Introduction to strong mixing conditions

By Richard C. Bradley. 2007

546 citations recorded by the source corpus.

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Citations in the corpus

Gaussian Approximation for High Dimensional Time Series (2015, background): “In the scalar case in which p = 1 or when p is ﬁxed, under suitable weak dependence conditions, we can have the central limit theorem (CLT) (1) See, for example, Rosenblatt (1956), Ibragimov and Linnik (1971), Wu (2005), Dedecker et al. (2007) and Bradley (2007) among others.”
Extensions of some classical methods in change point analysis (2014, methodology): “For surveys we refer to Berkes and Horváth (2002), Bradley (2007), Dedecker et al. (2007) and Doukhan (1994).”
Central limit theorem for stationary linear processes (2005, background): “(A∩B)−P(A)P(B)|;A∈F0 −∞,B∈F ∞ n }. We say that the strictly stationary sequence is strongly mixing if α(n)→0 as n→∞. Various examples of mixing sequences can be found in books by Rio [21] and Bradley [3], along with counterexamples showing that the conditions we use in the next corollary are sharp f”
Absolute regularity and ergodicity of Poisson count processes (2011, methodology): “Cox (1981) and later Davis, Dunsmuir and Wang (1999) classified these models into parameter-driven and observation-driven specifications.”
Split invariance principles for stationary processes. (2011, methodology): “This method yields very sharp results (for a complete account of the classical theory see Bradley [7]), but verifying mixing conditions of the above type is not easy and even when they apply (e.”
A new maximal inequality and invariance principle for stationary sequences (2004, unknown): “We observe that E[S2j (S2j+1 − S2j )|I] = E{E[S2j (S2j+1 − S2j )|F2j ]|I} [see, e.g., Proposition (2.2) in Bradley (2002), page 54].”
Space–time modelling of extreme events (2012, methodology): “For the definition of an α-mixing time series, see Bradley (2007), definition 1.6.”
Probability and moment inequalities under dependence (2013, methodology): “The latter quantities can be controlled by using various mixing coefficients in Bradley (2007), Rio (2000), and Dedecker et al. (2007).”
Breaking the curse of dimensionality in regression (2017, methodology): “Here, we assume that the data {(yt, xt, zt)}nt=1 is β-mixing with exponential decay; see Bradley (2007) for the definition of β-mixing conditions.”
Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes (2012, methodology): “We first choose a left matrix fraction description (P,Q) of the transfer function z 7→ C(zIN − A)−1B such that z 7→ detP (z) and z 7→ detQ(z) have no common zeros and z 7→ detP (z) has no multiple zeros.”
On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria (2007, background): “If X 0 is regular, the following orthogonal decomposition is valid: (2.1) Using the stationarity, we see that p n is a svf (see Theorem 8.13 in Bradley (2002)), and the same is true for This has been proved by Wu and Woodroofe (2004) if X 0 is M 0 -measurable, and extended to the general case by…”
Asymptotic results for the empirical process of stationary sequences (2009, unknown): “(See Berbee [4], Berkes and Philipp [9] and Bradley [17] for the case of β-, φ-and α-mixing, respectively.)”
Weakly dependent functional data (2010, background): “This method yields very sharp results (for a complete account of the classical theory (see Bradley [16]), but verifying mixing conditions of the above type is not easy, whereas the verification of L–m-approximability is almost immediate as our examples below show.”