Λ -coalescents arising in a population with dormancy * Λ -coalescents arising in a population with dormancy

Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with N dormant individuals waking up independently of each other according to a given distribution. Once an individual is awake, it starts reproducing at a constant rate. By the end of spring, all individuals are awake and continue reproducing independently as Yule processes during the whole summer. In the winter, N individuals chosen uniformly at random go to sleep until the next spring, and the other individuals die. We show that because an individual that wakes up unusually early can have a large number of surviving descendants, for some choices of model parameters the genealogy of the population will be described by a Λ -coalescent. In particular, the beta coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all Λ -coalescents that can arise in this framework.


Introduction
Dormancy is a widespread evolutionary strategy. A significant proportion of all living microorganisms are in some form of latent state of life [22]. The persistence of a latent subset of a population creates a buffer against selection [15,16] and contributes to the maintenance of biodiversity [18]. Recently, probabilistic models have been successful in explaining some aspects of this facet of evolution and opening interesting questions in the fields of population genetics and beyond. The impact of dormancy at different scales on the coalescent processes describing the genealogies of populations were investigated in [19,3,4,5]. Branching processes in random environment explain how dormancy can be selectively advantageous under fluctuating environmental conditions [6] while models from adaptive dynamics uncover that dormancy can arise from competition [7].
While some microbial organisms can last dormant for millions of years [27], most deactivation periods are much shorter. For example, mosquitos survive hostile environmental conditions by producing eggs which resist low temperatures and dry conditions [14]. Water triggers the eclosion of the eggs, so typically the newborns have favorable weather conditions and avoid the dry season. The earlier an individual reaches the reproductive state the higher its chances of having a large number of descendants in the next generation. This induces a selective pressure favoring mosquitos that are fortunate to be in contact with water soon after the rainy season starts.
The mechanisms for leaving a dormant state have been observed to be under selective pressure not only in the case of mosquitos [32] but also in different contexts such as experimental evolution; see, for example, figure 2 of [23] and [24]. In such experiments, individuals reproduce until the resources are depleted, and after that some of them are sampled and propagated to fresh identical media. The process of taking some bacteria from a depleted to a fully replenished environment induces a form of latency. Similar to the mosquitos, the bacteria undergoing the experiment are subjected to selective pressures resulting from the randomness of the activation time, and waking up rapidly from the dormant state provides an important advantage.
Wright and Vetsigian [33] postulated that the randomness in the times when individuals emerge from a dormant state could cause the distribution of the numbers of offspring produced by different individuals to become highly skewed. Indeed, they demonstrated in their bacterial experiments that "the heavy-tailed nature of the distribution of descendants can, in our case, be largely explained by phenotypic variability in lag time before exponential growth." It is well established in the probability literature [30,31] that heavy-tailed offspring distributions can affect the genealogical structure of the population. Whereas the genealogy of populations under a wide range of conditions can be modeled by Kingman's coalescent [21], in which only two lineages can merge at a time, the genealogies of populations with heavy-tailed offspring distributions are sometimes best described by coalescents with multiple mergers, also known as Λ-coalescents, which allow many ancestral lines to merge at once. The primary aim of this paper is to describe how the randomness in the times when individuals emerge from dormancy affects the genealogy of the population and, in particular, to understand the conditions under which the genealogy is best described by a Λ-coalescent. Wright and Vetsigian [33] report that "it is unlikely that the variance diverges with population size for the particular species and conditions we examined," indicating that Λ-coalescents probably do not arise in this instance. Nevertheless, we find that Λ-coalescent genealogies can appear if some rare individuals emerge from dormancy sufficiently early.

