Examples of Models Meeting the Assumptions

In this section we will present examples for which the assumptions in 3.6 hold.

First, we note that the reproduction mechanism defines our time rescaling but that the migration and coalescence mechanisms can otherwise be considered separately.

As we already noted in Examples 1.6: one way of modeling migration of individuals in the population model is to consider independent random walks on G. For the backwards-in-time spatial coalescent process we can just reverse those random walks to get new random walks which describe the migration of individuals backwards in time (special case of Remark 1.9).

Thus, for N PN we consider a family of i.i.d. random walks pX^i,NqiPrNs

on G with transition matrix P^N pP^Npx, yqqx,yPG. We assume that the random walks are irreducible and positively recurrent with stationary distri-bution π^N pπ^N_x qxPG. Furthermore we assume that the random walks are in equilibrium.

In this setting M^N and N0^N are given in the following way:

N0;x^N : ¸

iPrNs

1_tX^i,N

0 xu

and for x, y PG with xy and k PN0: Mk;x,y^M : ¸

iPrNs

1_tX^i,N

k x,X_k^i,N₁yu.

Proposition 3.8 (Migration Via Independent Random Walkers).

Set c^N :N^α and k_t^N : rt{c^Ns. Let P^N I

c^N ÑQ, π^N Ñπ (3.8)

as entrywise convergence of real numbers, with Q pQpx, yqqx,yPG a stable (i.e., finite diagonal entries), conservative (i.e. row sums are zero) generator matrix (i.e. nonegative off diagonal entries) with°

xPGQpx, xqπ_x   8where π is a distribution on G (represented as a row vector) with πQ0.

Then we have for all t PR and x, y PG:

R_t;x^N Ñπ_x and F_k^NN

t ;x,y Ñtπ_xQpx, yq (3.9) almost surely as N Ñ 8. Define Rt;x :π_x and Ft;x,y :tπ_xQpx, yq for all tPR , xyPG. Then 1. and 3. to 7. of Assumptions 3.6 are fulfilled. We even achieve almost sure pointwise convergence instead of just convergence of the finite dimensional distributions of the considered processes.

3.2. Examples of Models Meeting the Assumptions 31 The random variables 1_tX^i,N

t xu are mutually independent Binp1, π_x^Nq dis-tributed for iP rNs since the random walks are in equilibrium. We consider the array pE_N,iqNPN,iPrNs given by

E_N,i:1_tXt^i,N_xuπ_x^N.

Then the array is row-wise independent with EpE_N,iq 0 and |E_N,i| ¤ 1.

Thus we have by a strong law of large numbers for such arrays (see Theorem 4 in [28]) that

Next we show the convergence of F_k^NN

t ;x,y for x, y P G and t P R . We

It is noteworthy, that heuristically, due to the ergodic theorem for Markov chains, the inner sum will behave asymptotically likek^N_t π_x^NP^Npx, yqwhich would then converge to tπ_xQpx, yq forN Ñ 8. The issue with this approach is that we cannot apply the ergodic theorem since the transition matrix P^N and the stationary distribution π^N depend on N. Even worse, for large N the Markov chain X^N will move slower and therefore we cannot expect that the ergodic theorem could be applied uniformly in N. Alternatively we could try to apply the strong law for row-wise independent arrays again. Consider the array pZ_N,iqNPN,iPrNs given by

Due to independence of X^i,N the rows of pZ_N,iq are again independent and since the chains are in equilibrium:

PpX_j^i,N₁ x, X_j^i,N yq π^N_xP^Npx, yq (3.10)

and thusEpZ_N,iq 0 for all N PN, iP rNs. But we can no longer guarantee the uniform stochastic boundedness of Z_N,i which was necessary to apply Theorem 4 in [28].

Thus, we have to estimate the probabilities explicitely and show conver-gence from scratch. To ease notation we define for j P t1, . . . k_t^Nu:

A^i,N_j;x,y : tX_j^i,N₁ x, X_j^i,N yu.

Note that ifj1, . . . , jm P t1, . . . k_t^Nuare mutually different the event“_m

l1A^i,N_jl;x,y entails that the Markov chainX^i,N jumpes fromx toy on at leastm different occasions and thus

Moreover, if we expand this expression we get summands of the form that thej_lare mutually different. Taking expectations and applying the above estimation yields that each summand is bound by P^Npx, yq^|I|P^Npx, yq^k P^Npx, yq^m. Since there are m² summands we get

Ep|Y_j^i,N_1,...,jm;x,y|q ¤m²P^Npx, yq^m.

