Corollary 2.12. Let x ∈ [0, 1]. We have, for all λ ≥ 0,
E [1 + λ
Z
|X = x] = x1 − x
∞
X
a= 2
x
a− 1
+ 1 − x
a− 1
+∞
X
a= 1
a− 1
Y
k= 1
kk + 1 + 2λ
kk + 1
, 24
with the convention Q
;
= 1. We have PZ = 0|X = x = 2x1 − x, and, for all k ∈ N
∗
, P
Z = k|X = x =
2
k− 1
3 x
1 − x X
1a
k
···a
1
∞
a
1
+ 1a
1
+ 2
x
a
1
−2
+ 1 − x
a
1
−2
k
Y
i= 1
1 a
i
− 1a
i
+ 2 ·
25 We also have
E [Z|X = x] = 2 1 + x logx + 1 − x log1 − x
. 26
The second moment of Z conditionally on Y resp. X can be deduced from 21 resp. 24 or from 4 and 14 resp. 16.
Some elementary computations give: P
Z = 0|X = x = 2x1 − x, P
Z = 1|X = x = 1
3
x
2
+ 1 − x
2
− 2x1 − x lnx1 − x
, P
Z = 2|X = x = 2
3 11
6 x
2
+ 1 − x
2
− 1 − x ln1 − x − x lnx +
2 3
x 1 − x
2 −
π
2
3 + 2 lnx ln1 − x −
1 3
lnx1 − x
. We recover by integration of the previous equations the following results from [26]:
P Z = 0 =
1 3
, P
Z = 1 = 11
27 and
P Z = 2 =
107 243
− 2
81 π
2
.
3 Stationary distribution of the relative size for the two oldest families
3.1 Resurrected process and quasi-stationary distribution
Let E be a subset of R. We recall that if U = U
t
, t ≥ 0 is an E-valued diffusion with absorbing states ∆, we say that a distribution ν is a quasi-stationary distribution QSD of U if for any Borel
set A ⊂ R, P
ν
U
t
∈ A|U
t
6∈ ∆ = νA t ≥
0, where we write P
ν
when the distribution of U is ν. See also [31] 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: 1. U
is distributed according to ν and U
µ t
= U
t
for t ∈ [0, τ
1
, where τ
1
= inf{s ≥ 0; U
s
∈ ∆}. 789
2. Conditionally on τ
1
, {τ
1
∞}, U
µ t
, t ∈ [0, τ
1
, U
µ t+τ
1
, t ≥ 0 is distributed as U
µ
under P
µ
. According to Lemma 2.1 of [4], 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 [13] in a discrete setting. The uniqueness of quasi-stationary distributions is an open question in general. We will give a ge-
nealogical representation of the QSD for the Wright-Fisher diffusion and the Wright-Fisher diffusion conditioned not to hit 0, 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
t→∞
P
x
U
t
∈ A|U
t
6∈ ∆ = µA ∀A ∈ BR,
provided the limit exists and is independent of x ∈ E\∆.
3.2 The resurrected Wright-Fisher diffusion
From Corollary 2.3 and comments below it, we get that the relative proportion of one of the two oldest families at a time when a new MRCA is established is distributed according to the uniform
distribution over [0, 1]. Then the relative proportion evolves according to a Wright-Fisher WF diffusion. In particular it hits the absorbing state of the WF diffusion, {0, 1}, in finite time. At this
time one of the two oldest families dies out and there a new MRCA is again established.
The QSD distribution of the WF diffusion exists and is the uniform distribution, see [12, p. 161], or [18] for an explicit computation. From Section 3.1, we get that in stationary regime, for fixed
t and of course at time when a new MRCA is established the relative size, X
t
, of one of the two oldest families taken at random is uniform over 0, 1.
Similar arguments as those developed in the proof of Proposition 3.1 yield that the uniform distri- bution is the only QSD of the WF diffusion. Lemma 2.1 in [4] implies there is no other resurrection
distribution which is also the stationary distribution of the resurrected process.
3.3 The oldest family with the immortal line of descent