A model involving dormancy
We now describe a population model involving dormancy, which is very similar to the one introduced by Wright and Vetsigian in Section 2.7 of [33]. Note that we refer to the time period in the model as a day, which might be most appropriate when considering an evolutionary experiment involving bacteria, but in other contexts it will be more natural to think of this time period as lasting a year, and we will refer to different seasons. The model evolves as follows. We begin every day (or year) with a population of N dormant individuals. Each day (or year) has length T N and consists of three phases: • Activation phase (Spring): This phase has length t N . Each individual wakes up at some random time before t N . Once an individual is awake, it reproduces at rate λ N , meaning that, as in a Yule process, it gives birth to a new individual after an exponentially distributed time with rate λ N , and then the process repeats.
• Active phase (Summer): This phase has length T N − t N , and during this phase all individuals are awake and reproducing at rate λ N .
• Sampling phase (Winter): At time T N , we choose N individuals uniformly at random from the population to go to sleep until the start of the next day (or year), and all other individuals die.
Note that we have compressed the winter into a single time point, whereas it may be more realistic to think of the deaths as occurring gradually over a longer winter period. However, this will not substantially affect the results that follow.
The formalization is strongly inspired by the model introduced in [31]. For d ∈ Z and i ∈ [N ] := {1, . . . , N }, we denote by τ (d) i,N the random time in [0, t N ] when the ith individual at the start of day d becomes active. We assume that the random variables τ i,N do not depend on d, we will frequently drop the superscripts when we are concerned only with the distributions of these quantities. By a well-known fact about Yule processes, conditional on the activation time τ i,N , the random variable X i,N is geometrically distributed with parameter exp(−λ N (T N − τ i,N )). Moreover, conditional on (X 1,N , . . . , X N,N ), we see that ν i,N has a hypergeometric distribution with parameters S N , X i,N and N .
We now explain how we will represent the genealogy of this population. Let us assume that we sample n ∈ [N ] individuals at random on day 0. We define a discrete-time Markov chain (Ψ n,N (d)) ∞ d=0 taking its values on the set of partitions P n of [n], by letting Ψ n,N (d) be the partition of [n] such that i and j are in the same block if and only if the ith and jth individuals in the sample have the same ancestor on day −d. We will be interested in the asymptotic behaviour as N → ∞ of this ancestral process. The quantity where (·) n denotes the falling factorial, will play a crucial role. Note that c N is the probability that two individuals chosen uniformly at random from one generation have the same ancestor in the previous generation. It therefore establishes the appropriate time scale on which to study the process because after scaling time by 1/c N , the expected time for two randomly chosen individuals to trace their lineages back to a common ancestor will equal 1.
Tools for studying the limits as N → ∞ in models such as this one were developed in [30,26,31]. It is well-known that when the distribution of the family sizes ν 1,N is highly skewed, the genealogy can sometimes be described by a Λ-coalescent. The Λ-coalescents were introduced independently by Pitman [29] and Sagitov [30] and also appeared around the same time in work of Donnelly and Kurtz [12]. Whenever Λ is a finite measure on [0, 1], the Λ-coalescent is a P n -valued Markov process having the property that whenever there are b blocks, each possible transition that involves k of the blocks merging into one happens at rate When Λ is a unit mass at 0, we have λ b,2 = 1 and λ b,k = 0 whenever k ≥ 3, so we obtain Kingman's coalescent in which each pair of blocks merges at rate 1. When Λ({0}) = 0, the Λ-coalescent can be constructed from a Poisson point process on [0, ∞) × (0, ∞) with intensity dt × y −2 Λ(dy), in such a way that if (t, y) is a point of this Poisson point process, then at time t we have a merger event in which each lineage independently participates in the merger with probability y. This viewpoint is useful for understanding the results that follow. In particular, when Λ is a unit mass at one, we obtain a process known as the star-shaped coalescent in which all blocks merge after a waiting time whose distribution is exponential(1).
In Section 1.2, we consider a simple case in which there is no summer, and individuals can only wake up at the very beginning and at the very end of the spring. The genealogy of the population in this case is described in Theorem 1.1. In Section 1.3, we consider the case in which there is also no summer, but the rate at which individuals wake up from dormancy increases exponentially over time. The genealogy of the population in this case is stated in Theorem 1.3. In Section 1.4, we state Proposition 1.5, which gives conditions under which the summer period does not affect the genealogy of the population. In Section 1.5, we state Theorem 1.7, which characterizes the possible Λ-coalescents that can arise as limits in this model. After establishing some preliminary results in Section 2, we prove Theorem 1.1 in Section 3, Theorem 1.3 in Section 4, Proposition 1.5 in Section 5, and Theorem 1.7 in Section 6.

A two-point distribution for the exit time from dormancy
Here we describe the limiting genealogy in a simple instance of the model introduced above in which there is no summer, meaning that T N = t N , and the random variables τ i,N can take only the two values 0 and T N . We write Note that the progeny of individual i at time T N is given by . Let us assume in addition that for some κ, β > 0.
We denote by D S [0, ∞) the set of càdlàg functions from [0, ∞) to S, equipped with the usual Skorohod J 1 topology. We then have the following result. Throughout the paper, convergence of ancestral processes as N → ∞ refers to weak convergence of stochastic processes in D Pn [0, ∞). 1. If β > 1, the processes (Ψ n,N ( t/c N )) t≥0 converge as N → ∞ to the star-shaped coalescent.
This result can be understood as follows. Suppose an individual wakes up from dormancy unusually early, at time 0 rather than at time T N . Then this individual spawns a branching process in which individuals give birth at rate λ N , meaning that the number of descendants alive at time T N will be geometrically distributed with mean e λ N T N .
If instead (1.3) holds with β < 1, then the number of descendants at time T N of the individual who woke up early will be much smaller than N , making multiple mergers unlikely and giving rise to a Kingman's coalescent genealogy. On the other hand, if β > 1, then the number of descendants of this individual will be much larger than N , meaning that with high probability all of the sampled ancestral lineages merge, leading to a star-shaped genealogy.
Remark 1.2. The precise asymptotic behavior of (c N ) ∞ N =1 is characterized in Lemma 3.2.

