p := P (R = j R?1 = ) is equal to zeo fo 6, the focus will be on such pais ; 2 E n that. The elation implies that evey equivalence class of is eithe a union of seveal equivalence classes of o coincides with an equivalence class of. Reecting this obsevation wite a fo the numbe of -classes and b = b b a fo the numbe of -classes, whee b 1 : : : b g 2 ae odeed goup sizes fo meging -classes and b g+1 = : : : = b a = 1. Notice that g = 0 if = and g 1 if. In this notation the tansition pobability of the ancestal pocess is given by p = 1 (N) b N i 1 ;:::;ia=1 all distinct E(( i1 ) b1 ( ia ) ba ) = (N) a (N) b E(( 1 ) b1 ( a ) ba ); (4) whee (N) b := N(N? 1) (N? b + 1). Let c N denote the pobability that two individuals, chosen andomly without eplacement fom some geneation, have a common ancesto one geneation backwads in time, i.e. c N := 1 (N) 2 N E(( i ) 2 ) = E(( 1) 2 ) N? 1 = Va( 1) N? 1 = 1? E( 1 2 ): (5) This pobability, called the coalescence pobability is of fundamental inteest in the coalescent theoy as c?1 N is the pope time scale to get convegence to the coalescent (it is only natual to assume that c N > 0 fo suciently lage N because the case c N = 0 coesponds to the tivial epoduction law: P ( 1 = 1; : : : ; N = 1) = 1). The coalescence pobability is also impotant as it is diectly connected via c N = 1? 2 to the eigenvalue 2 := E( 1 2 ) of the tansition matix of the descendant pocess, i.e. the genealogical pocess looking fowads in time (see Cannings (1974)). Kingman (1982b) has shown that given sup N E( k 1 ) < 1, k 2 (this holds fo example fo the Moan model and the Wight-Fishe model) the convegence of nite-dimensional distibutions (R [t=cn ]) t0! (R t ) t0 ; N! 1 (6) takes place. The it pocess (R t ) t0, the so-called (standad) n-coalescent pocess, is a continuous time Makov pocess with state space E n, initial state (3) and innitesimal geneato Q = (q ) ;2En given by q := 8 < :?jj(jj? 1)=2 if =, 1 if, 0 othewise, (7) 2

3 whee means that, g = 1 and b 1 = 2, i.e. duing the tansition exactly two ancestal lines mege togethe. The convegence (6) is based on the asymptotic fomula: p = + c N q + o(c N ); N! 1 ; 2 E n which is often witten in matix notation P N = I + c N Q + o(c N ); N! 1; (8) whee P N := (p ) ;2En denotes the tansition matix of the ancestal pocess. In Mohle (1998, 1999) Kingman's esult was extended beyond the famewok of exchangeable population models and it was shown that (6) holds even in the sense of the weak convegence of stochastic pocesses. Recently a iche class of the coalescent geneatos Q allowing fo multiple meges with g = 1 and b 1 2 was found in Sagitov (19??). A membe Q of this class is chaacteized by a pobability measue F (dx) on the unit inteval [0,1] via the fomula q = 8 >< >:? [0;1] 1? (1? x) b?1 (1? x + bx) x 2 F (dx); if =, [0;1] x b1?2 (1? x) b?b1 F (dx); if, g = 1, 0 othewise. The geneato (7) of the standad n-coalescent is ecoveed fom (9) when the pobability measue F = 0 is concentated in zeo. The pesent pape is based upon an instumental development of Mohle and Sagitov (1998) of the method of Sagitov (19??). We establish a geneal coalescent stuctue allowing fo simultaneous meges of ancestal lines (g 1). Due to the main esult of this pape, Theoem 2.1, in geneal, a coalescent geneato Q = (q ) ;2En is chaacteized by a sequence of symmetic measues F, 2 IN, whee each F is concentated on the simplex (9) := f(y 1 ; : : : ; y ) 2 [0; 1] j y y 1g with 1 = F 1 ( 1 ) F 2 ( 2 ) : : : (10) If, then the coesponding enty has the fom q = [(a+g)=2] =g 1 x bg?2 g x b1?2 Hee the set of polynomials g;a?g( ; : : : ; x ) F (d ; : : : ; dx ): (11) j;s (; : : : ; x ); 1 j ; 1; s 0 (12) 3

