Osaka Journal of Mathematics
- Osaka J. Math.
- Volume 51, Number 1 (2014), 1-39.
On a localization property of wavelet coefficients for processes with stationary increments, and applications. II. Localization with respect to scale
Wavelet coefficients of a process have arguments shift and scale. It can thus be viewed as a time series along shift for each scale. We have considered in the previous study general wavelet coefficient domain estimators and revealed a localization property with respect to shift. In this paper, we formulate the localization property with respect to scale, which is more difficult than that of shift. Two factors that govern the decay rate of cross-scale covariance are indicated. The factors are both functions of vanishing moments and scale-lags. The localization property is then successfully applied to formulate limiting variance in the central limit theorem associated with Hurst index estimation problem of fractional Brownian motion. Especially, we can find the optimal upper bound $J$ of scales $1, \ldots, J$ used in the estimation to be $J = 5$ by an evaluation of the diagonal component of the limiting variance, in virtue of the scale localization property.
Osaka J. Math., Volume 51, Number 1 (2014), 1-39.
First available in Project Euclid: 8 April 2014
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G22: Fractional processes, including fractional Brownian motion 65T60: Wavelets 60F05: Central limit and other weak theorems
Secondary: 60G18: Self-similar processes 60G15: Gaussian processes 62F03: Hypothesis testing 62J10: Analysis of variance and covariance
Albeverio, Sergio; Kawasaki, Shuji. On a localization property of wavelet coefficients for processes with stationary increments, and applications. II. Localization with respect to scale. Osaka J. Math. 51 (2014), no. 1, 1--39. https://projecteuclid.org/euclid.ojm/1396966222