ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS

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1 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE Abstract. We extend some of the results of Pfaffelhuber and Walkobinger on the process of the most recent common ancestors (MRCA) in evolving coalescent by taking into account the size of the oldest family which contains the immortal individual. For example we give an explicit formula for the Laplace transform of the extinction time for the Wright-Fisher diffusion. We give also an interpretation of the quasy-stationary distribution of the Wright- Fisher diffusion using the process of the relative size of one of the two oldest families, which can be seen as a resurrected Wright-Fisher diffusion.. Introduction Many models have been introduced to describe population dynamics in population genetics. Fisher [5], Wright [35] and Moran [6] have introduced two models for exchangeable haploid populations of constant size. A generalization has been given by Cannings []. Looking backward in time at the genealogical tree leads to coalescent processes, see Griffiths [8] for one of the first papers with coalescent ideas. For a large class of exchangeable haploid population models of constant size, when the size N tends to infinity and time is measured in units of N generations, the associated coalescent process is Kingman s coalescent [3] (see also [8, 9, 5, 3] for general coalescent processes associated with Cannings model). One of the associated object of interest is the most recent common ancestor (MRCA) of the population currently alive, which is also the depth of their genealogical tree (see [3, ]). In the case of Kingman s coalescent, each couple of particle merges at rate one, which gives an MRCA of expectation, or an expectation equivalent to N generations in the discrete case (see [3] for more results on this approximation and [6] for exact coalescent for the Wright- Fisher Model). MRCAs have been studied for many other models (see e.g. [4] for a more relevant model for human population, where the MRCA of a population of N individuals is almost log N generations ago). In the special case of Fleming-Viot processes, which genealogies at a fixed time are given by Kingman s coalescent, [7] have introduced a treevalued process as solution of a martingale problem that represents the evolving genealogies. In Moran model (finite population size) and in Wright-Fisher model with infinite population size, only two lineages can merge at a time and the genealogy at a given time is given by Kingman s coalescent. At time t the population is divided in two oldest families each one born from one of the two children of the MRCA. Let X t and X t denote the relative proportion of those two oldest families. One of this two oldest families will disappear (in the future). Let Y t be the relative size of the oldest family which will fixate: either Y t X t or Date: May, 9. Mathematics Subject Classification. 6J7, 6K35, 9D5. Key words and phrases. Wright-Fisher diffusion, MRCA, Kingman coalescent tree, resurrected process, quasy-stationary distribution. This work is partially supported by the Agence Nationale de la Recherche, ANR-BLAN and ANR-8-BLAN-9.

2 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE Y t X t. In a sense Y t is the size of the oldest family to which belongs the immortal line of descent (or the immortal individual). Notice that X t can be estimated (for example using DNA analysis of neutral mutations) at time t whereas Y t is not observable at time t. When X t hits or, that is when Y t hits, one of the two oldest families disappears and there is a change of MRCA. At this time two new oldest families appear. This corresponds to a jump of the processes X (X t,t R) and Y (Y t,t R). For the Wright-Fisher model with infinite population size, in between two jumps the process X is a Wright-Fisher (WF) diffusion on [,]: dx t X t ( X t )db t, where B is a standard Brownian motion. The two absorbing states and are reached in finite time. In between two jumps the process Y is a WF diffusion on [,] conditioned not to hit : dy t Y t ( Y t )db t + ( Y t )dt. The WF diffusion and its conditioned version have been largely used to model allelic frequencies in a neutral two-types population, see [3, 9, ]. Using the look-down representation for the genealogy introduced by Donnelly and Kurtz [8, 9], we prove rigorously, see Corollary., that at a jump time the law of X, µ, is the uniform distribution on [,], and that the law of Y, µ, is the size-biased distribution of µ, that is the beta (,) distribution. The process X can be seen as a resurrected WF diffusion with resurrection distribution µ. It is then easy to check that µ is also the invariant distribution of X. Indeed, according to Lemma. of [5] (see also the pioneer work of [4] in a discrete setting), µ is a quasy-stationary distribution (QSD) of a process killed when it reaches a set if and only if µ is the stationary distribution of the corresponding resurrected process which jumps with resurrection distribution µ when it reaches the set. See Section 3. for a precise statement. Then the conclusion follows as µ (resp. µ ) is a QSD of the (resp. conditioned) WF diffusion, see [3, 9] and also [3]. The only QSD distribution of the WF diffusion is the uniform distribution, see [3, p. 6], or [9] for an explicit computation. In Section 3, we check that the distribution µ is the only QSD for the conditioned WF diffusion, see Proposition 3.. A similar result is also true for the Moran model. In this case also, the QSD can be seen as the distribution of the size of one of the two oldest families. There is no such interpretation for the WF model for finite population, see Remark 3.. To establish Corollary., we use the look-down process, which gives a representation of the genealogy for the WF model of a population with infinite size. Following Pfaffelhuber and Walkobinger [7], we are also interested in the distribution of the following quantities: A: the birth time of the MRCA for the current population. τ : the time to wait before a change of MRCA happens (the hitting time of {,} for X). L N : the number of living individuals which will have descendants at time τ. Z {,...,L}: the number of living individuals which will become MRCA in the future. Y (,) the relative size of the oldest family to which belongs the immortal individual. X (,) the relative size of one of the two oldest families taken at random (with probability one half it has the immortal individual). Recent papers give an exhaustive study of birth dates and death times of MRCA, see [7] and also [3] (see also [] for genealogies of continuous state branching processes). In particular the birth dates of MRCA, as well as the death times of MRCA for the WF model, are distributed according to a Poisson process, see [7] and [7]. The distribution of (τ,l,z) is given in [7]. In particular, τ is an exponential random variable with mean. We give, see Theorem. below, the joint distribution of (A,τ,L,Z)

