The Ordered Model and Exchangeability

We now define an ordered population model with the same family size distribution, extending the ideas of Donnelly and Kurtz [DK99] in an obvious way. This time each particle will be attached a “level” from {1,2, . . .} in such a way that we obtain a nested coupling of approximating (Ξ, B)-Moran models as N tends to infinity. It will be crucial to show that this ordered model retains initial exchangeability, so that the limit as N → ∞of the empirical measures of the particle systems, at each fixed time, exists by De Finetti’s Theorem (Theorem 1.2).

We will refer to this model as the (Ξ, B)-lookdown model. If the population size is N, it will be described at timetby theE^N-valued random vector

X^N(t) := X₁^N(t), . . . , X_N^N(t) .

The dynamics including the distribution of family sizes and the mutation processes for each particle works as in the (Ξ, B)-Moran model above.

Again, in each reproduction step for each family, a parental particle will be chosen.

This “parent” then superimposes its type upon its family. This time, however, the parental particle will not be chosen uniformly among the members of each family (as in the (Ξ, B)-Moran model). Instead, the parental particle will always be the particle with the lowest level among the members of a family. Hence each family memberlooks down to their relative with the lowest level. The attachment of types to levels is then rearranged as follows (see Figure 2.1 for an illustration):

a) All parental particles of every family (including the trivial ones) will retain their type and level.

b) All levels of members of families will assume the type of their respective parental particle.

c) All levels that are still vacant will assume the pre-reproduction types of non-parental particles retaining their initial order. Once all N levels are filled, the remaining types will be lost.

In this way the dynamics of a particle, at levell, say, will only depend on the dynamics of the particles withlower levels. This consistency property allows us to construct all approximating particle systems, as well as their limit asN → ∞,on the same probability space.

Exchangeability of the modified (Ξ, B)-lookdown model is crucial to pass to the De Finetti limit of the associated empirical particle systems. For each N, we will show that ifX^N(0) is exchangeable, thenX^N is exchangeable at fixed times and at stopping times. The proof will rely on an explicit construction of uniform random permutations Θ(t) that mapsX^N toY^N.

Theorem 2.6. If the initial distribution of the population vector X₁^N(0), . . . , X_N^N(0) in the(Ξ, B)-lookdown model is exchangeable, then X₁^N(t), . . . , X_N^N(t)

is exchangeable for each t≥0.

(a) Parental particles retain type and level.

(b) Family members copy type of parental particle.

(c) Remaining particles re-tain their order and surplus particles get killed.

Figure 2.1.: The reproduction mechanism in the (Ξ, B)-lookdown model. The particles at levels 2 and 5 belong to the “star” family, whereas the particles at levels 3, 6 and 8 belong to the “triangle” family. The particles on the remaining levels belong to no family.

For the rest of this section, we omit the superscript N on the population models in an attempt not to get lost in notation.

The proof of Theorem 2.6 follows that of Theorem 3.2 in [DK99]. We will construct a coupling via a permutation-valued process Θ(t) such that

Y1(t), . . . , Y_N(t)

= X_Θ1(t)(t), . . . , X_ΘN(t)(t)

holds, Θ(t) is uniformly distributed on all permutations of {1, . . . , N} for each t and independent of the empirical process up to time tand the “demographic information”

in the model (see (2.12) for a precise definition).

It suffices to construct the skeleton chain (θ_m)_m∈N₀ of Θ. As a guide through the following notation, it is useful to occasionally remember that Θ(t) (and its skeleton chain) is built to the following aim:

Θ maps a position of an individual in the vector Y ((Ξ, B)-Moran-model) to the level of the corresponding individual in the ordered vector X

((Ξ, B)-lookdown-model).

Notation and ingredients ForN >0 letS_N denote the collection of all permutations of {1, . . . , N},P_N =P({1, . . . , N}) be the set of all subsets of {1, . . . , N}, andP_N,k⊂ P_N be the subcollection of subsets with cardinality k. For a set M, M(i) will denote thei-th largest element in M.

At timem (for the skeleton chain) letc_m be the total number of children. Leta_m be the number of families and cⁱ_m the number of children born to family i, hence

am

X

i=1

cⁱ_m =cm.