4 is dened explicitly by the fomulae and ;s ( ; : : : ; x ) = (1??? x ) s ; (13)?j;s (; : : : ; x ) (14) = (?1) j+1 i j+1?2 i j =2j?1 : : : i 2?2 jy i 1=1 k=0?k i k (1? x i ) i k+1?i k?2 ; j = 1; : : : ; whee i 0 =?1, i j+1 = s + 1. Note that this implies The diagonal enties of Q ae calculated as: whee q =? [b=2] b?g g=1 a=g S () g;s( ; : : : ; x ) = [(a+g)=2] =g b 1 bg 2 b 1 ++bg =b?s?j;s 0 fo s < 2j. S g;a?g(x () 1 ; : : : ; x )F (d ; : : : ; dx ); (15) b b 1 : : : b g s x b1?2 1 : : : x bg?2 g T g;s () ( ; : : : ; x ): Obseve that in the case when F 2 ( 2 ) = 0 the fomulae (11), (15) and T (1) 1;s (x) = (1? x)s bing us back to (9) with F = F 1. 2 A weak convegence citeion This section pesents the main esult of the pape, Theoem 2.1, which shows that the fomulae (11) and (15) fully descibe the class of coalescent geneatos fo the population models with exchangeable epoduction. The cental condition of Theoem 2.1 equies the existence of the its E(( 1 ) k1 ( j ) kj ) N!1 N k1++kj?j = j (k 1 ; : : : ; k j ); k 1 k j 2 (16) c N fo all j 2 IN. To justify the denominato in the LHS of (16) tun to the chain of inequalities N i 1 ;:::;i j =1 all distinct ( i1 ) k1 ( ij ) kj (17) N i 1 ;:::;i l =1 all distinct N i 1 ;:::;i l =1 all distinct ( i1 ) m1 k1?m1 i 1 ( il ) ml k l?m l i l N i l+1 ;:::;i j =1 k l+1 i l+1 kj i j ( i1 ) m1 N k1?m1 ( il ) ml N k l?m l ( N ) k l+1++k j 4

5 = N k1++kj N m1++m l N i 1 ;:::;i l =1 all distinct ( i1 ) m1 ( il ) ml ; whee l j; k 1 m 1 1; : : : ; k l m l 1. With l = 1 and m 1 = 2 it entails that in geneal sup N!1 E(( 1 ) k1 ( j ) kj ) N k1++kj?j c N 1; k 1 k j 2: Relation (17) implies also that the set of the its (16) is monotone: j (k 1 ; : : : ; k j ) l (m 1 ; : : : ; m l ) wheneve l j; k 1 m 1 ; : : : ; k l m l : (18) Theoem 2.1 If the its (16) exist fo all j 2 IN, then fo each sample size n 2 IN the asymptotic fomula (8) holds with Q = (q ) ;2En dened by (11) and (15). The coesponding symmetic measues F, 2 IN ae uniquely detemined via thei moments x k1?2 1 x k?2 F (d ; : : : ; dx ) = (k 1 ; : : : ; k ); k 1 k 2: (19) If futhemoe, the it N!1 c N = c exists, then convegence (6) holds in the Skoohod sense. The it coalescent pocess (R t ) t0 is eithe (when c > 0) a discete time Makov chain with the initial state (3) and the tansition matix I + cq, o (when c = 0) a continuous time Makov chain with the initial state (3) and the tansition matix e tq. Convesely, if (8) holds, then all the its (16), j 2 IN exist. Condition (16) has anothe two equivalent vesions (cf. Section 4 fo the poof): one in tems of the cental moments E(( 1? 1) k1 ( j? 1) kj ) N!1 N k1++kj?j = j (k 1 ; : : : ; k j ); k 1 k j 2 (20) c N and the othe in tems of the tail distibutions N j P ( 1 > N ; : : : ; j > Nx j ) = N!1 c N x j F j (dy 1 ; : : : ; dy j ) y 2 1 y2 j ; (21) holding fo all points ( ; : : : ; x j ) of continuity fo the it measue. Vesion (21) bings the following pictue of the asymptotic coalescent stuctue. Call lage evey family whose size is of ode N. Obviously, evey lage family with a positive pobability embaces two o moe sampled ancestal lines (in othe wods begets a multiple mege). Due to the condition N!1 Nc?1 N P ( 1 > N ) = y?2 F 1 (dy) 5