In the case that the summands are not mutually different we assumej₁ j_m without loss of generality. Then, since

|1_tA^i,N

(note that the expression inside the square is a random variable taking values inr0,1s) and thus

|Y_j^i,N_1,...,jm;x,y| ¤ |Y_j^i,N_{1,...,jm1;x,y}|

3.2. Examples of Models Meeting the Assumptions 33

we get Ep|Y_j^i,N

1,...,jm;x,y|q ¤ pm1q²P^Npx, yq^m1. Applying this consecutively until there are only mutually different indeces we get

Ep|Y_j^i,N

1,...,jm;x,y|q ¤a²P^Npx, yq^a. where aP rms is the number of mutually different j_l.

We will now estimate theL⁴ distance betweenF_k^NN

t ;x,y andk_t^Nπ_x^NP^Npx, yq. We have using that the Z_N,i are i.i.d. in iP rNs and have mean zero. We use this to expand the 4th moment:

E

Our goal is to show that the expectations above are bounded for N Ñ 8 which then shows that the forth moment goes to 0 for N Ñ 8 with order 1{N².

We have for the first expectation where the sums are always taken over mutually different indexes in rk_t^Ns (we use multinomial coefficients):

E Z_N,1⁴

The right hand side of this inequality is bounded in N PN due to c^N Ñ0

Next, we have to estimate the second moment. We get similar to the fourth moment (again the sums are taken over mutually different indices):

E Z_N,1²

Again, the right-hand side is bounded in N PN . Thus there is a constant D¥0 that does not depend on N such that The Markov inequality yields for ε¡0:

Pp|F_k^NN

t ;x,y k_t^Nπ_x^NP^Npx, yq| ¡εq ¤ D N²ε⁴.

Since the probability is summable the Borel-Cantelli Lemma yields forN Ñ 8:

|F_k^NN

We show that the Assumptions 3.6 are fulfilled as claimed. 1. of As-sumptions 3.6 holds trivially. The functions t ÞÑ Ft;x,y tπ_xQpx, yq are linear int and thus absolutely continuous with differential Ft;x,y¹ π_xQpx, yq. Together with (3.9) we get 3. of Assumptions 3.6 with almost sure pointwise convergence of processes. Note that we have for all t¥0:

¸

3.2. Examples of Models Meeting the Assumptions 35

Thus 4. of Assumptions 3.6 is fulfilled. SinceR0;xπ_x is a distribution on G we have °

xPGR0;x 1 and thus 5. of Assumptions 3.6 is fulfilled. Let xP G, t¥0. Next, we show the almost sure convergence of the sum:

¸

We can argue completely analogously to the proof of the almost sure con-vergence Ft;x,y^N ÑtπxQpx, yqby simply replacing P^Npx, yqwith 1P^Npx, xq and replacing “hittingy” with “avoiding x” in all events, to get D¥0 such that for all ε¡0:

And again by Borel-Cantelli and due to k^N_t π_x^N 1P^Npx, xq

the almost sure convergence of R_t;x^N now implies for N Ñ 8:

¸

and thus we have almost surely for N Ñ 8:

¸

yPGztxu

Ft;x,y^N Ñ tπxQpx, xq ¸

yPGztxu

Ft;x,y

and ¸

yPGztxu

Ft;y,x^N Ñ tπ_xQpx, xq ¸

yPGztxu

Ft;y,y.

Thus 6. of Assumptions 3.6 holds in terms of almost sure pointwise conver-gence. Due to Rt;x π_x ¡0 for all x PG, t P R 7. of Assumptions 3.6 is also fulfilled.

One way to ensure that the requirements of Proposition 3.8 are met, especially if we want to attain a specific random walk in the limit, is by using lazy walkers.

Example 3.9 (Independent Lazy Walkers). LetQ pQpx, yqqx,yPGbe a conser-vative, stable generator matrix. More preciselyQis a matrix with nonnegative off-diagonal entries (interpretation as rates), row sums equal 0 (conservative) and finite, nonpositive diagonal entries (stable). andπ a distribution on G (represented as a row vector) with πQ0 and sup_xPGQpx, xq   8. Let α¡0. For large N P N we can define a stochastic matrix (a matrix with nonnegative entries and row sums equal 1)P^N pP^Npx, yqx,yPGqvia

P^Npx, yq Qpx, yq

N^α ¤1 for xy and

P^Npx, xq 1 ¸

yx

P^Npx, yq 1 Qpx, xq N^α ¥0.

The stochastic matrix P^N describes a discrete-time Markov chain and thus a random walk on G. We call this random walk lazy since the probability Ppx, xq to remain at a given state x converges to 1 for N Ñ 8. In words:

for large N the random walk will stay in its current state for long periods of time. Furthermore the random walks for differentN essentially only differ in this holding probability in the sense that the conditional probability of moving from x toy given that the walker does not stay in x is the same for allN. This Markov chain has the equilibrium distribution π since we have

Im Dokument Convergence of the Genealogy of the Spatial Cannings Model (Seite 38-45)