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Concentration of scalar ergodic diffusions and some statistical implications

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Concentration of scalar ergodic diffusions and some statistical implications. / Aeckerle-Willems, Cathrine; Strauch, Claudia.
In: Annales de l'institut Henri Poincare (B) Probability and Statistics, Vol. 57, No. 4, 11.2021, p. 1857-1887.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

Harvard

Aeckerle-Willems, C & Strauch, C 2021, 'Concentration of scalar ergodic diffusions and some statistical implications', Annales de l'institut Henri Poincare (B) Probability and Statistics, vol. 57, no. 4, pp. 1857-1887. https://doi.org/10.1214/20-AIHP1144

Vancouver

Aeckerle-Willems C, Strauch C. Concentration of scalar ergodic diffusions and some statistical implications. Annales de l'institut Henri Poincare (B) Probability and Statistics. 2021 Nov;57(4):1857-1887. doi: 10.1214/20-AIHP1144

Author

Aeckerle-Willems, Cathrine ; Strauch, Claudia. / Concentration of scalar ergodic diffusions and some statistical implications. In: Annales de l'institut Henri Poincare (B) Probability and Statistics. 2021 ; Vol. 57, No. 4. pp. 1857-1887.

Bibtex

@article{6814b05b872e44c09861052a5fd03254,

title = "Concentration of scalar ergodic diffusions and some statistical implications",

abstract = "We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method. The developed machinery is of independent probabilistic interest and can serve as a starting point for investigations of other processes such as more general Markov processes, in particular multivariate or discretely observed diffusions. As a first concrete statistical application, we analyse the sup-norm error of estimating the invariant density of an ergodic diffusion via the local time estimator and the classical nonparametric kernel density estimator, respectively.",

keywords = "Concentration of diffusions, Ergodic diffusion, Exponential inequalities, Local time estimator",

author = "Cathrine Aeckerle-Willems and Claudia Strauch",

year = "2021",

month = nov,

doi = "10.1214/20-AIHP1144",

language = "English",

volume = "57",

pages = "1857--1887",

journal = "Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "4",

}

RIS

TY - JOUR

T1 - Concentration of scalar ergodic diffusions and some statistical implications

AU - Aeckerle-Willems, Cathrine

AU - Strauch, Claudia

PY - 2021/11

Y1 - 2021/11

N2 - We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method. The developed machinery is of independent probabilistic interest and can serve as a starting point for investigations of other processes such as more general Markov processes, in particular multivariate or discretely observed diffusions. As a first concrete statistical application, we analyse the sup-norm error of estimating the invariant density of an ergodic diffusion via the local time estimator and the classical nonparametric kernel density estimator, respectively.

AB - We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method. The developed machinery is of independent probabilistic interest and can serve as a starting point for investigations of other processes such as more general Markov processes, in particular multivariate or discretely observed diffusions. As a first concrete statistical application, we analyse the sup-norm error of estimating the invariant density of an ergodic diffusion via the local time estimator and the classical nonparametric kernel density estimator, respectively.

KW - Concentration of diffusions

KW - Ergodic diffusion

KW - Exponential inequalities

KW - Local time estimator

UR - http://www.scopus.com/inward/record.url?scp=85106079973&partnerID=8YFLogxK

U2 - 10.1214/20-AIHP1144

DO - 10.1214/20-AIHP1144

M3 - Journal article

AN - SCOPUS:85106079973

VL - 57

SP - 1857

EP - 1887

JO - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

SN - 0246-0203

IS - 4

ER -