The Pólya sum process: Limit theorems for conditioned random fields

0.18 MB·English

by  Mathias Rafler

The Po´lya sum process: Limit theorems for
conditioned random fields
2 Mathias Rafler∗
1
0 TU Mu¨nchen
2 Zentrum Mathematik, M5

n Boltzmannstr. 3
a D-85747 Garching bei Mu¨nchen
J

0 January 24, 2012
2

]
R
P
h. In [16], Zessin constructed the so-called Po´lya sum process via partial in-
t tegration technique. This process shares some important properties with the
a
m Poisson process such as complete randomness and infinite divisibility. This
[ work discussesH-sufficient statistics for thePo´lya sumprocessas itwas done

for the Poisson process in [14].
1
v Keywords: Po´lya process, H-sufficient statistics, extreme points, Large devi-
0 ations
1
MSC: 60D05
4
4
.
1 1 Introduction
0
2
1 Giventhenumberofpointstobeplacedinaboundedregionofapolishspace,thePoisson
v: process places these points independently and identically distributed. In replacing this
i mechanism by a Po´lya urn-like scheme one obtains the Po´lya sum process. This point
X
process was constructed in [16] as the unique solution of the integral equation involving
r
a the Campbell measure
h(x,µ)CP(dx,dµ)= z h(x,µ+δx) ρ+µ (dx)P(dµ). (1.1)
Z ZZ
(cid:0) (cid:1)
With ρ instead of z(ρ+µ) this is Mecke’s characterisation of the Poisson process with
intensity measure ρ. The additional summand imports a reinforcement.
For the Poisson process with intensity measure ρ this means that independent of a
realized point configuration, a new point is added with intensity ρ. In contrast, the
mechanism of the Po´lya sum process rewards points contained in a configuration with
∗rafl[email protected]
1
their multiplicity, hence forces clustering of points. A similar effect show bosonic par-
ticles, and recently in [1], the distribution of the particles of the ideal Bose gas on the
possible states is connected with the Po´lya sum process. This point of view on the ideal
Bose gas is different from its position distribution derived in [6].
In [14], Nguyen and Zessin characterised the mixed Poisson processes as canonical
Gibbs states, and among those the Poisson process as the extremal ones. In the fol-
lowing we address a similar question for the Po´lya sum process for three different local
specifications yielding different families of mixed Po´lya sum processes. We identify the
set of extreme points as two one- and one two-parameter family of Po´lya sum processes.
This result paves the way to a Bayesian viewpoint on the mixed Po´lya sum processes
along the lines of [8]: Since the Gibbs states are mixed Po´lya sum processes directed by
some probability measure on a particular parameter set, one might estimate the distri-
bution of these parameters. Indeed, as will turn out, the distribution of the parameters
given a single observation, the posteriori measure is concentrated on a single point. This
is the characterization of ergodically decomposable priori measures obtained Glo¨tzl and
Wakolbinger.
We follow the approach to construct the Martin-Dynkin boundary as in Dynkin [3, 4]
and Foellmer [7]. Terms and notations are adopted to those in Dynkin [5].
In section 2 the basic setup is given including the Po´lya sum process and its repre-
sentation in 2.1. Thereafter in 2.2 the definition of local specification and H-sufficient
statistics is recalled. Finally in 2.3 we construct the local specifications and give the
main results, which are the Martin-Dynkin boundaries of the three local specifications.
Their proofs are contained in section 3.
2 P´olya sum process and results
2.1 P´olya sum process
Let X beapolish spaceand denoteby B = B(X) its Borel sets as well as by B = B (X)
0 0
the ringof boundedBorel sets of X. Furthermore let M(X) andM··(X) bethespace of
locally finitemeasuresandlocally finitepointmeasureson X, respectively, each of which
is vaguely polish, theσ-algebras F generated by the evaluation mappingsζ (µ)= µ(B),
B
B ∈ B . For B ∈ B denote by F the σ-algebra generated by ζ for all B′ ∈ B
0 0 B B′ 0
such that B′ ⊆ B, i.e. the σ-algebra of inside-B events. We call a probability measure
P on M(X) a random measure and if P is concentrated on M··(X) a point process.
