508 Electronic Communications in Probability
which, for n tending to infinity, since ˜ K
n−1,i
eventually reaches m with probability one, converges to
˜ A
θ ,σ
= θ
m
X
i=1 m
X
j=1
c
m ji
z
j
∂ ∂ z
i
+
m
X
i, j=1
z
i
δ
i j
− z
j
∂
2
∂ z
i
∂ z
j
. This is the neutral-alleles-model with m types, which can be dealt with as in [9]. In particular,
for m going to infinity and θ = mκ kept fixed, one obtains, under appropriate conditions, the
infinitely-many-neutral-alleles-model, whose stationary distribution is the one parameter Poisson- Dirichlet distribution. This is consistent with the fact that the same limit applied to 3 yields the
Blackwell-MacQueen urn scheme.
When the mutation is governed by 13 we have b
i j
= ν { j} − δ
i j
cf., e.g., [11]. Also, from 14 we have c
i j
= µ
∗ i
{ j} − δ
i j
= ˜ K
−1 n−1,i
− δ
i j
. When the distribution ν
of the allelic type of a mutant is diffuse, these parameters yield
A
θ ,σ n
= − θ
K
n
X
i=1
z
i
∂ ∂ z
i
+
K
n
X
i, j=1
z
i
δ
i j
− z
j
∂
2
∂ z
i
∂ z
j
+ σ
K
n
X
i=1
˜ K
n−1,i K
n
X
j=1
− δ
i j
− 1
˜ K
n−1,i
− δ
i j
z
j
∂ ∂ z
i
+ R
n
n which in turn equals
A
θ ,σ n
=
K
n
X
i, j=1
z
i
δ
i j
− z
j
∂
2
∂ z
i
∂ z
j
−
K
n
X
i=1
θ z
i
+ σ ∂
∂ z
i
+ R
n
n .
18 Alternatively we could take the mutation to be symmetric, that is b
K
n
ji
= K
n
− 1
−1
, for j 6= i, so that
X
1≤ j≤K
n
b
K
n
ji
z
j
= − X
1≤ j6=i≤K
n
b
K
n
i j
z
j
+ X
1≤ j6=i≤K
n
b
K
n
ji
z
j
= 1 − z
i
K
n
− 1 − z
i
This choice yields a different operator A
θ ,σ n
but is equivalent in the limit for n → ∞.
Proposition 3.2. Let Z
n
· be a ∆
n
-valued process with infinitesimal operator A
θ ,σ n
defined by 17 and 18. Then Z
n
· is a Feller Markov process with sample paths in D
∆
n
[0, ∞. Proof. Denote with P
θ ,σ n
the joint distribution of an n-sized vector whose components are se- quentially sampled from 3. Let X
n
· be the Markov process corresponding to 8. Then X
n
· has marginal distributions P
θ ,σ n
see also Corollary 4.2 below. Also, given x ∈ X
n
, from the exchangeability of the P
θ ,σ n
-distributed vector it follows that ˜ T t,
x, B = ˜ T t, ˜
πx, ˜ πB, for every
permutation ˜ π of {1, . . . , n} and B ∈ BX
n
. By Lemma 2.3.2 of [5], X
n
t is an exchangeable Feller process on X
n
. The result now follows from Proposition 2.3.3 of [5]. The remainder of the section is dedicated to prove the existence of a suitably defined limiting
process, which will coincide with that in [18], and the weak convergence of the process of ranked frequencies. In the following section we will then show that the limiting process is stationary and
ergodic with respect to the two-parameter Poisson-Dirichlet distribution.
Countable representation for two-parameter family of diffusions 509
For every z ∈ ∆
n
with K
n
positive components, define ρ
n
: ∆
n
→ ∇
∞
as ρ
n
z = z
1
, . . . , z
K
n
, 0, 0, . . . 19
where z
1
≥ · · · ≥ z
K
n
are the the descending order statistics of z and ∇
∞
is 5. Let also ∇
n
= z = z
1
, z
2
, . . . ∈ ∇
∞
: z
n+1
= 0 and define the operator
B
θ ,σ n
=
K
n
X
i, j=1
z
i
δ
i j
− z
j
∂
2
∂ z
i
∂ z
j
−
K
n
X
i=1
θ z
i
+ σ ∂
∂ z
i
+ R
n
n ,
with R
n
as in 18 and domain DB
θ ,σ n
= {g ∈ C∇
n
: g ◦ ρ
n
∈ C
2
∆
n
}.
