14,994 research outputs found
Brownian couplings, convexity, and shy-ness
Benjamini, Burdzy, and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in C-2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. (2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary
Coupling, local times, immersions
This paper answers a question of \'{E}mery [In S\'{e}minaire de
Probabilit\'{e}s XLII (2009) 383-396 Springer] by constructing an explicit
coupling of two copies of the Bene\v{s} et al. [In Applied Stochastic Analysis
(1991) 121-156 Gordon & Breach] diffusion (BKR diffusion), neither of which
starts at the origin, and whose natural filtrations agree. The paper commences
by surveying probabilistic coupling, introducing the formal definition of an
immersed coupling (the natural filtration of each component is immersed in a
common underlying filtration; such couplings have been described as co-adapted
or Markovian in older terminologies) and of an equi-filtration coupling (the
natural filtration of each component is immersed in the filtration of the
other; consequently the underlying filtration is simultaneously the natural
filtration for each of the two coupled processes). This survey is followed by a
detailed case-study of the simpler but potentially thematic problem of coupling
Brownian motion together with its local time at . This problem possesses its
own intrinsic interest as well as being closely related to the BKR coupling
construction. Attention focusses on a simple immersed (co-adapted) coupling,
namely the reflection/synchronized coupling. It is shown that this coupling is
optimal amongst all immersed couplings of Brownian motion together with its
local time at , in the sense of maximizing the coupling probability at all
possible times, at least when not started at pairs of initial points lying in a
certain singular set. However numerical evidence indicates that the coupling is
not a maximal coupling, and is a simple but non-trivial instance for which this
distinction occurs. It is shown how the reflection/synchronized coupling can be
converted into a successful equi-filtration coupling, by modifying the coupling
using a deterministic time-delay and then by concatenating an infinite sequence
of such modified couplings. The construction of an explicit equi-filtration
coupling of two copies of the BKR diffusion follows by a direct generalization,
although the proof of success for the BKR coupling requires somewhat more
analysis than in the local time case.Comment: Published at http://dx.doi.org/10.3150/14-BEJ596 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Return to the Poissonian City
Consider the following random spatial network: in a large disk, construct a
network using a stationary and isotropic Poisson line process of unit
intensity. Connect pairs of points using the network, with initial / final
segments of the connecting path formed by travelling off the network in the
opposite direction to that of the destination / source. Suppose further that
connections are established using "near-geodesics", constructed between pairs
of points using the perimeter of the cell containing these two points and
formed using only the Poisson lines not separating them. If each pair of points
generates an infinitesimal amount of traffic divided equally between the two
connecting near-geodesics, and if the Poisson line pattern is conditioned to
contain a line through the centre, then what can be said about the total flow
through the centre? In earlier work ("Geodesics and flows in a Poissonian
city", Annals of Applied Probability, 21(3), 801--842, 2011) it was shown that
a scaled version of this flow had asymptotic distribution given by the 4-volume
of a region in 4-space, constructed using an improper anisotropic Poisson line
process in an infinite planar strip. Here we construct a more amenable
representation in terms of two "seminal curves" defined by the improper Poisson
line process, and establish results which produce a framework for effective
simulation from this distribution up to an L1 error which tends to zero with
increasing computational effort.Comment: 11 pages, 2 figures Various minor edits, corrections to
multiplicative constants in Theorem 5.1. Version 2: minor stylistic
corrections, added acknowledgement of grant support. Version 3: three further
minor corrections. This paper is due to appear in Journal of Applied
Probability, Volume 51
From Random Lines to Metric Spaces
Consider an improper Poisson line process, marked by positive speeds so as to
satisfy a scale-invariance property (actually, scale-equivariance). The line
process can be characterized by its intensity measure, which belongs to a
one-parameter family if scale and Euclidean invariance are required. This paper
investigates a proposal by Aldous, namely that the line process could be used
to produce a scale-invariant random spatial network (SIRSN) by means of
connecting up points using paths which follow segments from the line process at
the stipulated speeds. It is shown that this does indeed produce a
scale-invariant network, under suitable conditions on the parameter; indeed
that this produces a parameter-dependent random geodesic metric for
d-dimensional space (), where geodesics are given by minimum-time
paths. Moreover in the planar case it is shown that the resulting geodesic
metric space has an almost-everywhere-unique-geodesic property, that geodesics
are locally of finite mean length, and that if an independent Poisson point
process is connected up by such geodesics then the resulting network places
finite length in each compact region. It is an open question whether the result
is a SIRSN (in Aldous' sense; so placing finite mean length in each compact
region), but it may be called a pre-SIRSN.Comment: Version 1: 46 pages, 10 figures Version 2: 47 pages, 10 figures
(various typos and stylistic amendments, added dedication to Burkholder,
added references concerning Lipschitz property and Sobolev space
Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables
We prove weak laws of large numbers and central limit theorems
of Lindeberg type for empirical centres of mass (empirical Fréchet means)
of independent nonidentically distributed random variables taking values in
Riemannian manifolds. In order to prove these theorems we describe and
prove a simple kind of Lindeberg–Feller central approximation theorem for
vector-valued random variables, which may be of independent interest and
is therefore the subject of a self-contained section. This vector-valued result
allows us to clarify the number of conditions required for the central limit
theorem for empirical Fréchet means, while extending its scope
Coupling of Brownian motions in Banach spaces
Consider a separable Banach space supporting a non-trivial
Gaussian measure . The following is an immediate consequence of the theory
of Gaussian measure on Banach spaces: there exist (almost surely) successful
couplings of two -valued Brownian motions and
begun at starting points and
if and only if the difference
of their initial positions belongs to
the Cameron-Martin space of corresponding to
. For more general starting points, can there be a "coupling at time
", such that almost surely
as
? Such couplings exist if there exists a Schauder basis of which is also a -orthonormal basis of
. We propose (and discuss some partial answers to) the
question, to what extent can one express the probabilistic Banach space
property "Brownian coupling at time is always possible" purely in
terms of Banach space geometry?Comment: 12 page
Coupling iterated Kolmogorov diffusions
The Kolmogorov (1934) diffusion is the two-dimensional diffusion generated by real Brownian motion B and its time integral integral B d t. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals integral integral B d s d t,... as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for (B, integral B d t, integral integral B d s d t); and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals
Perfect Simulation of Queues
In this paper we describe a perfect simulation algorithm for the stable
queue. Sigman (2011: Exact Simulation of the Stationary Distribution of
the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how
to build a dominated CFTP algorithm for perfect simulation of the super-stable
queue operating under First Come First Served discipline, with
dominating process provided by the corresponding queue (using Wolff's
sample path monotonicity, which applies when service durations are coupled in
order of initiation of service), and exploiting the fact that the workload
process for the queue remains the same under different queueing
disciplines, in particular under the Processor Sharing discipline, for which a
dynamic reversibility property holds. We generalize Sigman's construction to
the stable case by comparing the queue to a copy run under Random
Assignment. This allows us to produce a naive perfect simulation algorithm
based on running the dominating process back to the time it first empties. We
also construct a more efficient algorithm that uses sandwiching by lower and
upper processes constructed as coupled queues started respectively from
the empty state and the state of the queue under Random Assignment. A
careful analysis shows that appropriate ordering relationships can still be
maintained, so long as service durations continue to be coupled in order of
initiation of service. We summarize statistical checks of simulation output,
and demonstrate that the mean run-time is finite so long as the second moment
of the service duration distribution is finite.Comment: 28 pages, 5 figure
- …