A normal comparison inequality and its applications

Li, Wenbo V.; Shao, Qi-Man

doi:10.1007/s004400100176

Cited by 78 publications

(70 citation statements)

References 16 publications

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“…[184,185]. Rigorous lower and upper bounds for θ(2) have also been obtained, the best one being θ(2) ∈ (1/(4 √ 3), 1/4] proved in [187] and [188]. We also mention an explicit expression for Q 0 ([a, b], n) for d = 2 obtained in Ref.…”

Section: Connections With Random Polynomialsmentioning

confidence: 97%

Persistence and first-passage properties in nonequilibrium systems

Bray

Majumdar

Schehr

2013

Advances in Physics

519

669

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In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

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Section: Connections With Random Polynomialsmentioning

confidence: 97%

Persistence and first-passage properties in nonequilibrium systems

Bray

Majumdar

Schehr

2013

Advances in Physics

519

669

View full text Add to dashboard Cite

show abstract

“…Theorem 2.15 (Li and Shao, 2002) Suppose that (X 1i , 1 ≤ i ≤ n) are standard normal random variables with covariance matrix…”

Section: The Centered Normal Density Function With Covariance Matrixmentioning

confidence: 99%

Level crossings and other level functionals of stationary Gaussian processes

Kratz¹

2006

Probab. Surveys

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This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general m-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.

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“…For the covariance r 4 , Li and Shao (2002) established that b := 2q(0) lies in (0.5, 1]. Simulations of Dembo et al (2002) suggest that b = 0.76 ± 0.03.…”

Section: Estimation Of the Persistence Exponentmentioning

confidence: 99%

Numerical bounds for the distributions of the maxima of some one- and two-parameter Gaussian processes

Mercadier¹

2006

Adv. Appl. Probab.

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We consider the class of real-valued stochastic processes indexed on a compact subset of R or R 2 with almost surely absolutely continuous sample paths. We obtain an implicit formula for the distributions of their maxima. The main result is the derivation of numerical bounds that turn out to be very accurate, in the Gaussian case, for levels that are not large. We also present the first explicit upper bound for the distribution tail of the maximum in the two-dimensional Gaussian framework. Numerical comparisons are performed with known tools such as the Rice upper bound and expansions based on the Euler characteristic. We deal numerically with the determination of the persistence exponent.

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A normal comparison inequality and its applications

Cited by 78 publications

References 16 publications

Persistence and first-passage properties in nonequilibrium systems

Persistence and first-passage properties in nonequilibrium systems

Level crossings and other level functionals of stationary Gaussian processes

Numerical bounds for the distributions of the maxima of some one- and two-parameter Gaussian processes

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