Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

Nazarov, Alexander I.; Sheipak, I. A.

doi:10.1112/blms/bdr056

Cited by 7 publications

(7 citation statements)

References 29 publications

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“…Интересные результаты были получены для гауссовских полей в нормах, связанных с фрактальными множествами или мерами (см. [99][100][101][102][103][104]). Заметим, что и в этой задаче для гриновских процессов в гильбертовой норме удается исследовать более тонкую структуру логарифмической асимптотики малых уклонений (см.…”

Section: функциональный закон повторного логарифма (фзпл)unclassified

“…Заметим, что и в этой задаче для гриновских процессов в гильбертовой норме удается исследовать более тонкую структуру логарифмической асимптотики малых уклонений (см. [99][100][101]104], а также серию работ А. А. Владимирова и И. А. Шейпака).…”

Section: функциональный закон повторного логарифма (фзпл)unclassified

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To the history of Saint-Petersburg school of Probability and Statistics. II. Random processes and dependent variables

Zaporozhets¹,

Ибрагимов²,

Лифшиц³

et al. 2018

Vestnik SPbSU. Mathematics. Mechanics. Astronomy

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Вторая статья из серии обзоров о научных достижениях Ленинградской-Санкт-Петербургской школы теории вероятностей и математической статистики в период с 1947 по 2017 г. посвящена работам в области предельных теорем для зависимых величин, в частности, цепей Маркова, последовательностей, обладающих свойствами перемешивания, и последовательностей, допускающих мартингальную аппроксимацию, а также различным аспектам теории случайных процессов. Особое внимание уделено гауссовским процессам, включая изопериметрические неравенства, оценки вероятностей малых уклонений в различных нормах и функциональный закон повторного логарифма. Даны краткий обзор и библиография работ в области аппроксимации случайных полей с параметром растущей размерности и вероятностных моделей систем притягивающихся неупругих частиц, в том числе законы больших чисел и оценки вероятностей больших уклонений. Ключевые слова: предельная теорема для сумм зависимых величин, гауссовские процессы, вероятности малых уклонений, аппроксимация процессов растущей размерности, функциональный закон повторного логарифма, системы притягивающихся частиц.

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Section: функциональный закон повторного логарифма (фзпл)unclassified

To the history of Saint-Petersburg school of Probability and Statistics. II. Random processes and dependent variables

Zaporozhets¹,

Ибрагимов²,

Лифшиц³

et al. 2018

Vestnik SPbSU. Mathematics. Mechanics. Astronomy

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“…In , as a particular case of Proposition 5.1 (see also Theorem 8.2), this result was generalized to the case of absolutely continuous measure with arbitrary continuous density. Now we are going to obtain the logarithmic L 2 ‐small ball asymptotics in the case of the discrete, degenerately self‐similar measure (see ,, ). We recall briefly the construction of this measure (for

d = 1

).…”

Section: Small Ball Asymptotics Examplesmentioning

confidence: 99%

“…It is shown in [, Section ] that under these conditions there exists a unique function f such that

S (f) = f

, where

scriptS

is the simplest similarity operator

S [f] (t) = δ \cdot f (a^{- 1} (t - α_{m})) + \sum_{k = 1}^{M} β_{k} \cdot χ_{] α_{k}, α_{k + 1} [} (t) .

Moreover, f increases on [0, 1], and

f (0) = 0

f (1) = 1

. The derivative

f^{'}

in the sence of distributions is a discrete probability measure μ on [0, 1] with a unique singular point

\hat{x} = \frac{α_{m}}{1 - a}

.…”

Section: Small Ball Asymptotics Examplesmentioning

confidence: 99%

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

Karol

Nazarov

2013

Mathematische Nachrichten

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Key words Small deviations, slowly varying functions, smooth covariances, tensor product of operators, spectral asymptotics MSC (2010) 60G15, 60G60, 47A80We find the logarithmic L 2 -small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein-Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper [6], where the regular behavior at infinity of marginal eigenvalues was assumed.

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“…Thus, the main term of the spectral asymptotics does not depend on the boundary conditions. Also, it follows from [4, Theorem 3.2] (see also [5,Lemma 5.1] for a simple variational proof), that relatively compact perturbations of the operator (e.g. lower order terms) do not affect the main term of the asymptotics given by (2) below.…”

Section: Introductionmentioning

confidence: 99%

On the spectrum of the Sturm-Liouville problem with arithmetically self-similar weight

Rastegaev¹

2020

Preprint

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Spectral asymptotics of the Sturm-Liouville problem with an arithmetically selfsimilar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method to estimate the eigenvalue counting function. This allows to consider a much wider class of self-similar measures.

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Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes

Abstract: We find the logarithmic small ball asymptotics for the L2-norm with respect to degenerate selfsimilar measures of a certain class of Gaussian processes, including Brownian motion, Ornstein-Uhlenbeck process and their integrated counterparts.

Cited by 7 publications

References 29 publications

To the history of Saint-Petersburg school of Probability and Statistics. II. Random processes and dependent variables

To the history of Saint-Petersburg school of Probability and Statistics. II. Random processes and dependent variables

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

On the spectrum of the Sturm-Liouville problem with arithmetically self-similar weight

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