Germany, April{May 2009: Iterated Palm conditioning and some Slivnyak-type theorems for Cox and cluster processes | | | by Olav Kallenberg 1 Conditional expectations (Kolmogorov 1933): Given a random element » in (S; S): ¯ ¯ ¯ E[ ´; » 2 ds ]¯ ¡1 ¯ E[ ´ j » ]s = ¯ ; s 2 S a.e. P ± » : P f» 2 dsg ¯S Given a sub-σ-¯eld F in (­; A;P ): ¯ ¯ ¯ E[ ´; d! ]¯ ¯ EF ´ = E[ ´ j F ]! = ¯ ; s 2 S a.e. P; P (d!) ¯F ¯ ¯ ¯ P (A \ d!)¯ ¯ PF (A) = P [ A j F ]! = ¯ ; s 2 S a.e. P: P (d!) ¯F Interpretations of EF ´: ² projection of ´ onto L1(F) ² expected value of ´ when F is \known" 2 Conditional distributions (Doob 1938): For any random elements » and ´ in Borel spaces S and T , there exists a probability kernel ¹ from S to T with P [ ´ 2 B j » ]s = ¹(s; B); s 2 S; B 2 T : In other words, ¹(s; B) is ² a measurable function of s for ¯xed B, ² a probability measure on T in B for ¯xed s. Writing L(») = º gives the disintegration Z Z Ef(»; ´) = º(ds) ¹s(dt) f(s; t) ´ (º ­ ¹)f: Interchanging the roles of S and T gives the dual version: L(»; ´) = º ­ ¹ =» º0 ­ ¹0: 3 Palm distributions (Palm 1943): For any random measure » on S (kernel from ­ to S), we consider the Campbell measure with dual disintegrations » P ­ » = E» ­ P» (provided the intensity E» is σ-¯nite), de¯ning the Palm s distributions P [A k » ]s = P» (A). Conditioning approach: ² when » = ±¿ , take P [A k »]s = P [Aj¿]s, ² when »S is constant in (0; 1), choose ¿ with P [¿ 2 ¢ j »] = »=»S, and de¯ne P [A k »]s = P [Aj¿]s, ² in general, choose ¿ with P~[¿ 2 ¢ j »] = », and de¯ne P [A k »]s = P~[Aj¿]s. 4 Local conditioning (Matthes 1963, K83, K08): Say that » is a simple point process on S if X » = k ±¿k a.s. for some a.s. distinct random points ¿1;¿2;::: in S. Then P [A k »]s = lim P [A j »B > 0] a.e. B#s Though no such approximation is valid in general, it does hold in some interesting special cases, including ² local time random measures on R+, ² 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 ² EF EG = EGEF a.s. on L , ² F??G (conditional independence). F\G Then the product equals EF\G. Standard examples: ² F??G (independence), ² F ½ G (tower property), ² EFσ EF¿ = EFσ^¿ for F-optional times σ and ¿. 6 Iterated conditional distributions (K09): De¯ne the probability kernel (PF )G on ­ by (PF )G(!) = (P [ ¢ j F]!)[ ¢ j G]! (diagonal values !). Then a.s. (under regularity conditions), ² (PF )G = (PG)F = PF_G, ² PF = EF (PG)F , PG = EG(PF )G. This amounts to a change of conditioning σ-¯eld: PG ; PF_G ; PF ;PF ; PF_G ; PG 7 Iterated Palm conditioning (K09): De¯ne the kernels (PF )» and (P»)F from ­ £ S to ­ by (PF )»(!; s) = (P [ ¢ jF]!)[ ¢ k »]s; (P»)F (!; s) = (P [ ¢ k »]s)[ ¢ jF]!: Then a.e. (under regularity conditions), ² (PF )» = (P»)F = (P ­ »)F­S, ² P» = E»(PF )». Thus, (PF )» can be obtained directly from P», and we can get P» from P by ¯rst conditioning on F: P ; PF ; (PF )» ; P» 8 Poisson and Cox processes (1903{09, Cox 1955): For simple point processes » on S, these conditions are equivalent: ² » has independent increments, ² »B is a Poisson r.v. for every B. Then » is called a Poisson process on S with intensity ¹ = E». For any random measure ´ on S, we say that » is a Cox process directed by ´ if, conditionally on ´, » is a.s. Poisson with intensity ´. 