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On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter. (English) Zbl 1150.62058

A process X is said to have memory parameter dR (M(d) process), if for any integer K>d-1/2, the Kth-order difference process Δ K X is weakly stationary with spectral density function

f Δ K X (λ)=|1-e -iλ | 2(K-d) f * (λ),λ(-π,π),

where f * is some non-negative symmetric function. The authors derive an explicit expression for the covariance and spectral density of the wavelet coefficients of an M(d) process at a given scale. If f * belongs to a class of smooth functions (β,L) it is shown that the spectral density of the wavelet coefficients of an M(d) process can be approximated, at large scales, by the spectral density of the wavelet coefficients of a fractional Brownian motion.

An explicit bound for the difference between these two quantities is derived. The authors show that the relative L error between the spectral densities of the wavelet coefficients decreases exponentially fast to zero with a rate given by the smoothness exponent β of f * . Gaussian processes are considered and an explicit expression for the limiting variance of the estimator of d based on the regression of the log-scale spectrum is obtained.

62M15Spectral analysis of processes
42C40Wavelets and other special systems
62M10Time series, auto-correlation, regression, etc. (statistics)