By Jan Beran, Yuanhua Feng, Sucharita Ghosh, Rafal Kulik (auth.)

Long-memory procedures are recognized to play a major half in lots of parts of technological know-how and expertise, together with physics, geophysics, hydrology, telecommunications, economics, finance, climatology, and community engineering. within the final twenty years huge, immense development has been made in knowing the probabilistic foundations and statistical ideas of such methods. This booklet offers a well timed and accomplished assessment, together with an intensive dialogue of mathematical and probabilistic foundations and statistical equipment, emphasizing their sensible motivation and mathematical justification. Proofs of the most theorems are supplied and knowledge examples illustrate functional facets. This publication could be a necessary source for researchers and graduate scholars in information, arithmetic, econometrics and different quantitative components, in addition to for practitioners and utilized researchers who have to research facts within which lengthy reminiscence, strength legislation, self-similar scaling or fractal homes are relevant.

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**Extra resources for Long-Memory Processes: Probabilistic Properties and Statistical Methods**

**Example text**

Recall that Xˆ n+1 = nj=1 βnj Xn+1−j is the optimal linear prediction of Xn+1 given X1 , . . , Xn (also see Chap. 8). One may ask at this point why d = − 12 and 12 were excluded. The reason can be seen in the asymptotic behaviour of bj . 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.

If short memory is assumed but the actual value of d is larger than zero, then confidence intervals for μ = E(Xt ) will be too narrow by an increasing factor of nd , and the asymptotic level of tests based on this assumption will be zero. This effect is not negligible even for small sample sizes. 1 shows simulated rejection probabilities (based on 1000 simulations) for the t-test at the nominal 5 %-level of significance. 4 respectively (see Chap. 2, Sect. 4, for the definition of FARIMA models).

7 (Sect. 5). A different kind of nonstationarity is typical for financial time series. 17(a) shows daily values of the DAX index between 3 January 2000 and 12 September 2011. The series is nonstationary, but the first difference looks stationary (Fig. 17(b)), and the increments are uncorrelated (Fig. 17(c)). In this sense, the data resemble a random walk. However, there is an essential difference. Consider, as a measure of instantaneous volatility, the transformed series 1 Yt = | log Xt − log Xt−1 | 4 (see Ding and Granger 1996; Beran and Ocker 1999).