9 Slivnyak's theorem (Slivnyak -62, Mecke -67, K09): Given a simple point process » on S, let »s have distribu- tion P [» 2 ¢ k »]s. Then these conditions are equivalent: ² » is a Poisson process, d ² »s = » + ±s for a.e. s 2 S. More generally, for distinct s1; : : : ; sn 2 S: d »s1;:::;sn = » + ±s1 + ¢ ¢ ¢ + ±sn 0 (n) If » is Cox and directed by ´, then P [» 2 ¢ k » ]s is a.e. n 0 P Cox and directed by P [´ 2 ¢ k ´ ]s, where » = » ¡ k ±sk , (n) s = (s1; : : : ; sn) 2 S . 10 Randomizations (K97, K09): P Given a point process ´ = k ±σk on S and a probability kernel º from S to T , choose some conditionally inde- pendent random elements ¿k in T with distributions ºσk . P Then the point process » = k ±¿k on T is called a º- randomization of ´. If ´ is Poisson with E´ = ¹, then » is again Poisson with E» = ¹º, and similarly for Cox processes. If » is a º-randomization of ´, then for s 2 S(n) a.e., 0 (n) 0 (n) P [» 2 ¢ k » ]s is a º-randomization of P [´ 2 ¢ k ´ ]s, 11 In¯nitely divisible random measures (Jirina 1964): A random measure » on S is said to be in¯nitely divisible, if for every n there exist some i.i.d. »1;:::;»n such that d » = »1 + ¢ ¢ ¢ + »n. These conditions are equivalent: ² » is in¯nitely divisible, ² » = ® + R m ³(dm) a.s. for some ¯xed measure ® on S and a Poisson process ³ on MS. The characteristics ® and ¸ = E³ are then unique. Note P that if ³ = k ±´k , then Z X » ¡ ® = m ³(dm) = k ´k 12 Cluster ¯elds and processes (Matthes et al. 1974): P Given a point process ³ = k ±σk on S and a proba- bility kernel º from S to MT , form a º-randomization ~ P ³ = k ±´k on MT , and consider the cluster process R ~ P » = m ³(dm) = k ´k on T . If ³ is Poisson on S with intensity ¹, then ³~ is Poisson with intensity ¸ = ¹º, and so » is in¯nitely divisible with characteristics 0 and ¸. 13 Palm distributions of cluster processes (Mecke -67, Matthes et al., K09): Consider a Cox process ³ on S directed by ´ and a probability kernel º from S to MT , generating a cluster process » on T . When ´ = ¹ is ¯xed, n X O ~J E¹» = E¹» ; n 2 N; ¼2Pn J2¼ ~ ¼ where » has pseudo-distribution ¹º. Letting p¹ denote the associated relative densities, we have n X ¼ ~ ~J P¹[» 2 ¢ k » ]t = L¹(») ¤ p¹(t) (¤)P¹[ » 2 ¢ k » ]tJ : ¼2Pn J 2¼ In general, we only need to average ¹ with respect to L(´) or P [´ 2 ¢ k »n], 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 ¹. By suitable scaling, we get in the limit a measure-valued di®usion process »t, t ¸ 0, with »0 = ¹, called a Dawson{Watanabe superprocess. d When d ¸ 2, the random measures »t on R are a.s. dif- fuse, singular for t > 0 with Hausdor® 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 » be a DW-process starting at ¹. Then for every t > 0, the ancestors at time 0 contribut- P ing to »t form a Poisson process ³ = k ±σk with intensity ¡1 t ¹. The points σk generate conditionally independent P clusters ´k, and »t = k ´k. Thus, »t is a Poisson cluster process, and the previous theory applies. More generally, for any s 2 [0; t), the ancestors of »t at time s form a Cox ¡1 process directed by (t ¡ s) »s. Thus, for every s, »t is also a Cox cluster process with clusters of age t ¡ s. 16 Local conditioning in superprocesses (K09): Let » be a DW-process in Rd with d ¸ 2, starting at ¹. Consider the "-balls B1;:::;Bn around some distinct d points s1; : : : ; sn 2 R , and ¯x any t > 0. Then, condi- tionally on mink »Bk > 0 and asymptotically as " ! 0, the random measure »t has ² independent restrictions to the Bk and the exterior, ² stationary restriction to Bk, independent of ¹ and t, ² exterior governed by Palm distribution at (s1; : : : ; sn). 17.