Note that cⁱ_m = 0 is allowed for some, but not all, i. These are the trivial families where only the parental particle is below level N and all potential children are above.

We need to keep track of these “one-member families” to match the rates of the particle system to those of the Ξ-coalescent later on.

Let θ₀ be uniformly distributed over S_N. For each m∈N, pick (independently, and independent ofθ0)

• Φ_m: a random set, uniformly chosen fromP_N,cm+am,

• φ¹_m, . . . , φ^a_m^m

: a random ordered partition of Φ_m such that each φⁱ_m has size cⁱ_m+ 1, and

• σ_mⁱ , i = 1, . . . , a_m: random permutations such that each σⁱ_m is uniformly dis-tributed over S_ci

m+1, independently of Φ_m and theφⁱ_m. Denote

• µⁱ_m:= minφⁱ_m=φⁱ_m(1), i∈ {1, . . . , a_m}, and

• write ∆m for the set of the highestcm integers from {1, . . . , N} \Sam

i=1µⁱ_m. Proceeding inductively, assume thatθ_m−1has already been defined. The permutation θm is constructed as follows: Let

• ν_mⁱ :=θ⁻¹_m−1(µⁱ_m),

• Ψm :=θ_m−1⁻¹ (∆m), and

• ψ_m¹, . . . , ψ_m^am

be a random ordered “partition” of Ψ_m such that |ψ_mⁱ | = cⁱ_m, chosen independently of everything else.

In view of our intended application of θ_m to transfer from the Moran model to the lookdown model, we will later on interpret these quantities as follows: In them-th event, µⁱ_m will be the level of the parental particle of familyiin the lookdown-model, andν_mⁱ will give the corresponding index in the unordered Moran model. ∆_m will specify the levels in the lookdown-model at which individuals die. We do not just pick the highest c_m levels, because we wish to retain parental particles. Ψ_m will be the corresponding indices in the Moran model. φ¹_m, . . . , φ^a_m^m

describes the family decomposition (includ-ing the respective parents) for this resampl(includ-ing event in the lookdown model, and ψ_mⁱ are the indices of the children in the i-th family in the Moran model. Thus, θ_m will mapφⁱ_m toψ_mⁱ ∪ {ν_mⁱ } (in a particular order).

(a) Initial permuta-tionθ_m−1

(b) The families are added

(c) The completed permutation in Ex-ample 2.7

Figure 2.2.: The construction of the new permutation from the old permutation carried out in Example 2.7

Finally, defineθ_m as follows: Put Ψ_m :={ν_m¹, . . . , ν_m^am} ∪ψ_m. On Ψ_m, θm(ν_mⁱ ) :=φⁱ_m σⁱ_m(1)

, i= 1, . . . , am, (2.9) and

θ_m ψ_mⁱ (j)

:=φⁱ_m σ_mⁱ (j+ 1)

∀j∈ {1, . . . , cⁱ_m} (2.10) for each i ∈ {1, . . . , a_m} with cⁱ_m 6= 0. On {1, . . . , N} \Ψ_m let θ_m be the mapping onto {1, . . . , N} \Φ_m with the same order asθ_m−1 restricted to {1, . . . , N} \Ψ_m, that is, whenever θ_m−1(i) < θ_m−1(j) for some i, j ∈ {1, . . . , N} \Ψ_m, then θ_m(i) < θ_m(j) should also hold.

Example 2.7. We consider a realisation of the m-th event of a population of size N = 8, as illustrated in Figure 2.1. There are am = 2 families (depicted by “triangle”

and “star”, respectively). The first familyφ¹_m={3,6,8}has sizec¹_m+ 1 = 3, the second φ²_m={2,5}has size c²_m+ 1 = 2. Hence, the set of levels involved in this birth event is Φm ={2,3,5,6,8} and µ¹_m = 3, µ²_m = 2 are the levels of the parental particles. Since there are no parental particles among the highest three levels, the particles at levels

∆_m={6,7,8}“die”.