3 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 3 at time t conditionally on Y t or X t, where t is either fixed or an MRCA death time. The study of this conditional distribution is motivated by the fact that the relative size of the current two oldest families, X t, can be inferred from available DNA data at time t. By stationarity, for fixed t, this distribution does not depend on t. It is also the same, but for A, at the death time of an MRCA (the argument is the same as in the proof of Theorem in [7]). This property is the analogue of the so-called PASTA (Poisson Arrivals See Time Average) property in queuing theory, see [] for a review on this subject. We now state the main result of this paper. Let (E k,k N ) be independent exponential random variables with mean.we denote by T K k k(k+) E k and T T k k(k+) E k. Notice that T K has the law of the lifetime of Kingman s coalescent process. Theorem.. At a fixed time t or at the death time of an MRCA, we have: i) A is independent of (Y,X,τ,L,Z), and is distributed as T K at a fixed time and as T T at the death time of an MRCA. ii) Conditionally on Y, X and (τ, L, Z) are independent. iii) Conditionally on (Y, L), τ and Z are independent. iv) Conditionally on Y, we have X εy + ( ε)( Y ) where ε is an independent random variable such that P(ε ) P(ε ) /. v) Conditionally on Y, L is geometric with parameter Y. k(k+) E k, where (E k,k N ) are independent vi) Conditionally on (Y,L), τ kl exponential random variables with mean and independent of (Y,L). vii) For u [,], and a, if a E[u Z a Y,L a] u a + ( ) u + if a 3 a (k )(k + ) with the convention that. k We also give the first two moments of Z in Section conditionally on (Y,L), Y or X, see (8), () and (4), as well as its generating function conditionally on Y or on X, see Corollaries.9 and.. Our results also give a detailed proof of the heuristic arguments of Remarks 3. and 7.3 in [7]. From the conditional distribution of τ given in vi), we give its first two moments, see (9), and we recover the formula from Kimura and Ohta [, ] of its conditional expectation and second moment, see () and (3). See also (4) and (5) for the first and second moment conditionally on X. Notice the Laplace transform of τ conditionally on X x, given by (), solves the ODE: L X f λf, f() f(), where L is the generator of the WF diffusion: L X h(x) x( x)h (x) in (,). We also recover (Corollary.5) that τ is an exponential random variable with mean, see [7] or [7]. We shall end by a formula linking Z and τ. Proposition.. We have for all λ : [ ] [ ] E e λτ Y,L E e λt K E [ ( + λ) Z Y,L ]. In particular, we deduce that () E[e λτ X] E[e λt K ]E[( + λ) Z X].

4 4 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE Notice that we also immediately get the following relations for the first moments: () (3) E[τ Y,L] E[Z Y,L], E[τ Y,L] E[Z Y,L] 5E[Z Y,L] + 4π 3 8, using that E[T K ] for the first equality and that E[T K ] 4π 3 8 for the last. Remark.3. At the MRCA death time, we have a new MRCA which is born at A in the past and will die at τ in the future. On one hand, by looking at the death time of this new MRCA, Kingman s coalescent theory implies that A + τ is distributed as T K. As the coalescent times are independent of the structure of the coalescent tree, we get that A and τ are independent. On the other hand, there are Z living future MRCA and one new MRCA. We deduce that the new MRCA is the Z +-th point in the past of the birth dates of MRCA process. This latter is a Poisson point process with intensity, which is a direct consequence of the look-down representation of the genealogy. Intuitively, we could think that the new MRCA date of birth, A, is distributed as the sum of Z + independent exponential random variables with mean. This result is false, as one can easily check by computing Laplace transform; this is because the Poisson point process of the MRCA births is not independent of Z. However, this result is partially true at least for the conditional expectation thanks to (). The link between the distribution of T K and the joint distribution of τ and Z (which are independent conditionally on (Y,L)) is given by equation (). The rest of the paper is organized as follows. In Section, we state the results on the distribution of X,Y,L,τ using the look-down process and ideas of [7]. In Section 3, we present the QSD of the WF diffusion and the WF diffusion conditioned not to reach as the distribution of the relative size of one of the two oldest families at an MRCA change (Moran model and discrete WF are also considered). The proofs are postponed to Section 4.. Presentation of the main results on the conditional distribution.. The look-down process and notations. The look-down process and the modified look-down process have been introduced by Donnelly and Kurtz [8, 9] to give the genealogical process associated to a diffusion model of population evolution (see also [] for a detailed construction for the Fleming-Viot process). This powerful representation is now currently used We briefly recall the definition of the modified look-down process, without taking into account any spatial motion for the individuals. Consider an infinite size population evolving forward in time. Let E R N. Each (s,i) in E denotes the (unique) individual living at time s and level i. This level is affected according to the persistence of each individual: the higher the level is, the faster the particle will die. Let (N i,j, i < j) be independent Poisson processes with rate. At a jumping time t of N ij, the individual (t,i) reproduces and its unique child appears at level j. At the same time every particle having level at least j is pushed one level up (see Figure ). These reproduction events involving levels i and j are called look-down events (as j looks down at i). For any fixed time t, we can introduce the following family of equivalence relations R (t) (R (t ) s,s ): ir (t ) s j if the two individuals i and j living at time t have a common ancestor at time t s. It is then easy to show that the coalescent process on N defined by R (t) is the Kingman s coalescent. See Figure for a graphical representation.

