Exact spectral asymptotics of fractional processes

Chigansky, P.; Kleptsyna, M.; Marushkevych, D.

doi:10.48550/arxiv.1802.09045

Cited by 3 publications

(4 citation statements)

References 16 publications

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“…Example 2.7. If G is a fractional Brownian motion with Hurst exponent H ∈ (0, 1), then, by Chigansky and Kleptsyna [2018], the corresponding eigenvalues satisfy (2.3) with δ = 2H + 1:…”

Section: Methodology and Main Resultsmentioning

confidence: 99%

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Asymptotic efficiency for covariance estimation under noise and asynchronicity

Holtz

2018

Preprint

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The estimation of the covariance structure from a discretely observed multivariate Gaussian process under asynchronicity and noise is analysed under high-frequency asymptotics. Asymptotic lower and upper bounds are established for a general Gaussian framework which provides benchmark cases for various Gaussian process models of interest. The parametric bounds give rise to infinitedimensional convolution theorems for covariation estimation under asynchronicity, which is an essential estimation problem in finance.

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“…Example 2.7. If G is a fractional Brownian motion with Hurst exponent H ∈ (0, 1), then, by Chigansky and Kleptsyna [2018], the corresponding eigenvalues satisfy (2.3) with δ = 2H + 1:…”

Section: Methodology and Main Resultsmentioning

confidence: 99%

“…Similarly a fractional Brownian bridge and a fractional Ornstein-Uhlenbeck process seem to offer the same asymptotics as λ fBM p , cf. the (yet unpublished) drafts by Chigansky et al [2017] and Chigansky et al [2018].…”

Section: Methodology and Main Resultsmentioning

confidence: 99%

Asymptotic efficiency for covariance estimation under noise and asynchronicity

Holtz

2018

Preprint

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“…[7]), the Ornstein-Uhlenbeck process (cf. [8]), etc., can be solved in a similar way. Compared to [5], the profile of the eigenpair analyzed with the method in [6] is more complete and accurate.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Karhunen–Loève Eigenvalues for Sub-Fractional Brownian Motion and Its Application

Cai

Wang

2021

Fractal Fract

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In the present paper, the Karhunen–Loève eigenvalues for a sub-fractional Brownian motion are considered. Rigorous large n asymptotics for those eigenvalues are shown, based on the functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotical estimates for the small L2-ball probabilities for a sub-fractional Brownian motion are derived. Asymptotic analysis on the Karhunen–Loève eigenvalues for the corresponding “derivative” process is also established.

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“…Many processes such as Wiener process, Brownian bridge, Ornstein-Uhlenbeck process and their integrated counterparts are Green processes. Let us mention also an important series of works [21,22,23], where the exact asymptotics were obtained for non Green processes.…”

Section: Introductionmentioning

confidence: 99%

$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

Petrova¹

2019

Preprint

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In this article we study the small ball probabilities in L 2 -norm for a family of finite-dimensional perturbations of Gaussian functions. We define three types of perturbations: non-critical, partially critical and critical; and derive small ball asymptotics for the perturbated process in terms of the small ball asymptotics for the original process.The natural examples of such perturbations appear in statistics in the study of empirical processes with estimated parameters (the so-called Durbin's processes). We show that the Durbin's processes are critical perturbations of the Brownian bridge. Under some additional assumptions, general results can be simplified. As an example we find the exact L 2 -small ball asymptotics for critical perturbations of the Green processes (the processes which covariance function is the Green function of the ordinary differential operator).

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Exact spectral asymptotics of fractional processes

Cited by 3 publications

References 16 publications

Asymptotic efficiency for covariance estimation under noise and asynchronicity

Asymptotic efficiency for covariance estimation under noise and asynchronicity

Asymptotics of Karhunen–Loève Eigenvalues for Sub-Fractional Brownian Motion and Its Application

$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

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