6 a nite numbe of lage families is encounteed with a positive pobability while scanning N geneations in the population. A lage family caused a multiple mege might, in a sense, tigge a chain eaction of meges within the same geneation. To see this obseve that the total numbe of families in a geneation is equal to N and the elation N!1 NP ( 2 > Nx 2 j 1 > N ) = x 2 y?2 1 y?2 2 F 2(dy 1 ; dy 2 ) y x?2 F 1 1 (dy) indicates to the possibility that we might encounte anothe lage family outside the initial one povided F 2 ( 2 ) > 0. Futhemoe, if F 3 ( 3 ) > 0 the second lage family leaves oom fo the thid one: N!1 NP ( 3 > Nx 3 j 1 > N ; 2 > Nx 2 ) = x 2 x 3 y?2 x 2 y?2 1 y?2 2 F 2(dy 1 ; dy 2 ) 1 y?2 2 y?2 3 F 3(dy 1 ; dy 2 ; dy 3 ) an so on. This imaginay chain eaction of meges (thee is no eal ode fo the meges happening within one geneation) is bound to stop afte a andom numbe of ounds because the population of size N might host only a nite numbe of lage families (given F l+1 ( l+1 ) = 0 this numbe of ounds neve exceeds l). Remak. Accoding to Theoem 2.1 we have F ( ) = (2; : : : ; 2) so that (10) follows fom (18). Note that F l+1 ( l+1 ) = 0 in paticula when the andom vaiables ( 1 ) 2 ( l ) 2 and ( l+1 ) 2 ae not positively coelated, povided N!1 c N = 0. Indeed in this case E(( 1 ) 2 ( l+1 ) 2 ) E(( 1 ) 2 ( l ) 2 ) E(( 1 ) 2 ) N 2l c N (N) l Nc N and hence E(( 1 ) 2 ( l+1 ) 2 ) F l+1 ( l+1 ) = l+1 (2; : : : ; 2) = N!1 N l+1 c N N!1 c N = 0: 3 The poof of the citeion Lemma 3.2 If the its (16) exist fo some j 2 IN, then thee exists a measue F j uniquely detemined on the simplex j by its moments (19). Poof. If j (2; : : : ; 2) = 0 then (19) implies F j ( j ) = 0. In the case j (2; : : : ; 2) > 0 we have E(( 1 ) 2 ( j ) 2 ) > 0 fo suciently lage N. Let Y 1;j ; : : : ; Y j;j be andom vaiables with the joint distibution P (Y 1;j = i 1 ; : : : ; Y j;j = i j ) := (i 1 ) 2 (i j ) 2 E(( 1 ) 2 ( j ) 2 ) P ( 1 = i 1 ; : : : ; j = i j ); (22) 6

7 whee i 1 ; : : : ; i j 2 f2; : : : ; Ng. The epesentation E(Y k1 1;j Y kj j;j ) = (i 1 )k1 (i j )kj (i 1 ) 2 (i j ) 2 P ( 1 = i 1 ; : : : ; j = i j ) E(( i 1 ) 2 ( j ) 2 ) 1;:::;i j kj +2 kj +1 = E((k1+2 1? k1+1 1 ) (j? j )) E(( 1 ) 2 ( j ) 2 ) in view of the equation t k = P k l=1 (t) ls kl, t 2 IR, k 1 (S kl ae the Stiling numbes of the second kind) leads to N!1 Y1;j E(( N )k1 ( Yj;j N )kj ) (16) = j(k 1 + 2; : : : ; k j + 2) ; k 1 ; : : : ; k j 2 IN 0 : (23) j (2; : : : ; 2) This convegence of moments implies (see Felle (1971), Chapte 8, Section 1) the weak convegence of the pobability distibutions on j : P ( Y1;j N 2 dy 1; : : : ; Yj;j N 2 dy j)! P j (dy 1 ; : : : ; dy j ); N! 1: (24) Compaison between (23) with (24) shows that (19) holds with F j (d ; : : : ; dx j ) = j (2; : : : ; 2) P j (d ; : : : ; dx j ): The uniqueness of F j is due to the fact that the it moments (23) fully chaacteize the pobability measue P j. 2 Denition 3.3 Fo j 2 IN, k 1 ; : : : ; k j 2 and s 2 IN 0 dene j;s(k 1 ; : : : ; k j ) := E(( 1 ) k1 ( j ) kj j+1 j+s ) N!1 N k1++kj?j c N as long as this it exists. Lemma 3.4 The following ecusion ove s holds: j;s+1(k 1 ; : : : ; k j ) = j;s (k 1 ; : : : ; k j )? j j;s(k 1 ; : : : ; k i?1 ; k i + 1; k i+1 ; : : : ; k j )?s j+1;s?1 (k 1 ; : : : ; k j ; 2) fo all j 2 IN, k 1 ; : : : ; k j 2 and all s 2 IN 0. Poof. Take the LHS and the RHS in the following chain of equalities (N? j? s)e(( 1 ) k1 ( j ) kj j+1 j+s+1 ) (1) = E(( 1 ) k1 ( j ) kj j+1 j+s ( j+s N )) 7