Finally let F(X) be the set of bounded, non-negative and measurable functions on X
and F (X) ⊂ F(X) the subset of those functions in F(X) with bounded support.
b
By CP we denote the Campbell measure of P
CP(h) = h(x,µ)µ(dx)P(dµ).
ZZX×M(X)
CP determines P uniquely on M(X) \ {0} and therefore on M(X) since P is a law,
see e.g. [11] for details corresponding the Campbell measure. Of particular interest
have been disintegrations of the Campbell measure of the type of equation (2.1) below,
2
e.g. in [12], [10] and [13]. Recently, Zessin gave a construction method for the reverse
direction in [16]: Given a kernel η from M··(X) to M(X) is there a point process with
Papangelou kernel η, i.e. satisfies an integral equation of the type (2.1) with z(ρ+µ)
replaced by η? Aparticular example hegave is thePo´lya sumprocessS for ρ∈ M(X)
z,ρ
and z ∈ (0,1) which is the unique solution of the integral equation
CP(h) = z h(x,µ+δx) ρ+µ (dx)P(dµ), h ∈ F. (2.1)
ZZ
(cid:0) (cid:1)
Directly from equation (2.1) he showed that S firstly has independent increments
z,ρ
and secondly that the number ζ of points inside the bounded set B obeys a negative
B
binomial distribution. Also, only in usingrelation (2.1), one shows that S also satisfies
z,ρ
CP(h) = h(x,µ+ν)CL(dx,dν)P(dµ), (2.2)
ZZ
where L is the image of the measure ρ⊗τ on X ×N under the mapping (x,r) 7→ rδ
z x
and τ = zjδ . This shows that S is infinitely divisible with Levy measure L.
z j≥1 j j z,ρ
Again, the integral equation (2.2) determines S uniquely.
z,ρ
P
Since the intensity measure of S is
z,ρ
z
ν1 = ρ, (2.3)
S
z,ρ 1−z
directly from equation (2.2) we get that the Palm kernel Sx is given by, where ∆ = δ
z,ρ x δx
is the point process which realizes a point at x,
1−z
Sx = S ∗ zj∆∗j, (2.4)
z,ρ z,ρ z x
j≥1
X
i.e. the Palm kernel at x is the original process with an additional point with geo-
metrically distributed multiplicity at x. A second direct consequence of the integral
equation (2.2) or (2.1) is that the Laplace functional of S is given by
z,ρ
1−ze−f(x)
LS (f)= exp − log ρ(dx) . (2.5)
z,ρ 1−z
( )
Z
During the construction of the Po´lya sum process in [16], the image of the iterated
Po´lya sum kernels for B ∈ B , ρ denoting the restriction of ρ to B,
0 B
(m)
π (0,dx ,...dx )= ρ +δ +...+δ (dx )··· ρ +δ (dx )ρ (dx ) (2.6)
B 1 m B x1 xm−1 m B x1 2 B 1
(cid:0) (cid:1) (cid:0) (cid:1)
under the mapping (x ,...x )7→ δ +...+δ occurred. We keep terms of measures
1 m x1 xm
and write
M·· (N) = γ ∈ M··(N): jγ(j) = m ,
(m)
(cid:26) j≥1 (cid:27)
X
3
to denote the set of partitions of m elements represented by the number of families γ(j)
of size j. For given γ denote by the finite product
c(γ) = jγ(j)γ(j)!,
j∈N
Y
then m!/c(γ) is the number of permutations with cycle structure given by γ.
Main Lemma 1. Let ϕ be non-negative and F -measurable. Then
B
(m)
ϕ(δ +...+δ )π (0,dx ,...,dx )
x1 xm B 1 m
Z
m
m!
= ϕ(i δ +...+i δ )ρ(dx )···ρ(dx )
c(γ) 1 x1 k xk 1 k
Xk=1γ∈MX·(·m)(N) ZBk
γ(N)=k
m
m! 1
= ϕ(i′δ +...+i′δ )ρ(dx )···ρ(dx )
k! i′ ···i′ 1 x1 k xk 1 k
Xk=1 i′1,.X..,i′k≥1 1 k ZBk
i′+...+i′=m
1 k
where for each fixed γ with γ(N) = k, i ≤ ... ≤ i is the unique increasing sequence
1 k
with exactly γ(j) of the i ’s equal to j.
k
One recognises that if ϕ= 1, then the integral on the lhs equals ρ(B)[m] and the inner
sum on the rhs is apart from the weight ρ(B) the number of permutations with exacly
k cycles m , one recovers
k
(cid:2) (cid:3) m
m!