5 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 5 Figure. A look-down event between levels and 3. Each individual living at level at least 3 before the look-down event is pushed one level up after it. Figure. The look down process and its associated coalescent tree, started at time t for the 5 first levels. At each look-down event, a new curve is born. We indicate at which level this curve is at time t. The curve of the individual who is at level 5 at time t is bold. We can describe the path of an individual born at level j at time s as a curve in E G k N[s k,s k+ ) {j + k},

6 6 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE where for k N, s k is the first birth time after s k of an individual with level less than j + k +. In particular G describes the different levels occupied by the individual born at time s. In fact, we shall identify an individual with its curve. We shall write b G s for the birth time of individual G. We say that d G lim k s k is the death time of this individual. We say an individual or a curve is alive at time t if b G t < d G, and k is the level of G at time t if (t,k) G. The set of all the curves, G, is a partition of E R {,...}. We write G t for the set of all curves alive at time t. Notice that the individual at level is immortal and that by definition, its curve R {} is not in G. An individual at level j is pushed at rate ( j ( ) to level j + (since there are j ) possible independent look-down events which arrive at rate and which push an individual living at level j). Since j /( j ) <, we get that any individual but the one at level dies in finite time. In the study of MRCA, some curves will play a particular role. We say that a curve G is a fixation curve if (b G,) G: the corresponding individual is born at level ; the initial look-down event was from to. For a fixed time t, let G t be the living MRCA of the whole population living at time t. Notice the birth time of the MRCA is A t inf{b G ;G G t } b Gt. It corresponds to the birth time of the highest fixation curve living at time t. Let Z t + denote the number of fixation curves living at time t: Z t is the number of future MRCA living at time t. We denote by L (t) > L (t) > > L Zt (t) the decreasing levels of the fixation curves alive at time t. Notice L(t) L (t) is the number of living individuals at time t which will have descendants at the next MRCA change. The joint distribution of (Z t,l (t),l (t),...,l Zt (t)) is given in Theorem of [7], and the distribution of Z t, the number of future MRCA alive at fixed time t, is given in Theorem 3 of [7]. We consider the partition of the population into the two oldest families given by the equivalence relation R (t) t A t. This corresponds to the partition of individuals alive at time t whose ancestor is either G t or the immortal individual. We shall denote by Y t the relative proportion of the sub-population (i.e. the oldest family) whose ancestor at time A t is the immortal individual, that is the oldest family which contains the immortal individual. Let X t be the relative proportion of an oldest family picked at random: with probability / it is the one which contains the immortal individual and with probability / the other one. By stationarity, we have that the distribution of H t (X t,y t,z t,l(t),l (t),...,l Zt (t)) does not depend on t. In between two MRCA deaths, the process (X t,t R) is a Wright- Fisher diffusion with generator x( x) x and the process (Y t,t R) is a Wright-Fisher diffusion conditioned not to hit with generator L x( x) x + ( x) x, see [, 9]. Notice the distribution of Z t conditionally on X t is of interest, as the relative proportion of the two oldest families at time t, X t, can be well estimated by DNA analysis if the (neutral) mutation rate is strong enough. We are interested in the law of H t at (random) times where the MRCA changes, as well as the distribution of the labels of the individuals of the same oldest family. The distributions of H t is the same if we consider a fixed time t or this random time (the argument is the same as in the proof of Theorem of [7]). This is the so-called PASTA (Poisson Arrivals See Time Average) property, see [] for a review on this subject, where the Poisson process considered corresponds to the times where the MRCA changes. For this reason, we shall omit the subscript and write H, and carry out the proofs at the death time of an MRCA... Size of the new two oldest families. We are interested in the description of the population, and more precisely in the relative size of the two oldest families at the time of death of an MRCA. More precisely, let G be a fixation curve and G be the next fixation

7 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 7 curve: the individual G is the next MRCA after the MRCA G. Let s b G be the birth time of G and (s k,k N ) be the jumping times of G. Notice that s b G corresponds to the birth of the MRCA G. Let N. Notice that at time s N, only the individuals with level to N will survive up to the death time d G of G. They correspond to the ancestors at time s N of the population living at time d G. We consider the partition into subsets given by R (s N ) s N s which corresponds to the partition of individuals alive at time s N with labels to N whose ancestor is either G or the immortal individual. Consider the ancestor at time s of the individual at level k {,...,N} and time s N, and let σ N (k) if it is the immortal individual and σ N (k) if it is G. Let V N N k σ N(k) be the number of individuals at time s N whose ancestor at time s is the immortal individual, see Figure 3 for an example. Notice that lim N V N /N will be the proportion of the oldest family which contains the immortal individual at the death time of the MRCA G. By construction the process (σ N,N N ) is Markov. Figure 3. In this example, at time s 5, we have σ 6 (,,,,,) and V 6 4. In order to give the law of (V N,σ N ) we first recall some facts on Pólya s urns, see []. Let S (i,j) N be the number of green balls in an urn after N drawing, when initially there was i green balls and j of some other color in the urn, and where at each drawing, the chosen ball is returned together with one ball of the same color. The process (S (i,j) N,N N) is a Markov chain, and for l {,...,N} ( ) P S (i,j) N i + l ( ) N (i + l )!(j + N l )!(i + j )! l (i )!(j )!(i + j + N )!