Exponentially increasing rates of exit from dormancy
We now consider a possibly more realistic scenario in which the rate at which individuals exit from dormancy increases approximately exponentially over time. As in the model in Section 1.2, we assume there is no summer, so T N = t N . We assume that λ N = λ > 0 for all N . We also assume that is a sequence of i.i.d. random variables whose distribution does not depend on N . We assume there exist constants γ > 0 and c > 0 such that, using ∼ to denote that the ratio of the two sides tends to one, we have P (ζ 1 > y) ∼ ce −γy , as y → ∞,  The condition (1.7) ensures that the spring period is long enough to allow the individuals that emerge unusually early from dormancy to have a number of descendants which is a constant multiple of N , making multiple mergers of ancestral lines possible. We obtain the following result. 1. If a ≥ 2, the processes (Ψ n,N ( t/c N )) t≥0 converge as N → ∞ to Kingman's coalescent.
To understand this result, consider the distribution of the random variables X i,N . If we disregard the truncation at T N , which will turn out to have minimal effect, then conditional on ζ i = u, the distribution of X i,N is geometric with parameter e −λu . We assume for simplicity that the distribution of ζ i is exactly exponential with rate γ. Then, making the change of variables s = e −uλ and using Stirling's approximation as k → ∞ in the last step, we obtain Therefore, this result fits into the framework of Theorem 4 of [31], where it was established that beta coalescents describe the limiting genealogies in populations with these heavy-tailed offspring distributions. Note also that when 0 < a < 1, there will be multiple individuals in each day whose descendants comprise a substantial fraction of the population, which is why the limit of the ancestral processes is not a Λ-coalescent but rather a discrete-time process in which multiple groups of lineages merge in each time step.
Remark 1.4. The asymptotic behavior of (c N ) ∞ N =1 is considered in more detail in [31].
Lower bounds are given in Lemma 6 and Lemma 10 for the cases of a > 2 and a = 2 respectively. Lemma 13 yields lim N →∞ N a−1 c N = C for a suitable constant C when 1 < a < 2, and finally Lemma 16 states that lim N →∞ c N log(N ) = 1 for a = 1.

The effect of the summer on the genealogy
In Sections 1.2 and 1.3, we assumed there was no summer. We now consider how the inclusion of a summer period impacts the genealogy of the population. For this, we compare two populations whose spring has length t N and whose reproduction rate is λ N , which are subject to the same activation times. One population has a summer of length T N − t N > 0, and the other one has no summer, so that years have length t N . Let (Ψ n,N (d)) ∞ d=0 denote the ancestral process associated with a sample of n individuals at time 0 from the population without a summer, and let (Ψ n,N (d)) ∞ d=0 denote the corresponding ancestral process for the model with a summer. We obtain the following result.
When condition (1.8) holds, coalescence of lineages during the summer is sufficiently rare that the inclusion of the summer period does not affect the genealogy of the population in the limit. To understand why (1.8) is the correct condition, note that under the usual scaling ρ N = 1/c N , the probability that two randomly chosen lineages coalesce is c N = 1/ρ N . On the other hand, conditional on there being M individuals alive at the end of the spring, the probability that two randomly chosen lineages merge during the summer, regardless of the length of the summer, is bounded above by 2/(M + 1), as a consequence of Lemma 5.1. Therefore, condition (1.8) implies that the probability that two lineages coalesce in the summer is of smaller order than the probability that two lineages coalesce in the spring.
On the other hand, if (1.8) fails, then as long as the summer has a length that does not tend to zero as N → ∞, the summer period will cause additional pairwise mergers of ancestral lines. In the setting of Theorem 1.1, we therefore see from Lemma 3.2 that when β ≥ 1, condition (1.8) holds whenever lim N →∞ N 2 ω N = ∞, which ensures that early emergence from dormancy is not too rare. When β < 1, condition (1.8) holds when lim N →∞ N 2β ω N = ∞, which in view of (1.2) is only possible if β > 1/2. In the setting of Theorem 1.3, we can see from Remark 1.4 that condition (1.8) holds whenever a ≥ 2. In particular, when the genealogy is given by a Λ-coalescent, the multiple mergers happen rapidly enough that the inclusion of a summer period would not affect the genealogy of the population in the limit.
, and the distributions of the random variables . Therefore, to say that the Λ-coalescent can arise as a limit in this model means that there are choices of these parameters for which the rescaled ancestral processes converge to the Λ-coalescent. Theorem 1.7. It is possible for the Λ-coalescent to arise as the limit of the rescaled ancestral processes (Ψ n,N ( ρ N t )) t≥0 in the population model defined in Section 1.1 if and only if we can write where b 0 and b 1 are nonnegative real numbers and Λ is a measure on (0, 1) with density h with respect to Lebesgue measure, where for all y ∈ (0, 1) and We now make a few remarks concerning this result: 1. To relate this result to Theorem 1.1, note that the density of the random variable Y κ described in (1.4) is given by Therefore, if Λ has density h with respect to Lebesgue measure, where h is given by (1.10), then y −2 Λ (dy)/dy = ∞ 0 f κ (y) η(dκ). Note also that which is a constant multiple of the Beta(2 − a, a) density appearing in Theorem 1.3. 3. Typically, when rescaling time in the ancestral processes, we take ρ N = 1/c N , so that the expected time required for two randomly chosen lineages to merge equals 1. When ρ N = 1/c N , the measure Λ that appears in the limit must be a probability measure. Allowing arbitrary scaling constants (ρ N ) ∞ N =1 simply allows finite measures Λ that are not probability measure to arise in the limit. See Remark 6.1 for details.
and hence η =η. In combination with Theorem 1.7 we infer that {δ 0 , δ 1 } ∪ {Λ κ } κ>0 is the set of extremal points of the convex set of probability measures Λ appearing in Theorem 1.7. Note that (1.11) shows how any probability measure Λ appearing in Theorem 1.7 can be written as a mixture of the probability measures Λ κ .
We end this section with an alternate characterization of the measures Λ appearing in Theorem 1.7.