ρ(B)k = ρ(B)[m].
c(γ)
Xk=1 γ∈MX·(·m)(N)
γ(N)=k
The direct consequence is the fact that if z ∈ (0,1), then
zm
(m)
π (0,ϕ)
m! B
m≥1
X
1 zi1 ·...·zik
= ϕ(i δ +...+i δ )ρ(dx )···ρ(dx ).
k! i ·...·i 1 x1 k xk 1 k
Xk≥1 i1,.X..,ik≥1 1 k ZBk
2.2 H-sufficient statistics and local specifications
Equipped with ⊆, B is a partially ordered set. We assume that there is an increasing
0
sequence (Bn)n∈N of bounded sets such that n∈NBn = X and for each B ∈ B0 there
is a n ∈ N with B ⊆ B . Furthermore we construct a decreasing family E of σ-fields
0 n0 S E E
E ⊆ F indexed by the bounded sets. A family of Markovian kernels π = (π )
B B B∈B0
from M··(X) to M··(X) is called a local specification if
4
i) if B′ ⊆ B, then πEπE = πE;
B B′ B
E
ii) if f is F-measurable, then π f is E -measurable;
B B
E
iii) if f is E -measurable, then π f = f.
B B
E E
Given the family π , one is interested firstly in the convex set C(π ) of point processes
P with the property
P(ϕ|E ) = πE(·,ϕ) P-a.s. (2.7)
B B
E
and secondly in its extremal points. The former are the π -invariant measures. Since
B
E
B ⊆ B for each B ∈ B and some n depending on B, C(π ) agrees with the set of all
n0 0 0
point processes P such that equation (2.7) holds for each B .
n
A σ-field E is sufficient for C(πE) if the P ∈ C(πE) have a common conditional distri-
bution given E, i.e. there exists Q such that
µ
Q = P ·|E (µ) P-a.e. µ.
µ
(cid:0) (cid:1) E
According to [5], the tail-σ-field E = E is a H-sufficient statistic for C(π ), i.e. it
∞ n n
is a sufficient statistic and P µ : Q ∈ C(πE) = 1 for all P ∈ C(πE). In the situation of
µ
E E T
this work the family π = (π ) given by
(cid:0)n n (cid:1)
πE(µ,ϕ) =S (·|E )(µ)
n z,ρ Bn
E E
is a local specification and therefore E is H-sufficient for the set C(π ) of π -invariant
∞
E E
point processes. Its essential part ∆ is the set of extremal points of C(π ).
E
Thus the family π describes local laws and the aim is to determine global laws
consistent withthis description. Inparticular weget integral representations of elements
E E
in C(π ) in terms of the extremal points ∆ , since the latter set turns out to be a set
of Po´lya sum processes, we get a characterization of mixed Po´lya sum processes.
2.3 The tail-σ-fields and results
Since the multiplicity of points is rather the rule then the exception, there are several
possibilities to choose statistics. Having in mind a picture of building bricks, one may
measure on the one hand just the number of sites where they are placed, or on the next
handthetotalnumberofbrickswhichareplacedwithouttakingintoaccountthenumber
of sites, or on the third hand one may measure both. We call these three ensembles the
occupied sites ensemble, the total height ensemble and the size-and-height ensemble.
We start with the family Fˆ = (Fˆ ) of outside events,
B B∈B0
Fˆ = σ(ζ : B ⊆ B′ ∈ B )
B B,B′ 0
for B ∈ B , where ζ = ζ −ζ is the increment.
0 B,B′ B′ B
Of first interest is the family G = (G ) of σ-fields where G is generated by Fˆ
B B∈B0 B B
and σ(ξ ), where ξ µ := ζ µ∗ counts the number occupied sites of the configuration
B B B
µ ∈ M··(X) inside B ∈ B .
