The Law of the Maximum of a Bessel Bridge

Pitman, Jim; Yor, Marc

doi:10.1214/ejp.v4-52

Cited by 49 publications

(81 citation statements)

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“…The latter observation, as well as the following proposition, may be deduced from general calculations for the law of the maximum of diffusion bridges, which may be found for instance in Kiefer [14] or Pitman and Yor [18], [19].…”

Section: Proof Of Theoremmentioning

confidence: 64%

Path transformations of first passage bridges

Bertoin

Chaumont

Pitman

2003

Electron. Commun. Probab.

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Summary. We define the first passage bridge from 0 to λ as the Brownian motion on the time interval [0,1] conditioned to first hit λ at time 1. We show that this process may be related to the Brownian bridge, the Bessel bridge or the Brownian excursion via some path transformations, the main one being an extension of Vervaat's transformation. We also propose an extension of these results to certain bridges with cyclically exchangeable increments.

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Section: Proof Of Theoremmentioning

confidence: 64%

Path transformations of first passage bridges

Bertoin

Chaumont

Pitman

2003

Electron. Commun. Probab.

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“…2). An excursion in the time interval [0, τ i ], is a stochastic trajectory which is constrained to begin close to the velocity origin, at v(0) = → 0, end at v(τ i ) = 0, and never reach zero between (0, τ i ) (see e.g., [26][27][28]). …”

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confidence: 99%

Large Fluctuations for Spatial Diffusion of Cold Atoms

2017

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Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.PACS numbers: 05.40. Jc,46.65.+g In diffusion processes such as Brownian motion, the concentration of particles starting at the origin spreads out like a Gaussian, which is fully characterized by the mean squared-displacement. This is the result of the widely applicable Gaussian central limit theorem (CLT) [1]. Of no less importance is large-deviations theory [2], which deals with the rare fluctuations of processes such as simple coin tossing random walks (see e.g., [3]), extreme variations of the surface height in the KardarParisi-Zhang model [4] and the tails of the position distribution in single-file diffusion [5,6]. Mathematically, a prerequisite of the theory is that the cumulant generating function be "well behaved", i.e. smooth and differentiable. Large-deviations theory works when the decay of the probability of the observable of interest is exponential (see details in [2]). However many systems do not meet this requirement [2], for example Lévy fat-tailed processes [7][8][9], where the decay rate is a power-law. This is the case for a cloud of atoms undergoing Sisyphus laser-cooling [10], where both theoretically [11,12] and experimentally [13], it was shown that the central part of the spreading particle packet is described by the Lévy CLT [14]. The latter deals with the sum of independent identically-distributed random variables, but unlike the classical Gaussian CLT, here the summands' own distribution is heavy-tailed. As a result, Lévy's CLT yields an infinite mean squared-displacement for the sum [14], and consequently also the second cumulant. Largedeviations theory mainly deals with thin-tailed processes where extreme events are rare, but in Lévy processes these large fluctuations are dominant. To study the fluctuations in this system, we will show that the relevant tool is the asymptotic moment-generating function, which yields an infinite-covariant density (ICD) [12,15]. We will discuss the generality of this approach and its results below.In an experimental situation, diverging moments are unphysical. For example, although the experiment in [13] shows a nice fit of the particles' density to a symmetric Lévy distribution, clearly at finite times no particles traveling at finite velocities can ever be found infinitely far from their origin. The finiteness of all t...

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“…Letting T be ϕ 3 (κ), and noting that all positive integer moments of the maximum of a standard three-dimensional Bessel Bridge are finite (see [20,Corollary 7]) completes the proof of Lemma A.1.…”

Section: Monte Carlo Estimation Of Diffusion First Passage Time Densimentioning

confidence: 88%

Efficient Estimation of One-Dimensional Diffusion First Passage Time Densities via Monte Carlo Simulation

Ichiba

Kardaras²

2011

J. Appl. Probab.

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We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as the expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order 1/ √ N, where N is the sample size, is achieved, which is in sharp contrast to the slower nonparametric rates achieved by kernel smoothing of cumulative distribution functions.

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The Law of the Maximum of a Bessel Bridge

Cited by 49 publications

References 48 publications

Path transformations of first passage bridges

Path transformations of first passage bridges

Large Fluctuations for Spatial Diffusion of Cold Atoms

Efficient Estimation of One-Dimensional Diffusion First Passage Time Densities via Monte Carlo Simulation

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