By Jan Beran, Yuanhua Feng, Sucharita Ghosh, Rafal Kulik (auth.)
Long-memory approaches are recognized to play an incredible half in lots of components of technology and know-how, together with physics, geophysics, hydrology, telecommunications, economics, finance, climatology, and community engineering. within the final twenty years huge, immense development has been made in figuring out the probabilistic foundations and statistical rules of such methods. This e-book presents a well timed and entire evaluation, together with an intensive dialogue of mathematical and probabilistic foundations and statistical tools, emphasizing their functional motivation and mathematical justification. Proofs of the most theorems are supplied and information examples illustrate sensible features. This publication may be a invaluable source for researchers and graduate scholars in data, arithmetic, econometrics and different quantitative parts, in addition to for practitioners and utilized researchers who have to research info during which lengthy reminiscence, energy legislation, self-similar scaling or fractal homes are relevant.
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Extra info for Long-Memory Processes: Probabilistic Properties and Statistical Methods
The main problem with this model is that σt and εt are not directly observable. One would however like to be able to separate the components σt and εt even though only their product Xs (s ≤ t) is observed. g. Giraitis and Robinson 2001). One therefore often prefers to assume a recursive relation between vt and past values of Xt . The difficulty that arises then is to prove the existence of a stationary solution and to see what type of volatility dependence is actually achieved. For instance, in the so-called ARCH(∞) model (Robinson 1991; Giraitis et al.
For d = − 12 , the coefficients bj are 1 proportional to j − 2 , so that bj2 = ∞, and A−1 (e−iλ ) is no longer in L2 (Fε ). This means that Xt is no longer invertible, even though the process Xt = A(B)εt is well defined. The same comments apply to d = − 12 + m where m is a positive integer, since the mth difference of Xt is not invertible, and to d = − 12 + m with m a negative integer, since there Xt is the mth difference of a noninvertible process. 13) can be extended by replacing Zt = ϕ −1 (B)ψ(B)εt by any L2 -linear short-memory process.
15(d) was obtained for a patient suffering from Parkinson’s disease. The complete data set consists of patients with Parkinson’s disease (N = 15), Huntington’s disease (N = 20) and amyotrophic lateral sclerosis (N = 13), as well as a control group (N = 16) (Hausdorff et al. 1997, 2000). Both series in Figs. 15(a) and (d) contain a spiky, somewhat periodic but also irregular, component. A natural approach to analysing such data is to decompose them into a ‘spiky’ component and the rest. Here, kernel smoothing is not appropriate because it tends to blur sharp peaks.
Long-Memory Processes: Probabilistic Properties and Statistical Methods by Jan Beran, Yuanhua Feng, Sucharita Ghosh, Rafal Kulik (auth.)