0
5
Theorem 2 (Martin-Dynkin boundary of the occupied sites ensemble). Let z ∈ (0,1)
and ρ∈ M(X) be a diffuse and infinite measure. The tail-σ-field G is H-sufficient for
∞
the family
C(πG)= S V(dw)|V probability measure on [0,∞) ,
z,wρ
(cid:26)Z (cid:27)
and the set of its extremal points is exactly the family
∆G = {S |0 ≤ w < ∞}.
z,wρ
Therefore when estimating the number of occupied sites of the particles, we get a
one-parameter family. If we replace the number of occupied sites in B by the number of
particles in B, i.e. if H = (H ) is the collection of σ-algebras H generated by Fˆ
B B∈B0 B B
and σ(ζ ), we obtain
B
Theorem 3 (Martin-Dynkin boundaryof the total height ensemble). Let z ∈ (0,1) and
ρ∈ M(X) be a diffuse and infinite measure. The tail-σ-field H is H-sufficient for the
∞
family
C(πH) = S V(dz)|V probability measure on [0,1)
z,ρ
(cid:26)Z (cid:27)
and the essential part of the Martin-Dynkin boundary is exactly the family
∆H = {S |0 ≤ z <1}.
z,ρ
Inthiscase,byestimatingthenumberofparticlespervolume,weadjusttheparameter
z. Because of (2.4), this increases the average multiplicity of the points as well as
by (2.3) the average number of occupied sites. Each of the tail-σ-fields G and H is
∞ ∞
a H-sufficient statistic for an one-parameter family of Po´lya sum processes. Finally we
combine both statistics and obtain a two-parameter family. Let
E := Fˆ ∨σ(ξ )∨σ(ζ ).
B B B B
Theorem 4 (Martin-Dynkin boundary of the size-and-height ensemble). Let z ∈ (0,1)
and ρ ∈ M(X) be a diffuse and infinite measure. The tail-field E is H-sufficient for
∞
the family
C(πE)= S V(dz,dw)|V probability measure on (0,1)×(0,∞)∪{(0,0)} ,
z,wρ
(cid:26)Z (cid:27)
and its extremal points are given exactly by all Po´lya sum processes for the pairs (z,wρ),
∆E = {S |0 < z < 1,0 < w < ∞}∪{δ }.
z,wρ 0
Remark that if any of the parameters z and ρ vanishes, then the Po´lya sum process
realizes the empty configuration almost surely and in this case set both parameters zero.
6
3 Proofs
In the very first part of this proof section we show the representation of the iterated
Po´lya sum kernels from lemma 1. We then turn to the theorems from section 2.3.
The basic structure of the proofs of theorems 2 – 4 is similar and therefore we start
with the general part recalling the lines of [5, 7] in section 3.2. The basic problem is
to identify the limits Q. In section 3.3 theorem 2 is proved by direct computation,
theorems 3 and 4 are proven by means of large deviations in 3.4 and 3.5. Their common
part, which consists mainly of the proofs of propositions 8 and 10 can be found in
section 3.6.
3.1 The iterated Polya sum kernel
For any γ ∈ M·· (N) with γ(N)= k define a measure Mγ on M··(X) by
(m)
Mγ(ϕ) = ϕ(i δ +...+i δ )ρ(dx )···ρ(dx ), ϕ ≥ 0,
1 x1 k xk 1 k
Z
where i ≤ ... ≤ i is the unique increasing sequence with exactly γ(j) of the i ’s equal
1 k k
to j. Then immediatly we get for F -measurable, non-negative ϕ that
B
m
ϕ(δ +...+δ )π(m)(0,dx ,...dx )= α(m)Mγ(ϕ)
x1 xm B 1 m γ
Z Xk=1γ∈MX·(·m)(N)
γ(N)=k
(m)
for some constants α which we identify right after the following lemma
γ
(m)
Lemma 5. The family of constants (αγ )m≥1,γ∈M·· (N) satisfies the recursion
(m)
α(m+1) = j γ(j)+1 α(m) 1 +α(m) 1 (3.1)
γ γ+δj−δj+1 {γ(j+1)≥1} γ−δ1 {γ(1)≥1}
j≥1
X (cid:0) (cid:1)
(1)
with initial value α = 1.