Genealogies in Cannings models
The model introduced in Section 1.1 is an example of a Cannings model. Cannings models, first introduced in [9, 10], have discrete generations and a fixed population size N , and the distribution of the family size vector (ν 1,N , . . . , ν N,N ) is required to be exchangeable. For such models we always define the coalescence probability c N as in (1.1). There is by now a standard set of tools for studying the genealogy of such models. To prove convergence of the ancestral process in our model to a Λ-coalescent, we will mainly use the following result, which is essentially Theorem 3.1 of [30]; see also Proposition 3 of [31].

Remark 2.2.
In place of condition 1 in Theorem 2.1, Sagitov [30] has the condition that . Also, in place of condition 2, Sagitov has the condition that for all integers a ≥ 2, we have However, the equivalence of (16) and (20) in [26] implies that we can consider (ν k,N ) 2 in place of (ν k,N − 1) 2 , and equation (17) of [26] implies that the limit is zero for all a ≥ 2 if and only if the limit is zero when a = 2. These observations lead to the formulation of the result given above.

Remark 2.3.
In principle, one could enrich the Cannings models with neutral genetic types to obtain a forward model for the evolution of type frequencies. To this end, we assign types to each individual at generation zero and propagate them forwards in time by imposing the rule that each individual inherits the type of its parent. Assume now that we have two types, 0 and 1, and denote by Z N d the frequency of type 1 individuals in generation d ∈ N. Let A N d denote the number of blocks in the partition Ψ n,N (d). One can use sampling duality as described in [8,Thm. 3.1] to discover that Assume now that the ancestral process (Ψ n,N ( t/c N )) t≥0 converges to the Λ-coalescent as N → ∞ (this assumption is satisfied for example in the setting of Theorems 1.1 and 1.3). Relation (2.1) allows one to prove uniform convergence on compact time intervals of the semigroup of (Z N t/c N ) t≥0 to that of the two-types Λ-Fleming Viot process, when applied to polynomials. A standard density argument and Theorem [20,Thm. 19.28] yields the convergence of the involved processes. This line of thought was formalized in the diplom thesis [28] and can be generalized to the infinitely many types setting in the sense of [13].
For the rest of this section, we consider a subclass of Cannings models in which the family sizes in each generation are obtained in the following way. We consider a sequence of independent and identically distributed positive integer-valued random variables X 1,N , . . . , X N,N , where X k,N denotes the number of offspring produced by the kth individual. Note that we do not allow the random variables X k,N to take the value zero. We let S N = X 1,N + · · · + X N,N be the total number of offspring. We then sample N of the S N offspring without replacement to form the next generation, and denote by ν k,N the number of offspring of the kth individual that are sampled. Note that the model introduced in Section 1.1 fits into this framework.
In this section, we will use the notation f (N ) Lemma 2.4 is useful for translating properties of the family sizes after sampling to properties of the family sizes before sampling, and vice versa.
In particular, Proof. The identity (2.2) follows from equations (19) and (21) in the proof of Lemma 6 of [31]. Both sides give the probability that, if we sample k 1 + · · · + k r individuals, the first k 1 are descended from the first individual in the previous generation, the next k 2 are descended from the second individual in the previous generation, and so on. The result (2.3) is a special case of (2.2). It remains to prove (2.4). Using (2.3) for the first inequality and Jensen's inequality for the third, we have which ends the proof.
Lemma 2.5 shows that when c N → 0, the distribution of the total number of offspring is highly concentrated around some value a N when N is large. This result implies that, when an unusually large family arises that will produce multiple mergers of ancestral lines, the total size of the remaining N − 1 families in that generation can be treated as being essentially nonrandom, so that the size of the large family will determine the proportion of ancestral lineages that merge in this generation.
Using the fact that S N ≥ N followed by Markov's inequality, we have which tends to zero as N → ∞ by assumption. (2.10) and therefore by Markov's inequality, We now establish the result using a second moment argument. Given r = 1/ for some positive integer and an integer k such that 1 ≤ k ≤ , define Note that U N = U 1,1,N , and that U k,r,N is the sum of between rN − 1 and rN + 1 random variables. Therefore, and using that W 1,N ≥ 1, we have (2.15) Let Z 1,N , . . . , Z N,N be independent random variables whose distribution is the conditional distribution of X 1,N given X 1,N ≤ δm N . Note that . (2.16) LetV N −1 = Z 2,N + · · · + Z N,N . Then, using Jensen's inequality, followed by (2.16) and then (2.11), for N large enough that δm N > 1 we have It follows from (2.9) that lim N →∞ (N − 1)/(N P (X 1,N ≤ δm N )) = 1, and therefore for sufficiently large N , we have Λ-coalescents arising in a population with dormancy Combining this result with (2.14) and (2.15), we get and therefore . .
Let ε > 0. Taking r = 1 and using Chebyshev's inequality along with (2.9) and the assumption that lim N →∞ c N = 0, we get That is, we have S N /a N → p 1, which is part 1 of the result. To prove part 2, we take r = 1/3 and note that we can have only if at least two of the random variables U k,r,N , for k ∈ {1, 2, 3}, are less than δa N . Let 0 < δ < 1/3. By (2.12), (2.17), and Chebyshev's inequality, Therefore, the probability that at least two of the random variables U k,r,N , for k ∈ {1, 2, 3}, are less than δa N is bounded above by This expression tends to zero faster than c N because δ < 1/3.
Proof. Suppose Y is the number of red balls drawn, when n balls are chosen from an urn containing b balls, of which r are red. Chvátal [11] showed that if ε > 0, then Conditional on X k,N and S N , we see that ν k,N can be interpreted as the number of red balls drawn, when N balls are chosen from an urn containing S N balls, of which X k,N are red. By applying (2.18) on the event that S N > N , and noting that on the event that S N = N , we have ν k,N = X k,N = 1 for all k, we get Conditional on S N > N , the probability that two randomly chosen individuals have the same ancestor is at least 2/ (N (N + 1)), so The result follows.
Proof. Let 0 < ε < 1. Let Φ N be the set of all ordered pairs (j, k) with 1 ≤ j < k ≤ N such that ν j,N ≥ N ε and ν k,N ≥ N ε. Let |Φ N | be the cardinality of Φ N . Note that at most 1/ε of the random variables ν k,N can exceed N ε, and so (2.23) (2.24) and