8 8 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE In particular, for i, j and k {,N + }, we have (4) P(S (,) N k + ) Theorem.. Let N. k (N + )(N + ) () The process ( + V N+,N N) is a Pólya s urn starting at (,). In particular, V N has a size-biased uniform distribution on {,...,N }, i.e. P(V N k) k N(N ) () Conditionally on (V,...,V N ), σ N is uniformly distributed on the possible configurations: {σ {,} N ;σ() and N k σ(k) V N}. Notice that, in general, if N 3, the process (V N +N,N N) conditionally on σ N can not be described using Pólya s urns. Results on Pólya s urns, see Section of [], give that (V N /N,N N ) converges a.s. to a random variable Y with a beta distribution with parameters (, ). This gives the following result. Corollary.. When the MRCA changes, the relative proportion Y of the new oldest family which contains the immortal individual is distributed as a beta (,). If one chooses a new oldest family at random (with probability / the one which contains the immortal individual and with probability / the other one), then its relative proportion X is uniform on (,). This is coherent with the Remark 3. given in [7]. Notice that Y has the size biased distribution of X, which corresponds to the fact that the immortal individual is taken at random from the two oldest families with probability proportional to their size..3. Level of the next fixation curve. We keep notations from the previous section. Let L (N) + be the level of the fixation curve G when the fixation curve G reaches level N +, that is at time s N. Notice that L (N) belongs to {,...,V N }. The law of (L (N),V N ) will be useful to give the joint distribution of (Z,Y ), see Section.5. It also implies (6) which was already given by Lemma 7. of [7]. The process L (N) is an inhomogeneous Markov chain, see Lemma 6. of [7]. By construction, the sequence (L (N),N ) is non-decreasing and converges a.s. to L defined in Section.. Proposition.3. Let N. i) For i k N, we have (5) P(L (N) i,v N k) and for all i {,...,N }, (N i )! N! k! (k i)! (6) P(L (N) i) N + N (i + )(i + ) N k N, ii) The sequence ((L (N),V N /N),N N ) converges a.s. to a random variable (L,Y ), where Y has a beta (,) distribution and conditionally on Y, L is geometric with parameter Y.

9 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 9 A straightforward computation gives that for i N P(L i) (i + )(i + ) This result was already in Proposition 3. of [7]. The level L + corresponds to the level of the MRCA, just after a change of MRCA. Recall L (t) is the level at time t of the second fixation curve. We use the convention that L (t) if there is only one fixation curve i.e. Z(t). Just before the random time d G of the death of the fixation curve G, we have L (d G ) L (d G ) L +. At a fixed time t, by stationarity, the distribution of L (t) does not depend on t, and equation (3.4) from [7] gives that L (t) is distributed as L. In view of Remark 4. in [7], notice the result is also similar for M/M/k queue where the invariant distribution for the queue process and the queue process just before arrivals time are the same. This is known as the PASTA property..4. Next fixation time. We consider the time d G of death of the MRCA. At this time, Y is the proportion of the oldest family which contains the immortal individuals. We denote by τ the time we have to wait for the next fixation time. It is the time needed by the highest fixation curve alive at time d G to reach. Hence, by the look-down construction, we get that (7) τ kl k(k + ) E k where E k are independent exponential random variables with parameter and independent of Y and L. See also theorem in [7]. Proposition.4. Let a N. The distribution of the waiting time for the next fixation time is given by: for λ R +, ( ) k(k + ) (8) E[e λτ Y,L a]. k(k + ) + λ Its first two moments are given by: (9) E[τ Y,L a] a We also have: for y,x (,) and λ R +, () E[e λτ Y y] ( y) () E[e λτ X x] x( x) ka and E[τ Y,L a] 8 a + 8 k a y l l kl [x l + ( x) l ] l k ( ) k(k + ). k(k + ) + λ kl ( ) k(k + ). k(k + ) + λ We deduce from (9) that E[τ L a], which was already in Theorem in [7]. Notice a that using (), we recover the following result. Corollary.5. The random variable τ is exponential with mean.

10 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE Using (9) and the fact that L is geometric with parameter Y, we recover the well known results from Kimura and Ohta [, ] (see also [3]): ( y)log( y) () E[τ Y y], y ( ( y)log( y) ) E[τ log( z) (3) Y y] 8 dz. y y z The following Lemma is elementary. Lemma.6. Let Y be a beta (,) random variable and X εy + ( ε)( Y ) where ε is independent of Y and such that P(ε ) P(ε ) /. Then X is uniform on [,]. Furthermore, if W is integrable and independent of ε, then we have E[W X] Xg(X) + ( X)g( X) where g(y) E[W Y y]. (4) (5) We also get, thanks to the above Lemma that: E[τ X x] (xlog(x) + ( x)log( x)), and ( E[τ log( z) X x] 8 xlog(x) + ( x)log( x) x dz z x ( x) x log( z) z ) dz..5. Number of future MRCA already living. We keep notations from Sections. and.3. We set Z Z dg the number of future MRCA living at time d G of death of the MRCA G. Let L L(d G ) + and (L,L,...,L Z ) (L (d G ),...,L Z (d G )) be the levels of the fixation curves at the death time of G. Recall notations from Section.. The following Lemma and Proposition.3 characterize the joint distribution of (Y,Z,L,L,...,L Z ). Lemma.7. Conditionally on (L,Y ) the distribution of (Z,L,...,L Z ) does not depend on Y. Conditionally on {L N}, (Z,L,...,L Z ) is distributed as follows: () Z if N ; () Conditionally on {Z }, L is distributed as L (N) +. (3) For N {,...,N }, conditionally on {Z,L N + }, (Z,L,...,L Z ) is distributed as (Z,L,...,L Z ) conditionally on {L N }. We are now able to give the distribution of Z conditionally on Y or X. Proposition.8. Let a. We have P(Z L ) and for k, (6) P(Z k Y,L a) k a + k 3 a (a i )(a i + ) ; <a k < <a <a i for all u [,], if a, (7) E[u Z a Y,L a] u a + ( ) u + if a, 3 a (k )(k + ) k with the convention that. We also have (8) E[Z Y,L a] and E[Z Y,L a] 8 4π a 3 8 a + 8 k k a