Proposition 3.3. The closure in C∇
n
of B
θ ,σ n
generates a strongly continuous, positive, conserva- tive, contraction semigroup {T
n
t} on C∇
n
. Given ν
n
∈ P ∆
n
, let Z
n
· be as in Proposition 3.2, with initial distribution
ν
n
. Then ρ
n
Z
n
· is a strong Markov process corresponding to {T
n
t} with initial distribution
ν
n
◦ ρ
−1 n
and sample paths in D
∇
n
[0, ∞. Proof. Let {S
n
t} be the Feller semigroup corresponding to Z
n
·. Then the proof is the same as that of Proposition 2.4 of [9]. In particular, it can be shown that {S
n
t} maps the set of permutation-invariant continuous functions on ∆
n
into itself. This, together with the observation that for every such f there is a unique g ∈ C∇
n
such that g = f ◦ ρ
−1 n
and g ◦ ρ
n
= f , allows to define a strongly continuous, positive, conservative, contraction semigroup {T
n
t} on C∇
n
by T
n
t f = [S
n
t f ◦ ρ
n
] ◦ ρ
−1 n
. Then ρ
n
Z
n
· inherits the strong Markov property from Z
n
·, and is such that E[ f
ρ
n
Z
n
t + s|ρ
n
Z
n
u, u ≤ s] = T
n
t f ρ
n
Z
n
s. Define now the operator
B
θ ,σ
=
∞
X
i, j=1
z
i
δ
i j
− z
j
∂
2
∂ z
i
∂ z
j
−
∞
X
i=1
θ z
i
+ σ ∂
∂ z
i
20 with domain defined as
DB
θ ,σ
= {subalgebra of C∇
∞
generated by functions ϕ
m
: ∇
∞
→ [0, 1], where
ϕ
1
≡ 1 and ϕ
m
z =
X
i≥1
z
m i
, m ≥ 2}. 21
Here ∇
∞
is the closure of ∇
∞
, namely ∇
∞
= z = z
1
, z
2
, . . . ∈ [0, 1]
∞
: z
1
≥ z
2
≥ · · · ≥ 0,
∞
X
i=1
z
i
≤ 1 which is compact, so that the set C∇
∞
of real-valued continuous functions on ∇
∞
with the supremum norm
f = sup
x∈∇
∞
| f x| is a Banach space. Functions ϕ
m
are assumed to be evalu- ated in ∇
∞
and extended to ∇
∞
by continuity. We will need the following result, whose proof can be found in the Appendix.
510 Electronic Communications in Probability
Proposition 3.4. For M ≥ 1, let L
M
be the subset of DB
θ ,σ
given by polynomials with degree not higher than M . Then B
θ ,σ
maps L
M
into L
M
. Then we have the following.
Proposition 3.5. Let B
θ ,σ
be defined as in 20 and 21. The closure in C∇
∞
of B
θ ,σ
generates a strongly continuous, positive, conservative, contraction semigroup {T t} on C∇
∞
. Proof. For f ∈ C∇
∞
, let r
n
f = f |
∇
n
be the restriction of f to ∇
n
. Then for every g ∈ DB
θ ,σ
we have |B
θ ,σ n
r
n
g − r
n
B
θ ,σ
g| ≤ n
−1
R
n
, where R
n
is bounded. Hence B
θ ,σ n
r
n
g − r
n
B
θ ,σ
g → 0,
g ∈ DB
θ ,σ
22 as n → ∞. From Proposition 3.3 and the Hille-Yosida Theorem it follows that B
θ ,σ n
is dissipa- tive for every n ≥ 1. Hence 22, together with the fact that
r
n
g − g → 0 for n → ∞ for
all g ∈ C∇
∞
, implies that B
θ ,σ
is dissipative. Furthermore, DB
θ ,σ
separates the points of ∇
∞
. Indeed ϕ
m
z is the m − 1-th moment of a random variable distributed according to
ν
z
= P
i≥1
z
i
δ
z
i
+ 1 − P
i≥1
z
i
δ , for z ∈ ∇
∞
, and ϕ
m
z = ϕ
m
y for m ≥ 2 implies all moments are equal, hence z = y. The Stone-Weierstrass theorem then implies that DB
θ ,σ
is dense in C∇
∞
. Proposition 3.4, together with Proposition 1.3.5 of [10], then implies that the closure of B
θ ,σ
generates a strongly continuous contraction semigroup {T t} on C∇
∞
. Also, since B
θ ,σ
ϕ
1
= B
θ ,σ
1 = 0, {T t} is conservative. Finally, 22 and Theorem 1.6.1 of [10] imply the strong semigroup convergence
T
n
tr
n
g − r
n
T tg → 0,
g ∈ C∇
∞
23 uniformly on bounded intervals, from which the positivity of {T t} follows.
We are now ready to prove the convergence in distribution of the process of ranked relative fre- quencies of types.
Theorem 3.6. Given ν
n
∈ P ∆
n
, let {Z
n
·} be a sequence of Markov processes such that, for every n ≥ 2, Z
n
· is as in Proposition 3.2 with initial distribution ν
n
and sample paths in D
∆
n
[0, ∞. Also, let
ρ
n
: ∆
n
→ ∇
∞
be as in 19, Y
n
· = ρ
n
Z
n
· be as in Proposition 3.3, and {T t} be as in Proposition 3.5. Given
ν ∈ P ∇
∞
, there exists a strong Markov process Y ·, with initial distribution
ν, such that E
f Y t + s|Y u, u ≤ s = T t f Y s, f ∈ C∇
∞
, and with sample paths in D
∇
∞
[0, ∞. If also ν
n
◦ ρ
−1 n
⇒ ν, then Y
n
· ⇒ Y · in D
∇
∞
[0, ∞ as n → ∞.
Proof. The result follows from 23 and Theorem 4.2.11 of [10].
Remark 3.7. The statement of Theorem 3.6 can be strengthened. From [18] it follows that the sample paths of Y · belong to C
∇
∞
[0, ∞ almost surely. Then [3] cf. Chapter 18 implies that ρ
n
Z
n
· ⇒ Y · in the uniform topology.
Countable representation for two-parameter family of diffusions 511
4 Stationarity
Denote with P
θ ,σ n
dx = ν dx
1 n
Y
i=2
θ + σK
i−1
ν dx
i
+ P
K
i−1
k=1
n
k
− σδ
x
∗ k
dx
i
θ + i − 1 24
the joint law of an n-sized sequential sample from the Pitman urn scheme 3, and with p
n
dx
i
|x
−i
the conditional distribution in 7.
Proposition 4.1. For n ≥ 1, let X
n
· be the X
n
-valued particle process with generator 9. Then X
n
· is reversible with respect to P
θ ,σ n
. Proof. Let q
x, dy denote the infinitesimal transition kernel given by qx, dy = lim