Define the events
where δ is the constant from Lemma 2.5. Also, note that X k,N /S N > ε/2 is equivalent to X k,N > (S N − X k,N )ε/(2 − ε), which on B N implies that X k,N > εδa N /2. Therefore, To prove (2.21), note that By exchangeability and the fact that ν 1,N + · · · + ν N,N = N , It now follows, using (2.20), that Because ε > 0 was arbitrary, the result (2.21) follows.
Note that P (A c N ) N −2 c N by Lemma 2.6. Also, because (2.8) and (2.11) imply X 1,N /a N → p 0, it follows from Lemma 2.
The other direction is more involved. By exchangeability and the inclusion-exclusion formula, {ν k,N > N x} . (2.29) Λ-coalescents arising in a population with dormancy (2.32) We now show that the first four terms on the right-hand side of (2.32) are small. We have P (A c N ) N −2 c N by Lemma 2.6, and P (B c N ) c N by part 2 of Lemma 2.5. We have P (ν 1,N > N x, ν 2,N > N x) N −2 c N by Lemma 2.7. Recall also thatS N −1 /a N → p 1, and therefore P (S N −1 ≤ (1 − δ)a N ) → 0, while we have P (X 1,N ≥ θa N ) N −1 c N for all θ > 0 by (2.8) and (2.11). Thus, we get Therefore, condition 3 of Theorem 2.1 holds, and the proof is complete.

Results for two-point distributions
We would like to prove the results for the simple model with a two-point distribution for the activation times introduced in Section 1.2. To this end, we begin with an observation for the asymptotic behaviour of the moments of the geometric distributions that will govern the numbers of offspring of the early bird in each of the three regimes. The more general results from Section 2 will be very useful.
Proof. Note first that Assume now that β > 1 and let us show part 1 of the lemma. Note that, for any y ≥ 0

1.
Thus, part 1 of the lemma follows from the dominated convergence theorem.
Assume now that β = 1 and let us prove part 2. Note that, for any y ≥ 0 Thus, part 2 follows using the dominated convergence theorem and making the substitu- In the remainder of the proof we assume that β < 1. Making the change of variable y = x 1/n /(1 − x 1/n ) in (3.1) and using standard properties of the floor function, we obtain and part 3 follows since α N ∼ κ −β N 1−β as N → ∞.
Proof. We begin with a few general observations. LetS and note that, thanks to Lemma 2.4, we have Since the expectation in the previous expression is smaller than one, the first statement follows from the assumption that N ω N → 0 as N → ∞.
Let us now have a closer look at the expectation in (3.2). Splitting on the event that there exists b i,N = 1 with i > 1 and its complement, and defining Clearly, 0 ≤ I + N ≤ I 0 N , and because N ω N → 0 it follows that I 0 N is the leading term in (3.3) and In addition, we have 5) and the second term is smaller than 1/N . Note that we have not used any assumptions on the distribution of G 1,N up to this point.
Parts 1 and 2 of the lemma now follow directly using Lemma 3.1. In the remainder of the proof we assume that (1.3) holds for β < 1. In order to prove part 3, we use Lemma

