Basic properties of strong mixing conditions. A survey and some open questions

By Richard C. Bradley. 2005

1,049 citations recorded by the source corpus.

This is an update of, and a supplement to, a 1986 survey paper by the author on basic properties of strong mixing conditions.

Subfields: Geometry and Topology, Mathematical Physics, Probabilistic & Bayesian Methods

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Citations in the corpus

Stability and Generalization in Structured Prediction (2016, methodology): “1 Related Work One of the earliest explorations of generalization in structured prediction is by Collins (2001), who developed risk bounds for language parsers using various classical tools, such as the Vapnik-Chervonenkis dimension and margin theory.”
Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence (2013, positive): “This is because, according to Theorem 5.2 in Bradley (2005), ρZ(s)(k)≤ sup i≤n ρXi(k) = ρX1(k) for each k = 1, . . . , p and s ∈ (0,1).”
The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics (2012, background): “ of correlations of the dynamical system. This is where Dp(un)∗ is seen to be much more useful then Leadbetter’s D(un). While D(un) usually follows only from strong uniform mixing, like α-mixing (see [B05] for deﬁnitions), and even then only along THE COMPOUND POISSON DISTRIBUTION IN NON-UNIFORMLY H”
Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions (2014, background): “Recently, Cattaneo and Farrell (2013) proved that a local polynomial partitioning regression estimator can attain the optimal sup-norm rate under i.”
On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls (2019, background): “ing J = 1 and considering only i = 0. The proof that lJ 1 T 0 P t∈T ǫtǫ0t ∞ = op(1) follows from noting that, since ǫ0t and ǫit are β-mixing and independent, ǫ0tǫit is also β-mixing (Theorem 5.2 from Bradley (2005)). Moreover, from Assumption 3.2, we have, similar to equation 39, P t∈Hp E|˜ǫ0tǫ˜it| ”
Two-Stage Procedures for High-Dimensional Data (2011, background): “2 in Bradley (2005), we note from (A-iii) that {Yj/W} is a strictly stationary sequence and ρ-mixing.”
PCA CONSISTENCY IN HIGH DIMENSION, LOW SAMPLE SIZE CONTEXT (2009, background): “See Bradley [5] for a clear and insightful discussion.”
COPULAS AND TEMPORAL DEPENDENCE (2008, background): “Since {Zt} is a stationary Markov chain, it is known (see e.g. section 3.1 in Bradley, 2005) that its β-mixing coefficients satisfy βk = 1 2 sup {Ai},{Bj} I∑ i=1 J∑ j=1”