Now let us assume that θm−1 is given as in Figure 2.2(a). Thus, ν_m¹ = 4, ν_m² = 1, and ψ_m ={3,5,7}. The set of indicesψ_m of individuals in the Moran model who will be replaced by offspring in this event is partitioned according to the family sizes. For example let ψ_m¹ ={3,7} andψ_m² ={5}.

We constructθ_m as follows: Letσ_m¹ = ^{1 2 3}_{3 1 2}

andσ_m² = ^{1 2}_{2 1}

. For the restriction ofθ_m to Ψ_m ={1,3,4,5,7}, equation (2.9) yields thatθ_m(4) =φ¹_m(3) = 8,θ_m(1) =φ²_m(2) = 5 and from (2.10) that θm(3) = θm ψ_m¹(1)

= φ¹_m σ_m¹(1 + 1)

= φ¹_m(1) = 3, θm(7) = θ_m ψ¹_m(2)

=φ¹_m σ¹_m(2 + 1)

=φ¹_m(2) = 6 andθ_m(5) =θ_m ψ_m²(1)

=φ²_m σ_m²(1 + 1)

= φ²_m(1) = 2. This leads to the partial permutation given in Figure 2.2(b).

Restricted to the complementary set{2,6,8},θm is a mapping onto{1,4,7}with the same order as θ_m−1 restricted to {2,6,8}. The resulting permutation θ_m is given in

Figure 2.2(c).

For notational convenience, let

χ_m:= (ν_m¹, ψ_m¹, . . . , ν_m^am, ψ_m^am),

which summarises the combinatorial information generated in the m-th step, that is the family structure we would observe in the Moran model.

Lemma 2.8. For each m the random variables χ₁, . . . , χ_m, θ_m are independent. Fur-thermoreθ_m is uniformly distributed overS_N and

Υ_m:=

am

[

i=1

{ν_mⁱ } ∪ψⁱ_m

is uniformly distributed over P_N,cm+am, and givenΥ_m each χ_m is uniformly distributed on all ordered partitions of Υm with family sizes consistent withcⁱ_m.

Proof. We prove the statement by induction. Denoting Fm = σ(θ_k, χ_k : 0≤k ≤ m), we have

E[f(θ_m, χ_m)| Fm−1] =E[f(θ_m, χ_m)|θ_m−1], (2.11) sinceθ_m andχ_m are only based onθ_m−1 and additional independent random structure.

This implies for any choice off:S_N →Randh_k:∪^Nn=1 {1, . . . , N}×P({1, . . . , N})n

→Rthat E

"

f(θ_m) Ym

k=1

h_k(χ_k)

#

= E

"

E[f(θ_m)h_m(χ_m)| Fm−1]

m−1Y

k=1

h_k(χ_k)

#

= E

"

E[f(θm)hm(χm)|θm−1]

m−1Y

k=1

h_k(χ_k)

#

= E[f(θ_m)h_m(χ_m)]

m−1Y

k=1

E[h_k(χ_k)]

where we used equation (2.11) for the second equality and the induction hypothesis for the third equality. It remains to show that θ_m and χ_m are independent and have the correct distributions.

θ_m−1 is uniformly distributed by the induction hypothesis and independent of the distributions of the parental-levels µⁱ_m and the “death-levels” ∆_m by construction. It is immediate from the construction that Φ_m and Υ_m are uniformly distributed over P_N,cm+am and the family structure χ_m is uniformly distributed among all admissible configurations.

Furthermore, conditioning on χ_m and Φ_m, the permutation θ_m is uniformly dis-tributed over all permutations that map Υm onto Φm. This follows from the fact that Φ_m is uniform on P_N,cm+am and that this set is uniformly divided into the families φⁱ_m. Since uniform and independent permutationsσⁱ_m are used for the construction of θm and the non-participating levels remain uniformly distributed,θm is uniform under these conditions.

Finally, conditioning on χ_m does not alter the fact that Φ_m is uniformly distributed overP_N,cm+am. This implies that given χm,θm is also uniformly distributed overS_N. Since

L(θ_m|χ_m) = unif(S_N) =L(θ_m),

θ_m and χ_m are independent of each other.

Now we turn to the proof of exchangeability forX^N.