11 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS We deduce from ii) of Proposition.3 the next result. Corollary.9. Let y [,]. We have P(Z Y y) y, and, for all k N, (9) P(Z k Y y) k 3 ( y) for all u [,] () E[u Z Y y] ( y) + u y 3 <a k < <a < a (a + )(a + )y a a + a ya a l ( + with the convention that. We also have ( () E[Z Y y] + y ) log( y). y The next Corollary is a direct consequence of Lemma.6. k i ) u, (l )(l + ) (a i )(a i + ) ; Corollary.. Let x [,]. We have P(Z X x) x( x), and, for all k N, () P(Z k X x) k 3 x( x) for all u [,], (3) <a k < <a < E[u Z x( x) X x] x( x)+u 3 (a + )(a + ) ( x a + ( x) a ) k a with the convention that. We also have a + ( x a + ( x) a a ) a (4) E[Z X x] ( + xlog(x) + ( x)log( x)). l i (a i )(a i + ) ; ( ) u +, (l )(l + ) The second moment of Z conditionally on Y (resp. X) can be deduced from () (resp. (3)) or from (3) and (3) (resp. (5)). Some elementary computations give: P(Z X x) x( x), P(Z X x) [ x + ( x) x( x)ln(x( x)) ], 3 P(Z X x) [ ] 3 6 (x + ( x) ) ( x)ln( x) xln(x) + [ 3 x( x) π 3 + ln(x)ln( x) ] ln(x( x)). 3 We recover by integration of the previous equations the following results from [7]: P(Z ), P(Z ) 3 7 and P(Z ) π.

12 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE 3. Stationary distribution of the relative size for the two oldest families 3.. Resurrected process and quasy-stationary distribution. Let E be a subset of R. We recall that if U (U t,t ) is an E-valued diffusion with absorbing states, we say that a distribution ν is a quasy-stationary distribution (QSD) of U if for any Borel set A R, P ν (U t A U t ) ν(a) t, where we write P ν when the distribution of U is ν. See also [3] for QSD for diffusions with killing. Let µ and ν be two distributions on E\. We define U µ the resurrected process associated to U, with resurrection distribution µ, under P ν as follows: () U is distributed according to ν and U µ t U t for t [,τ [, where τ inf{s ;U s }. () Conditionally on (τ, {τ < },(U µ t,t [,τ ))), (U µ t+τ,t ) is distributed as U µ under P µ. According to Lemma. of [5], the distribution µ is a QSD of U if and only if µ is a stationary distribution of U µ. See also the pioneer work of [4] in a discrete setting. The uniqueness of quasy-stationary distributions is an open question in general. We will give a genealogical representation of the QSD for the Wright-Fisher diffusion and the Wright- Fisher diffusion conditioned not to hit, as well as for the Moran model for the discrete case. We also recall that the so-called Yaglom limit µ is defined by lim P x(u t A U t ) µ(a) A B(R), t provided the limit exists and is independent of x E\. 3.. The resurrected Wright-Fisher diffusion. From Corollary. and comments below it, we get that the relative proportion of one of the two oldest families at a change of MRCA is distributed according to the uniform distribution over [,]. Then the relative proportion evolves according to a Wright-Fisher (WF) diffusion with generator x( x) x. In particular it hits the absorbing state of the WF diffusion, {,}, in finite time. At this time one of the two oldest families dies out and there is (again) a change of MRCA. The QSD distribution of the WF diffusion exists and is the uniform distribution, see [3, p. 6], or [9] for an explicit computation. From Section 3., we get that in stationary regime, for fixed t (and of course at time when the MRCA changes) the relative size of one of the two oldest families taken at random, X t, is uniform over (,). Similar arguments as those developed in the proof of Proposition 3. yield that the uniform distribution is the only QSD of the WF diffusion. Lemma. in [5] implies there is no other resurrection distribution which is also the stationary distribution of the resurrected process The oldest family with the immortal individual. Recall that Y (Y t,t R) is the process of relative size for the oldest family containing the immortal individual. From Corollary., we get that Y at a change of MRCA is distributed according to the beta (,) distribution. Then Y evolves according to a WF diffusion conditioned not to hit ; its generator is given by L x( x) x + ( x) x, see [, 9]. Therefore Y is a resurrected Wright-Fisher diffusion conditioned not to hit, with beta (,) resurrection distribution. The Yaglom distribution of the Wright-Fisher diffusion conditioned not to hit exists and is the beta (,) distribution, see [9] for an explicit computation. In fact the Yaglom distribution is the only QSD according to the next proposition.

