Local hitting and conditioning in symmetric interval partitions

AbstractBy a symmetric interval partition we mean a perfect, closed random set Ξ in [0,1] of Lebesgue measure 0, such that the lengths of the connected components of Ξc occur in random order. Such sets are analogous to the regenerative sets on R+, and in particular there is a natural way to define a corresponding local time random measure ξ with support Ξ. In this paper, the author's recently developed duality theory is used to construct versions of the Palm distributions Qx of ξ with attractive continuity and approximation properties. The results are based on an asymptotic formula for hitting probabilities and a delicate construction and analysis of conditional densities

Asymptotically invariant sampling and averaging from stationary-like processes

AbstractGiven a process X on Rd or Zd, we may form a random sequence ξ1,ξ2,… by sampling from X at some independent points τ1,τ2,…. If X is stationary up to shifts (which holds for broad classes of Markov and Palm processes) and the distribution of (τn) is asymptotically invariant (as in the case of Poisson or Bernoulli sampling, respectively) then (ξn) is asymptotically exchangeable, and the associated empirical distribution converges to the corresponding product random measure

Some local approximations of Dawson--Watanabe superprocesses

Let $\xi$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d$ such that $\xi_t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed $t>0$, the singular random measure $\xi_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods of $\operatorname {supp}\xi_t$. When $d\geq3$, the local distributions of $\xi_t$ near a hitting point can be approximated in total variation by those of a stationary and self-similar pseudo-random measure $\tilde{\xi}$. By contrast, the corresponding distributions for $d=2$ are locally invariant. Further results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities. Our main proofs are based on a detailed analysis of the historical structure of $\xi$.Comment: Published in at http://dx.doi.org/10.1214/07-AOP386 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Domain Theory for Statistical Probabilistic Programming

We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixed-variance datatypes. Our semantics justifies important program equivalences including commutativity. Our new semantic model is based on `quasi-Borel predomains'. These are a mixture of chain-complete partial orders (cpos) and quasi-Borel spaces. Quasi-Borel spaces are a recent model of probability theory that focuses on sets of admissible random elements. Probability is traditionally treated in cpo models using probabilistic powerdomains, but these are not known to be commutative on any class of cpos with higher order functions. By contrast, quasi-Borel predomains do support both a commutative probabilistic powerdomain and higher-order functions. As we show, quasi-Borel predomains form both a model of Fiore's axiomatic domain theory and a model of Kock's synthetic measure theory.</p