to see that
To complete the proof, we need to show that the second term in (3.5) converges faster to 0. Note that, using Lemma 3.1 with n = 1 in the last step, we get as N → ∞. This completes the proof.
In particular, in this case, the processes (Ψ n,N (t/c N )) t≥0 converge to Kingman's coalescent.
Proof. The second statement follows from the first one using [25,Thm. 4(b)]. Let us prove the first statement. Thanks to Lemma 2.4, and using that ( we obtain for N ≥ 3, and the result follows using Lemma 3.1 with n = 3.
Proof of Theorem 1.1. The case β < 1 is already covered by Lemma 3.3.
We want to apply Theorem 2.1 for the case of β ≥ 1. Conditions 1 and 2 hold by Lemma 3.2 and Lemma 2.7 respectively. Hence we are left to check condition 3.
Using Lemma 2.6, we obtain that for any x ∈ (0, 1) and any ε ∈ (0, Note that P (X 1,N /S N > y | A c ) ≤ P (1/N > y) = 0 for any given y and N sufficiently large. In addition we can bound P ( Observe that we can rewrite , since G 1,N has a geometric distribution with parameter (κβ) −1 .

Exponentially increasing rates of exit from dormancy
This section is devoted to the proof of Theorem 1.3, which characterizes the asymptotic genealogies in the model described in Section 1.3.

A comparison between the genealogies of two models
The main ingredient in the proof of Theorem 1.3 is a result that allows us to compare the genealogies of two populations constructed from the same sequence (ζ i ) N i=1 of i.i.d. positive random variables. The first model is the one described in Section 1.1, with no summer (i.e. t N = T N ), with λ N = λ > 0, and where T N − τ i,N = ζ i ∧ T N (the model in Section 1.3 is the special case where ζ 1 is exponentially distributed with parameter γ). In the second model, the family sizes (X 1 , . . . ,X N ) at the end of the year are i.i.d. and such that, conditionally on ζ i ,X i is geometrically distributed with parameter e −λζi (i.e. days start at time −∞). The vector of family sizes (ν 1,N , . . . ,ν N,N ) is obtained by sampling N individuals without replacement among theS N :=X 1 + · · · +X N present at the end of the year.
The next result will be useful for comparing the genealogies of the previously described models.
Proof. It follows directly from the fact that (X 1,N , . . . , X N,N ) and (X 1 , . . . , Let us now set The next result provides sufficient conditions for the limiting genealogies of the two models to coincide.
implies that, for all k 1 , . . . , k r ≥ 2, in the sense that if either limit exists, then so does the other, and the limits are equal.
Proof. Using Lemma 2.4 for the two models we obtain Thus, using Lemma 4.1 with f defined via f (x 1 , . . . , Consider the function f : . Note that f (x 1 , . . . , x N ) gives the probability that, if we sample k 1 + · · · + k r balls from an urn containing, for each i ∈ [N ], x i balls with label i, the first k 1 of them have label 1, the next k 2 balls have label 2, and so on. In particular, f ∞ ≤ 1. Thus, applying Lemma 4.1 with f , we obtain, for N sufficiently large,

Exponential model
In this section, we come back to the model described in Section 4, i.e. we assume that ζ 1 satisfies (1.6), that its law has no mass at zero, and that T N satisfies (1.7).
The next result provides the asymptotic behaviour of the tails ofX 1 .
It remains to prove that that the processes (Ψ n,N (d)) ∞ d=0 and (Ψ n,N (d)) ∞ d=0 , in the appropriate time scale, have the same limiting genealogy. For this, note that, under (1.6) and (1.7), conditions (4.1) and (4.2) are satisfied. Hence, the result follows combining Proposition 4.2 with Theorem 2.1 in [26].
In this section we prove Proposition 1.5, which, informally speaking, states that under mild hypotheses, the length of the summer T N − t N does not change the genealogy of the population. The key to the argument will be the following lemma about Pólya urns. Proof. Let D be the event that both balls have the same color. For k ∈ {0, 1, 2}, let R k be the event that k out of the two balls chosen are among the M balls that were in the urn at the beginning. Then P (R 2 |D) = 0 because the initial M balls all have different colors. By symmetry, we have P (R 1 |D) = 1/M . Finally, the well-known exchangeability of Pólya urns (see, for example, [17]) implies that P (R 0 |D) is the same as the probability that the first two balls added to the urn are the same color, which is 2/(M + 1). It follows that P (D) = 2 k=0 P (R k )P (D|R k ) ≤ 2/(M + 1).
Proof of Proposition 1.5. For the population with a summer, we want to bound the probability q N that two individuals chosen at random at time T N have the same ancestor at time t N . Let M = Y 1,N + · · · + Y N,N be the number of individuals alive at time t N . Between times t N and T N , the sizes of the families started by the M individuals at time t N evolve as independent Yule processes. Using a well-known connection between Yule processes and Pólya urns, one can see that the sequence keeping track of the families into which successive individuals are born follows the same dynamics as the sequence keeping track of the colors of successive balls added to a Pólya urn starting from M balls of different colors. Thus, by Lemma 5.1, the probability, conditional on M , that two individuals chosen at random at time T N have the same ancestor at time t N is at most 2/(M + 1), which means the unconditional probability satisfies Consider the canonical coupling of Ψ n,N andΨ n,N using the same activation times τ (d) i,N and the same birth times during the spring for both processes. Let D N be the first day (or year) that at least one pair of individuals in the sample in Ψ n,N finds a common ancestor during the summer, that is, in (t N , T N ]. By the coupling, the processes Ψ n,N andΨ n,N coincide until time D N , and by the observations above and condition (1.8), for all K > 0 we have which implies the result.