8 (2) = E(( 1 ) k1 ( j ) kj j+1 j+s (N? 1?? j+s )) = E ( 1 ) k1 ( j ) kj j+1 j+s? N? (k1 + + k j )? s? j ( i? k i )? j+s i=j+1 ( i? 1) = (N? (k k j )? s) E(( 1 ) k1 ( j ) kj j+1 j+s )? j E(( 1 ) k1 ( i ) ki+1 ( j ) kj j+1 j+s )?se(( 1 ) k1 ( j ) kj ( j+1 ) 2 j+2 j+s ); and divide them by N k1++kj +1?j c N. Afte letting N! 1 we get the asseted ecusion equation. 2 Lemma 3.5 Polynomials (12) dened by elations (13) and (14) satisfy and fo j = 1; : : : ;? 1 j;s+1 (; : : : ; x ) = (1? ;s+1 (; : : : ; x ) = (1? j x i ) ;s ( ; : : : ; x ) (25) x i ) j;s (; : : : ; x )? s j+1;s?1 (; : : : ; x ): (26) Poof. Fomula (25) is obvious in view of (13). To veify (26) ewite it as?j?j;s+1 (; : : : ; x ) = (1? x i )?j;s (; : : : ; x )? s?j+1;s?1 (; : : : ; x ) and apply (14). 2 Lemma 3.6 If the its (16) exist fo all j 2 IN, then j;s(k 1 ; : : : ; k j ) = j x k1?2 j 1 x kj?2 fo all j 2 IN, k 1 ; : : : ; k j 2 and all s 2 IN 0. j;s (; : : : ; x ) F (d ; : : : ; x ); Poof. We use induction ove s. The case s = 0 follows fom the equality j;0(k 1 ; : : : ; k j ) = j (k 1 ; : : : ; k j ) (19) = x k1?2 1 x kj?2 j F j (d ; : : : ; dx j ): j Lemma 3.4 and Lemma 3.5 ensue that the induction assumption implies the asseted fomula j;s+1(k 1 ; : : : ; k j ) 8

9 L: 3:4 = j;s (k 1 ; : : : ; k j )? j?s j+1;s?1 (k 1 ; : : : ; k j ; 2) ind = x k1?2 1 x kj?2 j (1? j?s j+1 L: 3:5 = j x k1?2 x k1?2 j;s(k 1 ; : : : ; k i?1 ; k i + 1; k i+1 ; : : : ; k j ) j j 1 x kj?2 j 1 x kj?2 x i ) j;s (; : : : ; x ) F (d ; : : : ; dx ) j+1;s?1 (; : : : ; x ) F (d ; : : : ; dx ) To nish the poof of Theoem 2.1 tun to the equality j;s+1 (; : : : ; x ) F (d ; : : : ; dx ): p q = (27) N!1 c N (4) (N) a = E(( 1 ) b1 ( g ) bg g+1 a ) N!1 (N) b c N E(( 1 ) b1 ( g ) bg g+1 a ) = = N!1 N b?a g;a?g (b 1 ; : : : ; b g ) c N saying that fo any pai the LHS and the RHS exist o do not exist simultaneously and coincide when exist. Assume that the its (16) exist fo all j 2 IN. Then accoding to Lemma 3.6 and (27) the asymptotic fomula (8) with (11) is valid fo all. In view of the equality q =? : q =? [b=2] b?g g=1 a=g b 1 bg 2 b 1 ++bg=b?a+g fomula (11) entails (15). Afte the asymptotic fomula (8) is poved the weak convegence (6) is obtained as in Mohle (1999). Finally, the \only-if"-pat of Theoem 2.1 is a simple coolay of the equality j (b 1 ; : : : ; b a ) = g;0 (b 1 ; : : : ; b g ) (27) = q which holds povided a = g. 2 4 The equivalence of (16), (20), (21) Fix some j 2 IN. Hee we show the equivalence of conditions (16), (20), (21) with the measues F j and the its j being linked by (19). q 2 9