13 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 3 Proposition 3.. The only quasy-stationary distribution of the Wright-Fisher diffusion conditioned not to hit is the beta (,) distribution. Lemma. in [5] implies that the beta (, ) distribution is therefore the stationary distribution of Y. Furthermore, the resurrected Wright-Fisher diffusion conditioned not to hit, with resurrection distribution µ has stationary distribution µ if and only if µ is the beta (, ) distribution Resurrected process in the Moran model. The Moran model has been introduced in [6]. This mathematical model represents the neutral evolution of a haploid population of fixed size, say N. Each individual gives, at rate, birth to a child, which replaces an individual taken at random among the N individuals. Notice the population size is constant. Let ξ t denote the size of the descendants at time t of a given initial group. The process ξ (ξ t,t ) goes from state k to state k + ε, where ε {,}, at rate k(n k)/n. Notice that and N are absorbing states. They correspond respectively to the extinction of the descendants of the initial group or its fixation. The Yaglom distribution of the process ξ is uniform over {,...,N } (see [3, p. 6]). Since the state is finite, the Yaglom distribution is the only QSD. Let µ be a distribution on {,...,N }. We consider the resurrected process (ξ µ t,t ) with resurrection distribution µ. The resurrected process has the same evolution as ξ until it reaches or N, and it immediately jumps according to µ when it hits or N. The process ξ µ is a continuous time Markov process on {,...,N } with transition rates matrix Λ µ given by: Λ µ (,k) ( ) N µ(k) + {k} for k {,...,N }, N Λ µ k(n k) (k,k + ε) for ε {,} and k {,...,N }, N Λ µ (N,k) ( ) N µ(k) + {kn } for k {,...,N }. N We deduce from [4], that µ is a stationary distribution for ξ µ (i.e. µλ µ ) if and only if µ is a QSD for ξ, hence if and only if µ is uniform over {,...,N }. Using the genealogy of the Moran model, we can give a natural representation of the resurrected process ξ µ when the resurrection distribution is the Yaglom distribution. Since the genealogy of the Moran model can be described by the restriction of the look-down process to E (N) R {,...,N}, we get from Theorem. that the size of the oldest family which contains the immortal individual is distributed as the size-biased uniform distribution on {,...,N } when there is a change of MRCA. The PASTA property also implies that this is the stationary distribution. If, when there is a change of MRCA, we consider at random one of the two oldest families (with probability / the one with the immortal individual and with probability / the other one), then the size process is distributed as (ξ µ t,t R) under its stationary distribution, with µ the uniform distribution. Remark 3.. We can also consider the Wright-Fisher model (see e.g. []) in discrete time with a population of fixed finite size N, ζ (ζ k,k N). This is a Markov chain with state space {,...,N} and transition probabilities P(i,j) ( N j )( i N ) j ( i N ) N j.

14 4 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE There exists a unique quasy-stationary distribution, µ N (which is not the uniform distribution), see [6]. We deduce that the resurrected process ζ µ has stationary distribution µ if and only if µ µ N. Notice, that in this example there is no biological interpretation of µ N as the size of one of the oldest family at a change of MRCA. 4. Proofs 4.. Proof of Theorem.. We consider the set A N {(k,...,k N );k, for i {,...,N },k i+ {k i,k i + }}. Notice that P(V k,...,v N k N ) > if and only if (k,...,k N ) A N. To prove the first part of Theorem., it is enough to show that, for N and (k,...,k N+ ) A N+, (5) P(V N+ k N+ V N k N,...,V k ) For p and q in N such that q < p, we introduce the set: p,q {a (a,...,a p ) {,} p,a, { +k N N+ if k N+ k N, +k N N+ if k N+ + k N. p a i q}. Notice that Card ( p,q ) ( p q ). Hence to prove the second part of Theorem., it is enough to show that: for all (k,...,k N ) A N, and all a N,kN, i (6) P(σ N a V N k N,...,V k ) ( N k N ) We proceed by induction on N for the proof of (5) and (6). The result is obvious for N. We suppose (5) and (6) are true for a fixed N. We denote by I N and J N, I N < J N N +, the two levels involved in the look-down event at time s N. Notice that (I N,J N ) and σ N are independent. This pair is chosen uniformly so that, for i < j N+, P(I N i,j N j) (N + )N, (N i + ) P(I N i) (N + )N, (j ) P(J N j) (N + )N For a ( a(),...,a(n +) ) {,} N+ and j {,...,N +}, we set a j ( a(),...,a(j ),a(j + ),...,a(n + ) ) {,} N. Let us fix (k,...,k N+ ) A N+, and a ( a(),...,a(n + ) ) N+,kN+. Notice that {σ N+ a} {V N+ k N+ }. We first compute P(σ N+ a V N k N,...,V k ).