Classifying the possible limits
In this section, we prove Theorem 1.7 and Proposition 1.8. The first one classifies all possible Λ-coalescents that can arise as limits in the model introduced in Section 1.1.
The second one provides an alternate characterization of the measures Λ appearing in Theorem 1.7.
Remark 6.1. Suppose the ancestral processes (Ψ n,N ( ρ N t )) t≥0 converge to the Λcoalescent for all n, where Λ is a finite nonzero measure. Because Λ([0, 1]) is the rate at which two randomly chosen lineages merge in the Λ-coalescent and c N is the probability that two individuals have the same parent in the previous generation in the Cannings model, we must have ρ N c N ∼ Λ([0, 1]) as N → ∞. It follows that if we replace ρ N by 1/c N , then all of the transition rates in the limit will be multiplied by 1/Λ([0, 1]). Therefore, the ancestral processes (Ψ n,N ( t/c N )) t≥0 will converge for all n to theΛ-coalescent, wherẽ Λ = Λ/Λ([0, 1]) is a probability measure. Conversely, suppose the ancestral processes (Ψ n,N ( t/c N )) t≥0 converge for all n to the Λ-coalescent, where Λ is a probability measure. Let a > 0. Then, if we choose ρ N = a/c N , the ancestral processes (Ψ n,N ( ρ N t )) t≥0 will converge for all n to the aΛcoalescent. We can also, of course, obtain convergence to the zero measure by choosing In view of Remark 6.1, we may restrict our attention to the case in which ρ N = 1/c N , and to show that the possible Λ-coalescents that can arise as limits with this choice of scaling are precisely the probability measures satisfying (1.9) and (1.10). Proposition 6.2 shows how to obtain all of these Λ-coalescents as limits.
and distributions of the wake-up times (τ 1,N ) ∞ N =1 such that the ancestral processes (Ψ n,N ( t/c N )) t≥0 converge for all n to the Λ-coalescent.
Proof. We will construct the approximating Cannings model as a mixture of the simple two-point models discussed in Section 1.2 and deduce the convergence of their ancestral processes to the desired Λ-coalescent from Theorem 2.1 using the analogous observations for the simple models made in Section 3.   Like in (3.4) in the simple cases, we immediately see N the analogs of c N in the simple model with β = 1 + r, β = 1 and β = 1 − r, respectively, by Lemma 3.2 we know Assume more specifically that r ∈ (0, 1/2), and define In particular, N −2r α N → 0 as N → ∞.
We now distinguish different cases for the possible choices of Λ in (6.1). Let us first treat the case where b 0 > 0. Using X := d µ to denote that we are defining a random variable X to have the distribution µ, for sufficiently large N we can define If η is not the zero-measure, then α N is positive and the integral in the first term is an integral with respect to the probability measure 1 [N −r/2 ,N ] (κ)η(dκ)/α N . Otherwise the first term is simply zero.
Note that this σ 1,N is precisely a mixture of the simple cases discussed above. We now check the three conditions in Theorem 2.1 to obtain the desired convergence. Let G κ 1,N be a random variable having a geometric distribution on N with parameter (κN ) −1 . Conditioning on the possible values of σ 1,N in (6.2), and using (6.3) in the last step, which determines the asymptotic behavior of the last two summands, we then get Due to the integral, the first summand requires a bit more care. Equation (6.3) yields By (1.5), we have Y κ = κW/(κW + 1), where W has an exponential distribution with parameter 1. Therefore, using (1.4) and the fact that for any κ > 0. Like in the simple cases (3.5), we obtain , and the second term is smaller than 1/N . Using (6.4) Like in (3.1) and below, we see Using (6.6), for any N sufficiently large we can bound Standard calculus shows that κc(κ, N ) is decreasing in κ, and we can therefore estimate This allows us to estimate Since we assume ∞ 0 (κ 2 ∧ 1)η(dκ) < ∞, we have found an integrable upper bound. Lebesgue's dominated convergence theorem and the fact that c(κ, If we combine this with (6.7), we obtain It follows from (1.11) that and it was noted after (6.4) that N −2r α N → 0 as N → ∞. Therefore, plugging (6.9) into (6.5) yields We have P (X 1,N /S N > y | A c ) ≤ P (1/N > y) = 0 for N sufficiently large, whence we may ignore this term in further considerations. Also, since P (A ∩ B) ≤ N ω 2 N and we Let us consider the three limits separately. Reasoning as in the proof of Lemma 3.1, From Chebychev's inequality and (6.8), we get P (G κ 1,N /(G κ 1,N +N −1) > y) ≤ 12y −2 (κ 2 ∧1) when κ ∈ [N −r/2 , 1]. Therefore, we again obtain an integrable upper bound and can use Combining this, we obtain for every y ∈ (0, 1). Applying Lemma 2.6 as in (3.6) and using the identity As before, we calculate c N by conditioning on the possible values of σ 1,N and obtain N + 0.
