Iterated Palm Conditioning and Some Slivnyak-Type Theorems for Cox and Cluster Processes

Germany, April–May 2009:

Iterated Palm conditioning and some Slivnyak-type theorems for Cox and cluster processes

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by Olav Kallenberg

1 Conditional expectations (Kolmogorov 1933): Given a random element ξ in (S, S): ¯ ¯ ¯ E[ η; ξ ∈ ds ]¯ −1 ¯ E[ η | ξ ]s = ¯ , s ∈ S a.e. P ◦ ξ . P {ξ ∈ ds} ¯S Given a sub-σ-ﬁeld F in (Ω, A,P ): ¯ ¯ ¯ E[ η; dω ]¯ ¯ EF η = E[ η | F ]ω = ¯ , s ∈ S a.e. P, P (dω) ¯F ¯ ¯ ¯ P (A ∩ dω)¯ ¯ PF (A) = P [ A | F ]ω = ¯ , s ∈ S a.e. P. P (dω) ¯F

Interpretations of EF η: • projection of η onto L1(F) • expected value of η when F is “known”

2 Conditional distributions (Doob 1938): For any random elements ξ and η in Borel spaces S and T , there exists a probability kernel µ from S to T with

P [ η ∈ B | ξ ]s = µ(s, B), s ∈ S,B ∈ T . In other words, µ(s, B) is

• a measurable function of s for ﬁxed B, • a probability measure on T in B for ﬁxed s.

Writing L(ξ) = ν gives the disintegration Z Z Ef(ξ, η) = ν(ds) µs(dt) f(s, t) ≡ (ν ⊗ µ)f. Interchanging the roles of S and T gives the dual version: L(ξ, η) = ν ⊗ µ =∼ ν0 ⊗ µ0.

3 Palm distributions (Palm 1943): For any random measure ξ on S (kernel from Ω to S), we consider the Campbell measure with dual disintegrations

∼ P ⊗ ξ = Eξ ⊗ Pξ

(provided the intensity Eξ is σ-ﬁnite), deﬁning the Palm

s distributions P [A k ξ ]s = Pξ (A). Conditioning approach:

• when ξ = δτ , take P [A k ξ]s = P [A|τ]s, • when ξS is constant in (0, ∞), choose τ with P [τ ∈ · | ξ]

= ξ/ξS, and deﬁne P [A k ξ]s = P [A|τ]s, • in general, choose τ with P˜[τ ∈ · | ξ] = ξ, and deﬁne

P [A k ξ]s = P˜[A|τ]s.

4 Local conditioning (Matthes 1963, K83, K08): Say that ξ is a simple point process on S if

X ξ = k δτk a.s. for some a.s. distinct random points τ1, τ2,... in S. Then

P [A k ξ]s = lim P [A | ξB > 0] a.e. B↓s

Though no such approximation is valid in general, it does hold in some interesting special cases, including

• local time random measures on R+, • Dawson–Watanabe superprocesses in Rd, d ≥ 2.

5 Iterated conditional expectations (K09):

The product EF EG is again a linear contraction operator on L1. These conditions are equivalent:

1 • EF EG = EGEF a.s. on L , •F⊥⊥G (conditional independence). F ∩ G

Then the product equals EF∩G. Standard examples:

• F⊥⊥G (independence), • F ⊂ G (tower property),

• EFσ EFτ = EFσ∧τ for F-optional times σ and τ.

6 Iterated conditional distributions (K09):

Deﬁne the probability kernel (PF )G on Ω by

(PF )G(ω) = (P [ · | F]ω)[ · | G]ω

(diagonal values !). Then a.s. (under regularity conditions),

• (PF )G = (PG)F = PF∨G,

• PF = EF (PG)F , PG = EG(PF )G.

This amounts to a change of conditioning σ-ﬁeld:

PG ; PF∨G ; PF ,PF ; PF∨G ; PG

7 Iterated Palm conditioning (K09):

Deﬁne the kernels (PF )ξ and (Pξ)F from Ω × S to Ω by

(PF )ξ(ω, s) = (P [ · |F]ω)[ · k ξ]s,

(Pξ)F (ω, s) = (P [ · k ξ]s)[ · |F]ω.

Then a.e. (under regularity conditions),

• (PF )ξ = (Pξ)F = (P ⊗ ξ)F⊗S,

• Pξ = Eξ(PF )ξ.

Thus, (PF )ξ can be obtained directly from Pξ, and we can get Pξ from P by ﬁrst conditioning on F:

P ; PF ; (PF )ξ ; Pξ

8 Poisson and Cox processes (1903–09, Cox 1955): For simple point processes ξ on S, these conditions are equivalent:

• ξ has independent increments, • ξB is a Poisson r.v. for every B.

Then ξ is called a Poisson process on S with intensity µ = Eξ. For any random measure η on S, we say that ξ is a Cox process directed by η if, conditionally on η, ξ is a.s. Poisson with intensity η.

