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Approximation of a Random Process with Variable Smoothness
Actuarial Department, HEC Lausanne, University of Lausanne, 1015, Lausanne, Switzerland.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia .
Department of Mathematics and Mathematical Statistics, Umeå University,Umeå, Sweden.
2015 (English)In: Mathematical Statistics and Limit Theorems: Festschrift in Honour of Paul Deheuvels / [ed] Marc Hallin, David M. Mason, Dietmar Pfeifer and Josef G. Steinebach, Springer, 2015, 189-208 p.Chapter in book (Refereed)
##### Abstract [en]

We consider the rate of piecewise constant approximation to a locally stationary process X(t),t ε [0,1]  , having a variable smoothness index α(t). Assuming that α(⋅)  attains its unique minimum at zero and satisfies

$\small\alpha (t)=\alpha _0+b t^\gamma + \mathrm{o}(t^\gamma ) \quad {as} \ t \rightarrow 0,$

we propose a method for construction of observation points (composite dilated design) such that the integrated mean square error

$\small\int _0^1 {\mathbb E}\{(X(t)-X_n(t))^2\} dt \sim \frac{K}{n^{\alpha _0}(\log n)^{(\alpha _0+1)/\gamma }} \qquad {as} \ n\rightarrow \infty,$

where a piecewise constant approximation X n   is based on N(n)n  observations of X  . Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant K.

##### Place, publisher, year, edition, pages
Springer, 2015. 189-208 p.
##### National Category
Mathematics Mathematical Analysis Computational Mathematics
##### Identifiers
ISBN: 978-3-319-12441-4 (Print)ISBN: 978-3-319-12442-1 (online)OAI: oai:DiVA.org:liu-124315DiVA: diva2:897755
Available from: 2016-01-26 Created: 2016-01-26 Last updated: 2016-02-04

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