Time Series: Theory and Methods (2nd ed.).

By Rong Chen, Peter J. Brockwell, Richard A. Davis. 1992

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Uniform change point tests in high dimension (2015, background): “univariate case, we consider the polynomials Ah(z)= Xp j=0 α∗ j,hz j, B h(z)= Xq j=0 β∗ j,hz (5.2) j, 22 M. JIRAK where Ah(z) and Bh(z) are assumed to be relative prime. Then following, for instance, [11], one readily veriﬁes the following result. Proposition 5.3. If the associated polynomials Ch(z)”
Distribution free goodness-of-fit tests for linear processes (2006, unknown): “of j=0 in the deﬁnition of Gθ,T entails mean correction. That is, αθ,T is independent of both µand σ2. Under short-range dependence and H0, we have that max 1≤j≤Te E IX(λj) hθ0(λj) −Iε(λj) =o(1); see [7], Theorem 10.3.1, page 346. So, it is expected that αθ0,T will be asymptotically equivalent to Ba”
Szegö's theorem and its probabilistic descendants (2011, methodology): “nd the notation α = (αn) for the PACF and the coeﬃcients is justiﬁed: the two coincide. Indeed, the Szeg¨o recursion is known in the time-series literature as the Durbin- Levinson algorithm; see e.g. [BroDav], §§3.4, 5.2. The term Verblunsky coeﬃcient is from Simon [Si4], to which we refer repeatedl”
The Generalized Dynamic Factor Model (2005, unknown): “The model used in this article is an approximate factor model, in that the idiosyncratic component are allowed to be weakly correlated, as was done by Chamberlain (1983) and Chamberlain and Rothschild (1983) and in contrast to the approaches of Sargent and Sims (1977), Geweke (1977), and Quah and Sa”
Measures of serial extremal dependence and their estimation (2013, methodology): “For further spatio-temporal applications of max-stable random fields, see also Davis et al. [9].”
Continuous-time GARCH processes (2006, unknown): “ expression in the lower line of (3.9). Then (3.10) and, in particular, the existence of the limit variable Y∞ in the compound-Poisson case follow from (3.9) and a variant of Slutsky’s theorem (e.g., [11], Proposition 6.3.9), provided lim m→∞ limsup t→∞ (8.3) P(kYe t −Y t,mk B,r >ε)=0 ∀ε>0. Si”
Local polynomial estimation with a FARIMA-GARCH error process (2001, methodology): “Since X i de®ned in (1.2)±(1.3) is a zero-mean FARIMA process with innovations Ei following a GARCH model, we have X i X1 k 0 ckEiÿk (A:11) with ck (jø(1)j=jö(1)j)käÿ1 as for large K (see, for example, Brockwell and Davis 1991; Beran 1994).”
Modelling and Forecasting of Rainfall Time Series Using SARIMA (2017, unknown): “The mathematical formulation of a SARlMA(p,d,q)x(P, D, Q)s model can be written for time series yt as the following expression (Bockwell and Davis 1991, p. 323): Φ LS ϕ Lð ÞΔdΔsDyt ¼ θ0 þΘ LS θ Lð Þεt ð3Þ These models are specified as SARIMA(p, d, q) × (P, D, Q)S, where p is he order of nonseasonal…”
Filtering, Prediction and Simulation Methods for Noncausal Processes (2016, background): “The noncausal and mixed processes possess an infinite moving average representation: yt D C1X iD 1 ai"t i ; where ."t / is a sequence of i.i.d. variables and ai ; i D 1; : : : ;C1 a two-sided sequence of moving average coefficients (e.g. Brockwell and Davis, 1991).”
Futures pricing in electricity markets based on stable CARMA spot models (2012, unknown): “Brockwell and Davis [13]).”