Proof of Theorem 2.6. Suppose a realizationXof theN-particle (Ξ, B)-lookdown-model is given and let {tm} denote the times at which the birth events occur. The families involved in the m-th birth event are denoted by (φⁱ_m)1≤i≤am. Note that by definition of the lookdown-dynamics, the “ingredients” Φ_m, c_m, a_m, cⁱ_m, µⁱ_m,∆_m introduced earlier can be obtained from this, and their joint distribution is as discussed above.

Moreover, let the initial permutation θ₀ be independent of X and uniformly dis-tributed on S_N. Let σⁱ_m be independent of all other random variables and uniformly distributed onS_cⁱ_m+1, 1≤i≤a_m,m∈N.

Define θm as above, and

Θ(t) :=θm fortm ≤t < tm+1. Observe that, by Lemma 2.8,

Y1(t), . . . , YN(t)

:= X_Θ1(t)(t), . . . , X_ΘN(t)(t)

is a version of the (Ξ, B)-Moran-model. Note that in this construction “one-member families” are simply treated as non-participating individuals in the (Ξ, B)-Moran-model.

Y(t) depends only on Y(0), {χm}tm≤t and the the evolution of the type processes between birth and death events, so Θ(t), and hence Θ(t)⁻¹, is independent of

Gt:=σ (Y₁(s), . . . , Y_N(s)) :s≤t

∨σ(χ_m:m∈N) (2.12) due to Lemma 2.8. Therefore, we see from

X₁(t), . . . , X_N(t)

= Y_Θ−1

1 (t)(t), . . . , Y_Θ−1 N(t)(t) that X₁(t), . . . , X_N(t)

is exchangeable.

Corollary 2.9. Starting from the same exchangeable initial condition, the laws of the empirical processes of the(Ξ, B)-Moran-model and the(Ξ, B)-lookdown-model coincide.

The exchangeability property does not only hold for fixed times, but also for stopping times.

Theorem 2.10. Suppose that the initial population vectorsY^N(0)in the(Ξ, B)-Moran-model and X^N(0) in the (Ξ, B)-lookdown-model have the same exchangeable distri-bution, and let τ be a stopping time with respect to (Gt)_t≥0 given by (2.12). Then,

X₁^N(τ), . . . , X_N^N(τ)

is exchangeable.

Proof. We show that Θ(τ) is independent of theσ-algebraGτ (theτ-past) and uniformly distributed over S_N.

First, assume that τ takes only countable many values t_k, k ∈ N. Let A ∈ Gτ and h:S_N →R₊, then

E

h Θ(τ)

1A

=EX^∞

k=1

h Θ(t_k)

1A∩{τ=tk}

= X∞

k=1

Eh Θ(t_k)E1

A∩{τ=tk}

= Z

h(Θ)U(dΘ) X∞

k=1

E1A∩{τ=tk}

= Z

h(Θ)U(dΘ)E1A,

(2.13)

whereUdenotes the uniform distribution onSN. To see that the second equality holds, observe that for fixedt_k Θ(t_k) is independent of Gtk defined in equation (2.12).

By approximating an arbitrary stopping time from above by a sequence of discrete stopping times, we see that (2.13) holds in the general case as well. Now, exchangeability of X₁^N(τ), . . . , X_N^N(τ)

follows as in the proof of Theorem 2.6.

Remark 2.11. One can also define a variant of the (Ξ, B)-lookdown model that is more in the spirit of the ‘classical’ lookdown construction from [DK96]. Instead of a)–c) on page 26, at a jump time each particle simply copies the type of that member with the lowest level in the family to which it belongs(and no types get shifted upwards). This variant, up to a renaming of levels by the points of a Poisson process on R, has been considered by Taylor & V´eber (2008, personal communication) adapting [KR08] to the

‘simultaneous multiple merger’-scenario.

The same results as above hold for this variant with only minor modifications of the proofs. Note that the flavour of the lookdown process described above is easily adaptable to a set-up with time-varying total population size, which is not obvious for the other variant.

Im Dokument Multiple Merger Coalescents and Population Genetic Inference (Seite 34-40)