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Transportation of diffuse random measures on Rd
... This raises the question of either characterizing pairs of random measures (ξ, η) admitting a point process factor χ or deriving complementary conditions ensuring the existence of balancing factor allocations, e.g. see Haji-Mirsadeghi and Khezeli (2016); Last and Thorisson (2023). In this article, we concentrate on the latter question and derive conditions on ξ such that for any η, such that (ξ, η) are jointly stationary and ergodic, there is a balancing factor allocation. ...
... We say that T is a factor allocation, if T is measurable w.r.t. to σ(ξ, η), the sigma algebra generated by ξ and η (see Hoffman et al. (2006); Huesmann and Sturm (2013); Last and Thorisson (2023) for further disussion). ...
We give an algorithm to construct a translation-invariant transport kernel
between ergodic stationary random measures and on ,
given that they have equal intensities. Our algorithm is deterministic given
realizations and of the measures. The existence of such a
transport kernel was proved by Thorisson and Last. Our algorithm is a
generalization of the work of Hoffman, Holroyd and Peres, which settles the
case that is the Lebesgue measure and is a point process. To do
so, we limit ourselves to mild transport kernels which are obtained from a
measure whose density w.r.t. is bounded by 1. We give a
definition of stability of mild transport kernels and introduce a construction
algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable
mild transport kernels, we study existence, uniqueness, monotonicity w.r.t. the
measures and boundedness. As a result, our algorithm yields a construction of a
shift-coupling of an ergodic random measure and its Palm version.
This paper is concerned with various aspects of the {\em Slepian process}
derived from a one-dimensional Brownian motion
. In particular, we offer an analysis of the local structure
of the {\em Slepian zero set} , including a path
decomposition of the {\em Slepian process} for . We also
establish the existence of a random time T such that T falls in the the
Slepian zero set almost surely and the process
is standard Brownian bridge.
This paper is devoted to the study of couplings of the Lebesgue measure and
the Poisson point process. We prove existence and uniqueness of an optimal
coupling whenever the asymptotic mean transportation cost is finite. Moreover,
we give precise conditions for the latter which demonstrate a sharp threshold
at d=2. The cost will be defined in terms of an arbitrary increasing function
of the distance. The coupling will be realized by means of a transport map
("allocation map") which assigns to each Poisson point a set ("cell") of
Lebesgue measure 1. In the case of quadratic costs, all these cells will be
convex polytopes.
An occupation-time density is identified for a class of absolutely continuous functions x(t) in terms of and the number of times that x(t) assumes the values in its range. This result is applied to stationary random processes with a finite second spectral moment. As a by-product, a generalization of Rice's formula for the mean number of crossings is obtained.
For d>=3, we construct a non-randomized, fair and translation-equivariant allocation of Lebesgue measure to the points of a standard Poisson point process in R^d, defined by allocating to each of the Poisson points its basin of attraction with respect to the flow induced by a gravitational force field exerted by the points of the Poisson process. We prove that this allocation rule is economical in the sense that the "allocation diameter", defined as the diameter X of the basin of attraction containing the origin, is a random variable with a rapidly decaying tail. Specifically, we have the tail bound: P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d = (d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are positive constants that depend on d and alpha_d. This is the first construction of an allocation rule of Lebesgue measure to a Poisson point process with subpolynomial decay of the tail P(X>R).
Let be a discrete set in . Call the elements of centers. The well-known Voronoi tessellation partitions into polyhedral regions (of varying sizes) by allocating each site of to the closest center. Here we study ``fair'' allocations of to in which the regions allocated to different centers have equal volumes. We prove that if is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale--Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than . We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in .
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].
This new, thoroughly revised and expanded 3rd edition of a classic gives a comprehensive coverage of modern probability in a single book. It is a truly modern text, providing not only classical results but also material that will be important for future research. Much has been added to the previous edition, including eight entirely new chapters on subjects like random measures, Malliavin calculus, multivariate arrays, and stochastic differential geometry. Apart from important improvements and revisions, some of the earlier chapters have been entirely rewritten. To help the reader, the material has been grouped together into ten major areas, each arguably indispensable to any serious graduate student and researcher, regardless of their specialization.
Each chapter is largely self-contained and includes plenty of exercises, making the book ideal for self-study and for designing graduate-level courses and seminars in different areas and at different levels. Extensive notes and a detailed bibliography make it easy to go beyond the presented material if desired.