δ1
The recursion (3.1) states that for given family composition γ ∈ M·· (N), α(m+1)
(m+1) γ
is determined by all possibilities to introduce a new member to a population of size m
weighted with the size of families in the smaller population.
(m+1)
Proof. Letm ≥ 1andϕbeF -measurableandnon-negative. π putsanotherpoint
B B
(m)
totherealisation ofπ eitherintroducinganewoneorputtingittoanalreadyexisting
B
point. The first case leads to the first summand in the pre-last line with an additional
family of size 1. In the second case, putting the new point to an existing one means to
7
add it to an existing family. There are exactly jγ(j) members in families of size j, hence
exactly this number of possibilities to add the new point to a family of size j.
(m+1)
ϕ(δ +...+δ )π (0,dx ,...,dx )
x1 xm+1 B 1 m+1
Z
(m)
= ϕ(δ +...+δ ) ρ +δ +...+δ (dx )π (0,dx ,...,dx )
x1 xm+1 B x1 xm m+1 B 1 m
ZZ
m (cid:0) (cid:1)
= α(m) ϕ(δ +...+δ )Mγ(dx ,...,dx )ρ(dx )
γ x1 xm+1 1 m m+1
Xk=1γ∈MX·(·m)(N) ZBk+1
γ(N)=k
m m
+ α(m) ϕ(δ +...+δ +δ )Mγ(dx ,...,dx )
γ x1 xm xl 1 m
Xl=1Xk=1γ∈MX·(·m)(N) ZBk
γ(N)=k
m m
= α(m)Mγ+δ1(ϕ)+ jγ(j)α(m)Mγ−δj+δj+1(ϕ)
γ γ
Xk=1γ∈MX·(·m)(N) Xk=1γ∈MX·(·m)(N)Xj≥1
γ(N)=k γ(N)=k
m+1
= α(m+1)Mγ(ϕ),
γ
Xk=1 γ∈MX·(·m+1)(N)
γ(N)=k
where the coefficients satisfy the recursion (3.1).
Since α(1) = 1, one checks that α(m) = m! satisfies the recursion (3.1), which is the
δ1 γ c(γ)
first equation in Main Lemma 1. The second is an immediate consequence by noting
that
m
m!
Mγ(ϕ)
c(γ)
Xk=1γ∈MX·(·m)(N)
γ(N)=k
m
m! k 1
= ϕ(i δ +...i δ )ρ(dx )···ρ(dx )
k! γ i ···i 1 x1 k xk 1 k
Xk=1 γ∈MX·(·m)(N)(cid:18) (cid:19) 1 k ZBk
γ(N)=k
m!
= ϕ(i′δ +...i′δ )ρ(dx )···ρ(dx )
k! 1 x1 k xk 1 k
Xk≥1 i′1,.X..,i′k≥1 ZBk
i′+...+i′=m
1 k
where in the second line the i ’s are ordered and given by γ, which drops in the last line
j
since ϕ is symmetric when changing the order of that summation.
8
3.2 The main frame
We follow the programme given in [7] and give the basic common structure for the local
specifications related to G, H and E. Here weuseG and πG to representany of thethree
setups. ThefamilyπG = (πG) withπG = S (·|G )isafamilyofMarkoviankernels
B B∈B0 B z,ρ B
since for A ∈ Fˆ, S (A|G ) is σ(ζ )-measurable and furthermore a local specification
z,ρ B B
G G
since π clearly satisfies i) –iii). We want to determinethe setC(π )of pointprocesses
P which are locally given by
P(A|G ) = πG(·,A) P-a.s.
B B
for all B ∈ B and A ∈ F. S ∈ C(πG) in each ensemble ensures the non-emptiness of
0 z,ρ
G
C(π ). Following F¨ollmer and Dynkin, the Martin-Dynkin boundary is constructed in
G
the following way: If C (π ) is the set of all limits
∞
G
lim π (µ , ·),
B k
k→∞ k
where(µk)k∈N is asequencein M··, thenC∞(πG)is complete in thesetof all probability
measures on M··, therefore Polish. The measurable space C (πG) equipped with the
∞
G
Borel-σ-field is the Martin-Dynkin boundary of π . Since (Bk)k∈N is an increasing
sequence, we have for each P ∈ C(πG) and P-integrable ϕ
P(ϕ|G ) = lim πG (·,ϕ) P-a.s.
