Wavelets for time series analysis - a survey and new results

In the paper we review stochastic properties of wavelet coefficients for time series indexed by continuous or discrete time. The main emphasis is on decorrelation property and its implications for data analysis. Some new properties are developed as the rates of correlation decay for the wavelet coefficients in the case of long-range dependent processes such as the fractional Gaussian noise and the fractional autoregressive integrated moving average processes. It is proved that for such processes the within-scale covariance of the wavelet coefficients at lag k is O(k^2(H-N)-2), where H is the Hurst exponent and N is the number of vanishing moments of the wavelet employed. Some applications of decorrelation property are briefly discussed

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 2(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c0(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω1) (resp. C([1,ω1])) contains a complemented copy of 2(ω1) (resp. c0(ω1)). As consequence, we give examples (J(ω1), C([1,ω1]), C(V ), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i. e., for every subspace Y ⊆ X we have that Dens (Y) = w∗ –Dens (Y ∗)), in spite of these spaces are not weakly Lindelof determined (WLD)