15 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 5 st case: k N+ k N +. We have: (7) P(σ N+ a V N k N,...,V k ) P(I N i,j N j,σ N+ a V N k N,...,V k ) i<j N+ i<j N+,a(i)a(j) i<j N+,a(i)a(j) i<j N+,a(i)a(j) (N + )N ( N k N (k N + )!(N k N )!, (N + )! P(I N i,j N j,σ N a j V N k N,...,V k ) P(I N i,j N j)p(σ N a j V N k N,...,V k ) (N + )N ) k N+(k N+ ) ( N k N where we used the independence of (I N,J N ) and σ N for the third equality, the uniform distribution of σ N conditionally on V N for the fourth, and that k N+ k N + for the sixth. Hence, we get P(V N+ k N + V N k N,...,V k ) P(σ N+ a V N k N,...,V k ) a N+,kN+ (8) nd case: k N+ k N. Similarly, we have: (9) P(σ N+ a V N k N,...,V k ) Hence, we get (3) P(V N+ k N V N k N,...,V k ) ) ( ) N (kn + )!(N k N )! k N+ (N + )! + k N N + i<j N+,a(i)a(j) (N + )N ( N k N (N + )N ( N k N ) (N + k N)(N k N ) (N k N)(k N )!(N k N + )!. (N + )! P(σ N+ a V N k N,...,V k ) a N+,kN+ ( ) N (N kn )(k N )!(N k N + )! k N+ (N + )! + k N N + )

16 6 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE Equalities (8) and (3) imply (5). Moreover, we deduce from (7) and (9) that, for k N+ {k N,k N + }, P(σ N+ a V N+ k N+,...,V k ) P(σ N+ a,v N+ k N+ V N k N,...,V k ) P(V N+ k N+ V N k N,...,V k ) ), ( N k N+ which proves that (6) with N replaced by N + holds. This ends the proof. 4.. Proof of Proposition.3. Theorem. shows that the distribution of σ N conditionally on V N is uniform. Then, if V N k, we can see L (N) as the number of draws (without replacement) we have to do in a two-colored urn of size N with k black balls until we obtain a white ball. Hence, for k {,...,N } and i {,...,k}, P(L (N) i V N k) k k N N k i + N k N i + N i (N i )! (k )! (N k). (N )! (k i)! This and Theorem. give (5). It is easy to prove by induction on j that for all j N, (3) i+j ki k! (i + j + )! (k i)! j!(i + ) Summing (5) over k {i,...,n } gives: P(L (N) i) (N i )! N!(N ) (N i )! N!(N ) (N i )! N!(N ) N + N N ki k! (N k) (k i)! [ ] N N k! (N + ) (k i)! (k + )! ((k + ) (i + ))! ki ki [ ] (N + )! (N i )!(i + ) (N + )! (N i )!(i + ) (i + )(i + ), where we used (3) twice in the third equality.

17 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS 7 Since (L (N),n N ) is non-decreasing, we deduce from Theorem. that the sequence ((L (N),V (N) /N),N N ) converges a.s. to a limit (L,Y ). Let i and v [,). We have: ( P L (N) i, V ) Nv N N v ( ) P L (N) i,v N k v which converges to ki Nv ki N Nv ki (N i )! N! k! (k i)! N k N k k N N k i + N i ( k ), N y i ( y)dy as N goes to infinity. We deduce that P(L i,y v) v yi ( y)dy for i N and v [,). Thus Y has a beta (,) distribution and conditionally on Y, L is geometric with parameter Y Proof of Proposition.4. The Laplace transform (8) comes from (7). To get the moments, we set g(λ) E[e λτ Y,L a] c k c k + λ, with c k k(k + ). We get k a g (λ) g(λ) k a c k + λ, and thus We also have Thus we get E[τ Y,L a] g () k a g (λ) g(λ) k a E[τ Y,L a] g () 4 (c k + λ) + g(λ) k(k + ) a l,k a k (k + ) k a l,k a 8 k(k + ) l(l + ) k a l k 8 k (k + ) k a k 8 k(k + ) k a 8 k a 8 k a k 8 a c k + λ k(k + ) c l + λ l(l + ) We get () from (8) and Proposition.3. We get () from () and Lemma.6.

18 8 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE 4.4. Proof of Corollary.5. We give a direct proof. We set c k k(k+) and b k c k (k )(k + ). Notice that c k + λ b k + ( + λ). We have from () E[e λτ ] a y dy ( y)y a a k a (a + )(a + ) k a b a a + (λ + ) ( + λ + λ, a k a+ c k c k + λ b a b a + ( + λ) c k c k + λ b k b k + ( + λ) ) where we used for the sixth equality that lim a k a+ k a+ b k b k + ( + λ) b k b k +(+λ) Proof of Lemma.7. Let us fix N. We have introduced L (N) + as the level of the fixation curve G when the fixation curve G reaches level N +, that is at time s N. We denote by Z N the number of other fixation curves alive at this time, and L (N) > L (N) > > L (N) Z N their levels. By construction of the fixation curves, the result given by Lemma.7 is straightforward for (V N /N,Z N,L (N),L (N),L (N),...,L (N) Z N ) instead of (Y,Z,L,L,...,L Z ). Now, using similar arguments as for the proof of the second part of Proposition.3, we get that ( (V N /N,Z N,L (N),L (N),L (N),...,L (N) Z N ),N ) converges a.s. to (Y,Z,L,L,...,L Z ) which ends the proof Proof of Propositions.8. Since conditionally on (Y,L), Z does not depend on Y thanks to Lemma.7, it is enough to compute the quantities P(Z k L a). Those quantities are given in [7], but we recall their proofs. By definition of L and Z, P(Z L ) and P(Z L a) for a. We suppose that a. We get P(Z k L a) by induction on k: for k, P(Z k L a) P(Z k,l a + L a) <a <a <a <a <a <a <a k < a <a P(Z k L a +,L a)p(l a + L a) P(Z k L a )P(L (a) a ) P(L (a k) )P(L (a k ) a k ) P(L (a) a ), where we have used Lemma.7 for the third and last equalities. Using (6), equation (6) follows. An expansion of ( ) a u k + (k )(k+) and (6) immediately give (7). The result on the first two moments (8) follows from (9) and Proposition..

