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Let <bold>X</bold> = {X(t)}<subscrpt>t ≥ 0</subscrpt> be a stochastic process with a stationary version <bold>X<supscrpt>*</supscrpt></bold>. It is investigated when it is possible to generate by simulation a version <bold>X˜</bold> of <bold>X</bold> with lower initial bias than <bold>X</bold> itself, in the sense that either… (More)

We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S ⊂ R d. The algorithm is independent of the origin of coordinates. We show that (1) the graph has one topological end —that is, from any point there is exactly one infinite self-avoiding path;… (More)

We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu's exchange formula. The second main result is a simple sufficient and necessary criterion for… (More)

Let B = (B t) t∈R be a two-sided standard Brownian motion. An unbiased shift of B is a random time T , which is a measurable function of B, such that (B T +t − B T) t∈R is a Brownian motion independent of B T. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of B. For any probability distribution ν on R we… (More)

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This note presents four sets of problems. The first suggests the possibility of a limit theory for null-recurrent renewal processes similar to the theory in the positive recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread out. The third concerns the coupling characterization… (More)

This note presents four independent sets of open problems. The first set suggests an extension of the limit theory for positive recurrent renewal processes to the null recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread-out. The third concerns the coupling characterization of… (More)

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