∞ B
k→∞ k
Therefore we firstly have to compute the P-a.s. existing weak limit
G G
Q = lim π (µ, ·),
µ B
k→∞ k
which is contained in C (πG) for P-a.e. µ by construction and in C(πG) for P-a.e. µ by
∞
the H-sufficiency. We will see that QG is a Po´lya sum process for P-a.e. µ, which implies
µ
that
P(ϕ|G ) = S (ϕ) P-a.s. (3.2)
∞ Z,Wρ
or
P(ϕ) = S (ϕ)P(dµ)
Z(µ),W(µ)ρ
Z
for suitable, possiblya.s.constant randomvariables Z andW on (M··(X),F) (even G -
∞
G
measurable) and that C(π ) consists of mixed Po´lya sum processes. Finally we identify
G G
the extremal points ∆ of C(π ) as the Po´lya sum processes among the mixed ones.
G
The important step is to determine the limits Q . In the first ensemble Q can be
µ µ
identified in showing the convergence of Laplace functionals, in the other two we use a
large deviation principle and a conditional minimisation procedure.
9
3.3 Occupied sites ensemble
In this first case we are only interested in the number of sites which are occupied, the
multiplicity does not matter. The infinite divisibility of S admits direct computations.
z,ρ
For B ∈ B , recall that ξ µ = ζ µ∗ is the number of points of the support of µ. Then
0 B B
G
the kernel π is given by
B
πBG(µ,ϕ) = Sz,ρ ϕ|GB (µ) = Sz,ρ ϕ(· +µBc)|ξB = ξBµ .
(cid:0) (cid:1) (cid:0) (cid:1)
On {ξ = n}, for F -measurable ϕ this is because of the diffuseness of ρ
B B
n
1
G
π (·,ϕ) = (3.3)
B −log(1−z)ρ(B)
(cid:18) (cid:19)
zi1+...+in
× ϕ(i δ +...+i δ ) ρ(dx )···ρ(dx ).
1 x1 n xn i ···i 1 n
ZBni1,.X..,in≥1 1 n
G
InsideB,π placespointsatexactly ξ siteswithindependentmultiplicities. Wedenote
B B
by W the number of occupied sites in B normalised by −log(1−z)ρ(B )
k k k
ξ µ
B
W µ := k .
k
−log(1−z)ρ(B )
k
Let
ξ µ
M·· = µ ∈M··(X) : lim Bk exists
ρ (cid:26) k→∞−log(1−z)ρ(Bk) (cid:27)
and set W on M·· the corresponding limit and 0 otherwise.
ρ
Proposition 6. Let ρ ∈ M(X) be diffuse and infinite and z ∈ (0,1). For any P ∈
C(πG), P(M ) = 1 and furthermore for ϕ ∈L1(P)
ρ
P ϕ|G (µ) = S (ϕ) P-a.e. µ. (3.4)
∞ z,W(µ)·ρ
Proof. Let (cid:0) (cid:1)
M·· = µ ∈ M·· : lim πG (µ, ·) exists .
G B
k→∞ k
(cid:26) (cid:27)
On{ξ = n}wehave for f ∈ F (X), andfor B ∈ B large enough suchthat suppf ⊆ B,
B b 0
πG ·,e−ζf = −log(1−z)ρ(B) −n
B
(cid:16) (cid:17) zi1+...+in
(cid:2) × e(cid:3)−i1f(x1)−...−inf(xn) ρ(dx )···ρ(dx )
1 n
i ···i
ZBni1,.X..,in≥1 1 n
n
1
= log(1−ze−f(x))ρ(dx)
−log(1−z)ρ(B)
(cid:20) ZB (cid:21)
log(1−z)ρ(B) n
1 1−ze−f(x) log(1−z)ρ(B)
= 1− log ρ(dx) .
−log(1−z)ρ(B) 1−z
" #
Z
10