9 Slivnyak’s theorem (Slivnyak -62, Mecke -67, K09):

Given a simple point process ξ on S, let ξs have distribution P [ξ ∈ · k ξ]s. Then these conditions are equivalent:

• ξ is a Poisson process, d • ξs = ξ + δs for a.e. s ∈ S.

More generally, for distinct s1, . . . , sn ∈ S:

d ξs1,...,sn = ξ + δs1 + ··· + δsn

0 (n) If ξ is Cox and directed by η, then P [ξ ∈ · k ξ ]s is a.e.

n 0 P Cox and directed by P [η ∈ · k η ]s, where ξ = ξ − k δsk , (n) s = (s1, . . . , sn) ∈ S .

10 Randomizations (K97, K09):

P Given a point process η = k δσk on S and a probability kernel ν from S to T , choose some conditionally independent random elements τk in T with distributions νσk . P Then the point process ξ = k δτk on T is called a ν- randomization of η. If η is Poisson with Eη = µ, then ξ is again Poisson with Eξ = µν, and similarly for Cox processes. If ξ is a ν-randomization of η, then for s ∈ S(n) a.e.,

0 (n) 0 (n) P [ξ ∈ · k ξ ]s is a ν-randomization of P [η ∈ · k η ]s,

11 Inﬁnitely divisible random measures (Jirina 1964): A random measure ξ on S is said to be inﬁnitely divisible, if for every n there exist some i.i.d. ξ1, . . . , ξn such that d ξ = ξ1 + ··· + ξn. These conditions are equivalent:

• ξ is inﬁnitely divisible, • ξ = α + R m ζ(dm) a.s. for some ﬁxed measure α on

S and a Poisson process ζ on MS.

The characteristics α and λ = Eζ are then unique. Note

P that if ζ = k δηk , then

Z X ξ − α = m ζ(dm) = k ηk

12 Cluster ﬁelds and processes (Matthes et al. 1974):

P Given a point process ζ = k δσk on S and a probability kernel ν from S to MT , form a ν-randomization ˜ P ζ = k δηk on MT , and consider the cluster process R ˜ P ξ = m ζ(dm) = k ηk on T . If ζ is Poisson on S with intensity µ, then ζ˜ is Poisson with intensity λ = µν, and so ξ is inﬁnitely divisible with characteristics 0 and λ.

13 Palm distributions of cluster processes (Mecke -67, Matthes et al., K09): Consider a Cox process ζ on S directed by η and a probability kernel ν from S to MT , generating a cluster process ξ on T . When η = µ is ﬁxed,

n X O ˜J Eµξ = Eµξ , n ∈ N, π∈Pn J∈π ˜ π where ξ has pseudo-distribution µν. Letting pµ denote the associated relative densities, we have

n X π ˜ ˜J Pµ[ξ ∈ · k ξ ]t = Lµ(ξ) ∗ pµ(t) (∗)Pµ[ ξ ∈ · k ξ ]tJ . π∈Pn J ∈π In general, we only need to average µ with respect to L(η) or P [η ∈ · k ξn], respectively.

14 Superprocesses (Watanabe 1968, Dawson 1975): Consider a critical branching process in Rd with exponen- tial life-lengths, binary splitting, and spatial motion given by independent Brownian motions, starting from a Pois- son process with intensity µ. By suitable scaling, we get in the limit a measure-valued diﬀusion process ξt, t ≥ 0, with ξ0 = µ, called a Dawson–Watanabe superprocess.

d When d ≥ 2, the random measures ξt on R are a.s. dif- fuse, singular for t > 0 with Hausdorﬀ dimension 2. They represent the random evolution of a population at times t ≥ 0.

15 Cluster structure of superprocesses (Dawson, Perkins, Le Gall 1991): Let ξ be a DW-process starting at µ. Then for every t > 0, the ancestors at time 0 contribut-

P ing to ξt form a Poisson process ζ = k δσk with intensity −1 t µ. The points σk generate conditionally independent

P clusters ηk, and ξt = k ηk. Thus, ξt is a Poisson cluster process, and the previous theory applies. More generally, for any s ∈ [0, t), the ancestors of ξt at time s form a Cox

−1 process directed by (t − s) ξs. Thus, for every s, ξt is also a Cox cluster process with clusters of age t − s.

16 Local conditioning in superprocesses (K09): Let ξ be a DW-process in Rd with d ≥ 2, starting at

µ. Consider the ε-balls B1,...,Bn around some distinct

d points s1, . . . , sn ∈ R , and ﬁx any t > 0. Then, conditionally on mink ξBk > 0 and asymptotically as ε → 0, the random measure ξt has

• independent restrictions to the Bk and the exterior,

• stationary restriction to Bk, independent of µ and t,

• exterior governed by Palm distribution at (s1, . . . , sn).