19 ON THE TWO OLDEST FAMILIES FOR THE WRIGHT-FISHER PROCESS Proof of Theorem.. The proof of i) is a direct consequence of Kingman s coalescent (for fixed t) or of [33] (for the death time of MRCA) and the fact that the coalescent times (and thus the birth time of the MRCA A) does not depend on the coalescent tree shape. This last property can be deduced from [34], Section 3, see also [7]. In particular, A does not depend on (X, Y, L, Z) neither on τ which conditionally on the past depends only on the coalescent tree shape (see Section.4). Properties ii) and iv) are straightforward by construction of X. We deduce iii) from (7), as the exponential random variables are independent of the past before d G. Proposition.3 implies v). Proposition.4 implies vi) and Proposition.8 implies vii). The properties ii)-vi) are proved at time d G, but arguments as in the proof of Theorem in [7] yields that the results also holds at fixed time Proof of Proposition.. We set c k k(k + ) and b k c k (k )(k + ). Using (7), we have for a 3 E[( + λ) Z Y,L a] + λ a a + b k + ( + λ) 3 a b k k + λ a a + c k + λ 3 a b k k a k c k + λ c k. This equality is also true for a. And for a, we have E[( + λ) Z Y,L a]. The conclusion is then clear from vi) of Theorem. (see also (8)) as E[e λt K c k ] c k + λ Proof of Proposition 3.. Let µ be the beta (,) distribution. Using [5], it is enough to prove that µ is the only probability distribution µ on [,) such that µ is invariant for Y µ. Since x E x [τ] is bounded (see (4)), we get that E µ [τ] <. For a measure µ and a function f, we set µ,f f dµ when this is well defined. As E µ [τ] <, it is straightforward to deduce from standard results on Markov chain having one atom with finite mean return time (see e.g. [4] for discrete time Markov chains) [ that Y µ has a unique invariant probability τ ] measure π which is defined by π,f E µ f(y s ) ds /E µ [τ]. Hence we have [ τ ] (3) E µ f(y s )ds E µ [τ] π,f. Let τ n be the n-th resurrection time (i.e. n-th hitting time of ) after of the resurrected process Y µ : τ τ and for n N, τ n+ inf{t > τ n ;Y µ t }. The strong law of large numbers implies that for any real measurable bounded function f on [,), P µ a.s. τ n τn f(y s )ds n For g any C function defined on [,], the process M t g(y t ) π,f. t k Lg(Y s )ds is a martingale. Since M t g + t( g + g ) and E µ [τ] <, we can apply the optional stopping

20 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE theorem for (M t,t ) at time τ to get that [ τ ] g() E µ Lg(Y s )ds µ,g If a C function g λ is an eigenvector with eigenvalue λ (with λ > ) such that g λ (), we deduce from (3) that (33) µ,g λ λe µ [τ] µ,g λ. Let (a λ n,n ) be defined by a λ and, for n, a λ n(n + ) λ n+ (n + )(n + ) aλ n. The function n aλ nx n solves Lf λf on [,). For N N N(N + ) and λ, notice that P N (x) n aλ n xn is a polynomial function of degree N. By continuity at, P N is an eigenvector of L with eigenvalue N(N + )/, and such that P N () (as Lf() for any C function f). Notice that P (x) x, which implies that µ,p >. We deduce from (33) that E µ [τ] and µ,p N for N. As P N () for all N, we can write P N (x) ( x)q N (x), where Q N is a polynomial function of degree N. For the probability distribution µ(dx) x µ,p µ(dx), as µ,p N+ µ,p, we get that: (34) µ,q N, for all N. Since µ is a probability distribution on [, ], it is characterized by (34). To conclude, we just have to check that µ satisfies (34). In fact, we shall check that µ,g λ for any C function g λ eigenvector of L with eigenvalue λ such that g λ () and λ. Indeed, we have λ µ,g λ λ xg λ (x)dx x ( x)g λ (x)dx + x( x)g λ (x)dx [ x ( x)g λ (x)] (x( x) x )g λ (x)dx + x( x)g λ (x)dx x g λ (x)dx [ x g λ (x) ] xg λ (x)dx µ,g λ, which implies µ,g λ unless λ. References [] P. Brémaud, R. Kannurpatti, and R. Mazumdar. Event and time averages: a review. Adv. in Appl. Probab., 4():377 4, 99. [] C. Cannings. The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Advances in Appl. Probability, 6:6 9, 974. [3] P. Cattiaux, P. Collet, A. Lambert, S. Martinez, S. Méléard, and J. S. Martin. Quasi-stationarity distributions and diffusion models in population dynamics, 6. [4] J. T. Chang. Recent common ancestors of all present-day individuals. Adv. in Appl. Probab., 3(4): 38, 999. With discussion and reply by the author.

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22 JEAN-FRANÇOIS DELMAS, JEAN-STÉPHANE DHERSIN, AND ARNO SIRI-JEGOUSSE Université Paris-Est, École des Ponts, CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, Marne La Vallée, France. address: MAP5, Université Paris Descartes, 45 rue des Saints Pères, 757 Paris Cedex 6, France address: MAP5, Université René Descartes, 45 rue des Saints Pères, 757 Paris Cedex 6, France address: