7,002 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
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 LindebergâFeller 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 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
N-fold way simulated tempering for pairwise interaction point processes
Pairwise interaction point processes with strong interaction are usually difficult to
sample. We discuss how Besag lattice processes can be used in a simulated tempering
MCMC scheme to help with the simulation of such processes. We show how
the N-fold way algorithm can be used to sample the lattice processes efficiently
and introduce the N-fold way algorithm into our simulated tempering scheme. To
calibrate the simulated tempering scheme we use the Wang-Landau algorithm
Rayleigh random flights on the Poisson line SIRSN
We study scale-invariant Rayleigh Random Flights (âRRFâ) in random environments given by planar Scale-Invariant Random Spatial Networks (âSIRSNâ) based on speed-marked Poisson line processes. A natural one-parameter family of such RRF (with scale-invariant dynamics) can be viewed as producing ârandomly-broken local geodesicsâ on the SIRSN; we aim to shed some light on a conjecture that a (non-broken) geodesic on such a SIRSN will never come to a complete stop en route. (If true, then all such geodesics can be represented as doubly-infinite sequences of sequentially connected line segments. This would justify a natural procedure for computing geodesics.) The family of these RRF (âSIRSN-RRFâ), is introduced via a novel axiomatic theory of abstract scattering representations for Markov chains (itself of independent interest). Palm conditioning (specifically the Mecke-Slivnyak theorem for Palm probabilities of Poisson point processes) and ideas from the ergodic theory of random walks in random environments are used to show that at a critical value of the parameter the speed of the scale-invariant SIRSN-RRF neither diverges to infinity nor tends to zero, thus supporting the conjecture
MEXIT: Maximal un-coupling times for stochastic processes
Classical coupling constructions arrange for copies of the \emph{same} Markov
process started at two \emph{different} initial states to become equal as soon
as possible. In this paper, we consider an alternative coupling framework in
which one seeks to arrange for two \emph{different} Markov (or other
stochastic) processes to remain equal for as long as possible, when started in
the \emph{same} state. We refer to this "un-coupling" or "maximal agreement"
construction as \emph{MEXIT}, standing for "maximal exit". After highlighting
the importance of un-coupling arguments in a few key statistical and
probabilistic settings, we develop an explicit \MEXIT construction for
stochastic processes in discrete time with countable state-space. This
construction is generalized to random processes on general state-space running
in continuous time, and then exemplified by discussion of \MEXIT for Brownian
motions with two different constant drifts.Comment: 28 page
Initial radio-frequency gas heating experiments to simulate the thermal environment in a nuclear light bulb reactor
Initial radio frequency gas heating experiments to simulate thermal environment in nuclear light bulb reacto
Coupling polynomial Stratonovich integrals : the two-dimensional Brownian case
We show how to build an immersion coupling of a two-dimensional Brownian motion (W1,W2) along with (n2)+n=12n(n+1). integrals of the form â«Wi1Wj2âdW2, where j=1,âŠ,n and i=0,âŠ,nâj for some fixed n. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)) and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also multiple selected integral functionals of the diffusions
Coupling polynomial Stratonovich integrals : the two-dimensional Brownian case
We show how to build an immersion coupling of a two-dimensional Brownian motion (W1,W2) along with (n2)+n=12n(n+1). integrals of the form â«Wi1Wj2âdW2, where j=1,âŠ,n and i=0,âŠ,nâj for some fixed n. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)) and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also multiple selected integral functionals of the diffusions
Data processing method for a weak, moving telemetry signal
Method of processing data from a spacecraft, where the carrier has a low signal-to-noise ratio and wide unpredictable frequency shifts, consists of analogue recording of the noisy signal along with a high-frequency tone that is used as a clock to trigger a digitizer
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