Bounds for expected maxima of Gaussian processes and their discrete approximations

Borovkov, Konstantin; Mishura, Yuliya; Novikov, Alexander; Zhitlukhin, Mikhail

doi:10.1080/17442508.2015.1126282

Cited by 36 publications

(43 citation statements)

References 26 publications

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“…This is so since it follows from Sudakov-Fernique's inequality (see e.g. Proposition 1.1 and Section 4 in [5]) that the expected maximum M H is a non-increasing function of H. Remark 6. Our new upper bound for M H can be used to improve Shao's upper bound from [12] for Pickands' constant H H , which is a basic constant in the extreme value theory of Gaussian processes and is of interest in a number of applied problems.…”

Section: Resultsmentioning

confidence: 90%

“…Although this step is common with the proof of Theorem 3.1 in [5], the rest of the argument uses a different idea. Namely, we apply Chatterjee's inequality ( [6]; see also Theorem 2.2.5 in [1]) which, in its general formulation, states the following.…”

Section: Proofsmentioning

confidence: 99%

“…On the other hand, it follows from Sudakov-Fernique's inequality (see e.g. Proposition 1.1 in [5]) that, for any fixed n ≥ 1, the quantity M H n is non-increasing in H, and it follows from Lemma 4.1 in [5] that…”

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confidence: 99%

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New and refined bounds for expected maxima of fractional Brownian motion

Borovkov

Mishura

Novikov

et al. 2018

Statistics & Probability Letters

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For the fractional Brownian motion B H with the Hurst parameter value H in (0,1/2), we derive new upper and lower bounds for the difference between the expectations of the maximum of B H over [0,1] and the maximum of B H over the discrete set of values in −1 , i = 1, . . . , n. We use these results to improve our earlier upper bounds for the expectation of the maximum of B H over [0, 1] and derive new upper bounds for Pickands' constant.

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Section: Resultsmentioning

confidence: 90%

Section: Proofsmentioning

confidence: 99%

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New and refined bounds for expected maxima of fractional Brownian motion

Borovkov

Mishura

Novikov

et al. 2018

Statistics & Probability Letters

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“…In the sense of finite-dimensional distributions, asymptotic for large N behaviour of the process ξ N t is close to behaviour of the fractional Brownian motion B H t with small parameter H 1. Indeed, in Lemma 4.1 of [5] it is pointed out that…”

Section: Finite Dimensional Distributions and Motivation Behind Theormentioning

confidence: 99%

A Functional Limit Theorem for the Sine-Process

Bufetov

Dymov

2018

International Mathematics Research Notices

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The main result of this paper is a functional limit theorem for the sine-process. In particular, we study the limit distribution, in the space of trajectories, for the number of particles in a growing interval. The sine-process has the Kolmogorov property and satisfies the Central Limit Theorem, but our functional limit theorem is very different from the Donsker Invariance Principle. We show that the time integral of our process can be approximated by the sum of a linear Gaussian process and independent Gaussian fluctuations whose covariance matrix is computed explicitly. We interpret these results in terms of the Gaussian Free Field convergence for the random matrix models. The proof relies on a general form of the multidimensional Central Limit Theorem under the sine-process for linear statistics of two types: those having growing variance and those with bounded variance corresponding to observables of Sobolev regularity 1/2. Contents

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“…The authors of [1] obtain an upper bound for the error ∆ N of approximation (2). Namely, for N ≥ 2 1/H ,…”

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confidence: 99%

Simulation paradoxes related to a fractional Brownian motion with small Hurst index

Makogin

2016

Modern Stoch. Theory Appl.

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We consider the simulation of sample paths of a fractional Brownian motion with small values of the Hurst index and estimate the behavior of the expected maximum. We prove that, for each fixed N , the error of approximation E max t∈grows rapidly to ∞ as the Hurst index tends to 0.Keywords Fractional Brownian motion, Monte Carlo simulations, expected maximum, discrete approximation 2010 MSC 65C50, 60G22 IntroductionA fractional Brownian motion {B H (t), t ≥ 0} is a centered Gaussian stochastic process with covariance functionwhere H ∈ (0, 1) is the Hurst index. The fractional Brownian motion is a self-similar process with index H, that is, for any a > 0,= means the equality of finite-dimensional distributions.

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Bounds for expected maxima of Gaussian processes and their discrete approximations

Cited by 36 publications

References 26 publications

New and refined bounds for expected maxima of fractional Brownian motion

New and refined bounds for expected maxima of fractional Brownian motion

A Functional Limit Theorem for the Sine-Process

Simulation paradoxes related to a fractional Brownian motion with small Hurst index

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