From the reviews of the first edition:
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“…great edifice of material, clearly and ingeniously presented, without any non-mathematical distractions. Readers … are in very capable hands.” F. B. Knight, Mathemtical Reviews
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From the reviews of the second edition:
“This … edition presents … more material in the concise and elegant style of the former edition which by now has become a highly praised standard reference book for many areas of probability theory.” M. Reiß, zbMATH
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We consider two jointly stationary and ergodic random measures and on the real line with equal intensities. An allocation is an equivariant random mapping from to . We give sufficient and partially necessary conditions for the existence of allocations transporting to . An important ingredient of our approach is to introduce a transport kernel balancing and , provided these random measures are mutually singular. In the second part of the paper, we apply this result to the path decomposition of a two-sided Brownian motion into three independent pieces: a time reversed Brownian motion on , an excursion distributed according to a conditional It\^{o}'s law and a Brownian motion starting after this excursion. An analogous result holds for Bismut's excursion law.
Shift-coupling means pasting together the paths of two processes modulo a random shift. This concept can be related to the invariant [sigma]-field in a similar way as ordinary coupling is related to the tail [sigma]-field. We give an expository account of this relationship, implicit in work of Berbee and Greven. In developing these relations we introduce the concept of coupling with respect to a sub-[sigma]-field.
We study couplings of two equivariant random measures
and on a Riemannian manifold (M,d,m). Given a
cost function we ask for minimizers of the mean transportation cost per volume.
In case the minimal/optimal cost is finite and we prove
that there is a unique equivariant coupling minimizing the mean transportation
cost per volume. Moreover, the optimal coupling is induced by a transportation
map, We show that the optimal
transportation map can be approximated by solutions to classical optimal
transportation problems on bounded regions. In case of cost the optimal
transportation cost per volume defines a metric on the space of equivariant
random measure with unit intensity.
Let be a two-sided standard Brownian motion. An
\emph{unbiased shift} of B is a random time T, which is a measurable
function of B, such that is a Brownian motion
independent of . We characterise unbiased shifts in terms of allocation
rules balancing additive functionals of B. For any probability distribution
on we construct a stopping time with the above properties
such that has distribution . In particular, we show that if we
travel in time according to the clock of local time we always see a two-sided
Brownian motion. A crucial ingredient of our approach is a new theorem on the
existence of allocation rules balancing jointly stationary diffuse random
measures on . We also study moment and minimality properties of unbiased
shifts.
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 the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary. Comment: Published in at http://dx.doi.org/10.1214/08-AOP420 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and in this space agree on invariant sets if and only if there is a random transformation such that has the same distribution as . Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.
We consider the following problem:given an i.i.d. family of Bernoulli random variables indexed by , find a random occupied site such that relative to X, the other random variables are still i.i.d. Bernoulli. Results of Thorisson imply that such an X exists for all d. Liggett proved that for, there exists an X with tails of order ct^(-1 /2}, but none with finite 1/2th moment. We prove that for general d there exists a solution with tails of order , while for there is none with finite first moment. We also prove analogous results for a continuum version of the same problem. Finally we prove a result which strongly suggests that the tail behavior mentioned above is the best possible for alld.
Thorisson and others have proved results that imply the following: given an i.i.d. family of Bernoulli random variables indexed byZ
d
there exists an occupied siteX ∈Z
d
with the property that relative to its location, the other variables are still i.i.d. We raise the question of how large such anXmust be. Ford =1, we prove that anyXwith this property satisfies E | X |½= ∞. Moreover, there does exist such anXwith tails P(|X| 1 ≥ n) of orderCn
-½so these results are essentially best possible. Analogous results for the Poisson process in one dimension are given. The corresponding problem for stationary ergodic sequences is considered also. This project was motivated by some tagged particle problems.
- Holroyd, Alexander E. and Liggett, Thomas M.
Lectures on the Poisson process
- G Last
- M Penrose
Last, G. and Penrose, M. Lectures on the Poisson process, volume 7 of Institute of Mathematical
Statistics Textbooks. Cambridge University Press, Cambridge (2018). ISBN 978-1-107-45843-7;
978-1-107-08801-6. MR3791470.
Grundlehren der mathematischen Wissenschaften
- C Villani
Villani, C. Optimal transport: Old and new, volume 338 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin (2009). ISBN 978-3-540-71049-3. MR2459454.