10 (16),(20). The poof of this equivalence is based on the decomposition ( 1 ) k1 : : : ( j ) kj = k 1 i 1=1 k j i j =1 i1;:::;i j ( 1? 1) i1 ( j? 1) ij ; (28) whee i1;:::;i j ae some nite coecients and k1;:::;k j = 1. It suces to veify that E(( 1? 1) i1 ( j? 1) i l ) = o(n k1++kj?j c N ); N! 1 (29) fo all (i 1 ; : : : ; i l ) 2 [1; k 1 ] [1; k l ] n f(k 1 ; : : : ; k l )g; 1 l j; k 1 : : : k j 2: To pove (29) notice st that E( 1? 1) = 0; E(( 1? 1) 2 ) = E(( 1 ) 2 ) = (N? 1)c N : Tuning to a countepat of (17) fo Ej( 1? 1) i1 ( j? 1) i l j we see that (29) is tue when at least one i is geate o equal 2. In the emaining case i 1 = : : : = i l = 1 the equality chain (N? l + 1)E( 1? 1) : : : ( l? 1) (1) = E( 1? 1) : : : ( l?1? 1)[( l? 1) + + ( N? 1)] (2) =?E( 1? 1) : : : ( l?1? 1)[( 1? 1) + + ( l?1? 1)] =?(l? 1)E( 1? 1) 2 ( 2? 1) : : : ( l?1? 1): ends with a tem of ode o(nc N ) in accodance with the pevious agument. Thus (29) holds and we can conclude fom (28) that fo any xed set of indices k 1 : : : k j 2 two its (16) and (20) ae equal when exist with the existence of one entailing the existence of the othe. This conclusion is slightly stonge than the asseted equivalence. 2 (16))(21). To aive at the weak convegence (21) multiply P ( 1 > N ; : : : ; j > Nx j ) = (22) = N j E(( 1 ) 2 ( j ) 2 ) : : : : : : P ( 1 x j N 2 dy 1 : : : j N 2 dy j) Y P ( 1 N 2 dy 1 : : : Yj N 2 dy j) x j y 1 (y 1? 1 N ) y j(y j? 1 N ) ; by N j c?1 N and apply (24) and (16). 2 (21))(20). Condition (21) implies the weak convegence of measues N j P ( 1? 1 N!1 c N N > ; : : : ; j? 1 N > x j) = x j F j (dy 1 ; : : : ; dy j ) y 2 1 y2 j 10

11 which in tun implies the convegence of integals E(( 1? 1) k1 ( j? 1) kj ) N!1 N k1++kj?j c N = N j x k1 1 c xk P ( 1? 1 N N 2 d; : : : ; j? 1 N 2 dx j) = x k1?2 1 x k?2 F (d ; : : : ; dx ) (19) = (k 1 ; : : : ; k ): 2 5 The Wight-Fishe model as a it Recall that the Wight-Fishe model descibes a population of a xed size (say l), whee evey individual chooses its paent at andom among l individuals constituting the pevious geneation. Hee we discuss a simple exchangeable population model whose time-scaled ancestal pocess conveges to the ancestal pocess of the Wight-Fishe model. Take a xed constant 1 l N=2 and conside such an exchangeable population model that in each geneation exactly l families ae of size [N=l] while othe family sizes ae zeos and ones. In this case P ( 1 = : : : = l = [N=l]; l+1 = : : : = l+l1 = 1; l+l1+1 = : : : = N = 0) = l!l 1! (N) l+l1 whee l 1 := N? l[n=l]. It follows that and hence This entails E(( 1 ) k1 ( j ) kj ) (l) j (N) j ( N l ) k 1 ( N l ) k j ; N! 1 j (k 1 ; : : : ; k j ) = c := N!1 c N = 1=l: E(( 1 ) k1 ( j ) kj ) N!1 N k1++kj?j = (l) j l 1?k1??kj : c N Thus fo this paticula model the it measue F j assigns its total mass j (2; : : : ; 2) = (l) j l 1?2j to the single point (1=l; : : : ; 1=l) 2 IR j being a zeo measue fo j > l. Now using Lemma 3.4 and induction ove s we can show that j;s(k 1 ; : : : ; k j ) = (l) j+s l 1?s?k1??kj : (30) The case s = 0 follows fom j;0 (k 1 ; : : : ; k j ) = j (k 1 ; : : : ; k j ) = (l) j l 1?k1??kj. The step fom s to s + 1 is given by j;s+1(k 1 ; : : : ; k j ) L: 3:4 = (l) j+s l 1?s?k? j (l) j+s l?s?k? s(l) j+s l?s?k = (l) j+s l?s?k (l? j? s) = (l) j+s+1 l 1?(s+1)?k ; 11

