Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function

Harper, Adam J.

doi:10.1214/12-aap847

Cited by 52 publications

(90 citation statements)

References 17 publications

(29 reference statements)

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“…Numerous papers have considered the calculation of Pickands constants, with particular focus on the case δ = 0; see for instance Shao (1996); Hüsler and Piterbarg (1999); Dȩbicki et al (2003); Dȩbicki (2005); Dȩbicki and Kisowski (2008); Harper (2013Harper ( , 2015. Recently, the seminal contribution Dieker and Yakir (2014) The principal advantage of Dieker-Yakir representation (3) is that it is given as an expectation rather than as a limit, which is particularly useful for Monte Carlo simulations of H δ W .…”

Section: Introductionmentioning

confidence: 99%

Generalized Pickands constants and stationary max-stable processes

2017

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Abstract. Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

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Section: Introductionmentioning

confidence: 99%

Generalized Pickands constants and stationary max-stable processes

2017

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“…In the following proof, we have adapted the method of [, Section 2] which relies crucially on Harper's work . We refer to [, pp.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…As in [, Lem. 1], we can modify the proof of [, Corollary 2] to show that

\begin{matrix} true \\ left sup_{\begin{matrix} 1 ⩽ t ⩽ 2 {(log log x)}^{2} \\ false| 1 - 2^{- i t} false| ⩾ 1 / 4 \end{matrix}} ((Re (0true \sum_{p ⩽ x} z false(p false) p^{- 1 / 2 - 1 / log x - i t})) + () \frac{1}{2 α} Re (0true \sum_{p ⩽ x} z {(p)}^{2} p^{- 1 - 2 / log x - 2 i t})) \\ left ⩾ prefixlog prefixlog x - prefixlog prefixlog prefixlog x + O ((), {(log log log x)}^{3 / 4}) \end{matrix}

with probability

1 - o (1)

x \to \infty

. To achieve this, we add a minor technical detail: In the part of the argument that follows [, Section 6], we only take into account integers

1 ⩽ n ⩽ {(log log x)}^{2}

, such that

\underset{2 n + 1 ⩽ t ⩽ 2 n + 2}{trueprefixmin} false| 1 - 2^{- i t} false| ⩾ 1 / 4,

noting that the number of such

n

is bounded below by a constant times

{false(prefixlog prefixlog x false)}^{2}

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For the proof of (17), we may assume that 2k < q < 2(k + 1) since q 2. We use Lemma 4 (in reverse) with = 2k/q and (13) to conclude that…”

Section: Hardy-littlewood Inequalitiesmentioning

confidence: 99%

“…Using this terminology, we may think of Z N as a sum of random multiplicative functions. A probabilistic approach to problems of computing integral moments, based on this viewpoint, can be found in recent work of Harper [11,13] and Harper, Nikeghbali and Radziwi l l [14]. In many situations, both approaches apply equally well, but sometimes one viewpoint is more illuminating and profitable than the other, and occasionally it is useful to combine them.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Pseudomoments of the Riemann zeta function

Bondarenko

Brevig

Saksman

et al. 2018

Bull. London Math. Soc.

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The 2kth pseudomoments of the Riemann zeta function ζ(s) are, following Conrey and Gamburd, the 2kth integral moments of the partial sums of ζ(s) on the critical line. For fixed k>1/2, these moments are known to grow like false(prefixlogNfalse)k2, where N is the length of the partial sum, but the true order of magnitude remains unknown when k⩽1/2. We deduce new Hardy–Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when k→∞. In the case k<1/2, we consider pseudomoments of ζαfalse(sfalse) for α>1 and the question of whether the lower bound false(prefixlogNfalse)k2α2 known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali and Radziwiłł and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by logN to a power larger than k2α2 when k<1/e and N is sufficiently large.

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Gaussian Process-Based Confidence Estimation for Hybrid System Falsification

Zhang

Arcaini

2021

Lecture Notes in Computer Science

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Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function

Cited by 52 publications

References 17 publications

Generalized Pickands constants and stationary max-stable processes

Generalized Pickands constants and stationary max-stable processes

Pseudomoments of the Riemann zeta function

Gaussian Process-Based Confidence Estimation for Hybrid System Falsification

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