Using (6.3), (6.9), and (6.10), we get which converges to 0 as N → ∞ and therefore the first condition in Theorem 2.1 holds. Again, the second condition follows directly from Lemma 2.7. To obtain the third condition, as before, we condition on the different values of σ 1,N and obtain Using (6.12) and (6.13) we obtain for every y ∈ (0, 1) as in (6.14) and therefore, for every x ∈ (0, 1), as in (6.16). With this we have verified the third condition of Theorem 2.1, which completes the proof.
Proof of Theorem 1.7. Proposition 6.2 and Remark 6.1 establish that all measures Λ that can be written as in (1.9), with the density of Λ being given by (1.10), can arise as limits of the ancestral processes in the model introduced in Section 1.1. It remains to show that these are the only measures that can be obtained.
In view of Remark 6.1, it suffices to consider the scaling in which ρ N = 1/c N . Note that if we denote by µ N the distribution of exp(−λ N (T N − τ 1,N )), then for all m ∈ N. (6.17) That is, the distribution of X 1,N is a mixture of geometric distributions. We need to show that if (6.17) holds, then the measure Λ that appears on the right-hand side of (2.26) must satisfy (1.9) and (1.10).
Note that a N ≥ N , which can be seen from (2.10) because the X k,n , and therefore the W k,N , are always at least 1. Hence, for all N and all p ≥ N −3/4 , we have   for all x ∈ (0, 1) such that Λ({x}) = 0. We claim that this convergence must hold for all x ∈ (0, 1). To see this, we assume, seeking a contradiction, that Λ({x}) = b > 0 for some x > 0. Choose u and v such that 0 < u < v < z. Choose C 1 and C 2 such that e − v ≤ C 1 e − u for all ≥ 0 and ∞ 0 e − u χ N (d ) ≤ C 2 for sufficiently large N . Choose 0 < δ < min{z − v, b/(4x 2 C 1 C 2 )}. Then, (6.20) implies that for sufficiently large N , we which is a contradiction. Therefore, Λ({x}) = 0 for all x > 0, and thus by (6.20) the Laplace transforms of the measures χ N converge pointwise to a limit on (0, ∞). By Theorem 8.5 of [1], it follows that the measures χ N converge vaguely to a limit measure χ, and the pointwise limit of the Laplace transforms of χ N is the Laplace transform of χ. That is, for all x ∈ (0, 1), we have We now use (6.15) and change the order of integration to get Now letting η be the push-forward of χ by the map x → 1/x, we see that the restriction of Λ to (0, 1) must have density h, as given in (1.10). To obtain the integrability condition, we note that because we are assuming ρ N = 1/c N , the measure Λ must be a probability measure, and therefore 1 ≥ Letting W have an exponential distribution with mean 1, the inner integral is E κW κW + 1 2 , which is easily seen to be bounded between C 3 (1 ∧ κ 2 ) and C 4 (1 ∧ κ 2 ) for some positive constants C 3 and C 4 for all κ > 0. This implies that ∞ 0 (1 ∧ κ 2 ) η(dκ) < ∞. It remains only to establish (6.19). For 0 < p < 1, the Taylor expansion log(1 − p) = − ∞ n=1 p n /n yields −p/(1 − p) ≤ log(1 − p) ≤ −p, and therefore e −p/(1−p) ≤ 1 − p ≤ e −p .
It follows that    If, on the other hand, N 1/4 /a N < p ≤ N −3/4 , note that the assumption a N > N 5/4 implies, by part 1 of Lemma 2.5, that P (S N ≥ 1 2 N 5/4 ) ≥ 1/2 for sufficiently large N , and therefore there is a positive constant C 5 such that c N ≥ C 5 /N . Therefore, such that either all elements are greater or all elements are less or equal than N 5/4 . Finally, to handle the case of general (a N ) ∞ N =1 , note that any subsequence will in turn have a subsequence which falls into one of the two categories we have proven to converge.