12 whee k := k k j. We conclude that fo (27) q = g;a?g (b 1 ; : : : ; b g ) (30) = (l) a l 1?(a?g)?b1??bg = (l) a l 1?b so that the tansition matix = I + cq fo the it Makov chain has enties = (l) a l?b fo and the esulting coalescent pocess coincides with the ancestal pocess fo the Wight-Fishe model with the population size l. As l tends to innity the geneato Q conveges to the geneato of the standad n-coalescent in ageement with the weak convegence of the measue F 1 to the point measue in zeo. Fo j > 1 the total mass of F j conveges to zeo as l tends to innity. To genealize ou example take an intege valued andom vaiable L N with P (1 L N N=2) = 1; N 2 IN and conditional on fl N = lg, l 2 IN dene a population model as befoe. Assuming that L N conveges weakly as N tends to innity to some andom vaiable L, we deduce and E(( 1 ) k1 ( j ) kj ) j (k 1 ; : : : ; k j ) = N!1 N k1++kj?j c N N=2 = N!1 = 1 l=1 l=1 E(( 1 ) k1 ( j ) kj j L N = l) N k1++kj?j c N P (L N = l) (l) j l 1?k1??kj P (L = l) = E((L) j L 1?k1??kj ) c := N!1 c N = E(1=L): Note that the last expectation is positive even if we allow fo the possibility 0 P (L = 1) < 1. In paticula, if L? 1 has a Poisson distibution with paamete > 0, then c = 0 E(x L?1 ) dx = 0 e (x?1) dx = 1? e? Fo the genealized example it follows that the enties of the it geneato Q ae given by q = E((L) a L 1?b ) and the tansition matix = I + cq fo the it Makov chain has enties = E(1=L)E((L) a L 1?b ) fo. The esulting coalescent pocess depends on the obseved value of the it andom vaiable L. If L = l < 1, the coalescent is the ancestal pocess of the Wight- Fishe model with the population size l. When L = 1 the sampled ancestal lines neve mege. : 12

13 Refeences [1] Cannings, C. (1974). The latent oots of cetain Makov chains aising in genetics: a new appoach, I. Haploid models. Adv. Appl. Pob. 6, 260 { 290. [2] Cannings, C. (1975). The latent oots of cetain Makov chains aising in genetics: a new appoach, II. Futhe haploid models. Adv. Appl. Pob. 7, 264 { 282. [3] Felle, W. (1971). An Intoduction to Pobability Theoy and Its Applications. Volume I, Second Edition, Wiley. [4] Kingman, J.F.C. (1982a). On the Genealogy of Lage Populations. J. Appl. Pob. 19A, 27{43. [5] Kingman, J.F.C. (1982b). Exchangeability and the Evolution of Lage Populations. in: Koch, G. and Spizzichino, F.: Exchangeability in Pobability and Statistics, Noth{Holland Publishing Company, pp. 97{112. [6] Kingman, J.F.C. (1982c). The Coalescent. Stoch. Pocess. Appl. 13, 235{ 248. [7] Mohle, M. (1998). Robustness Results fo the Coalescent. J. Appl. Pob. 35, 438 { 447. [8] Mohle, M. (1999). Weak convegence to the coalescent in neutal population models. J. Appl. Pob. 36 (to appea June 1999). [9] Mohle, M. and Sagitov, S. (1998). A Chaacteisation of Ancestal Limit Pocesses Aising in Haploid Population Genetics Models. Beichte zu Stochastik und vewandten Gebieten, Johannes Gutenbeg-Univesitat Mainz, Novembe 1998, ISSN [10] Sagitov, S. (1999). The Geneal Coalescent with Asynchonous Meges of Ancestal Lines. J. Appl. Pob. 36 (to appea Decembe 1999) Matin Mohle Johannes Gutenbeg-Univesity Mainz Depatment of Mathematics Saastae Mainz, Gemany moehle@mathematik.uni-mainz.de Seik Sagitov Chalmes and Gotebogs Univesities Depatment of Mathematical Statistics Gotebog, Sweden seik@math.chalmes.se 13