A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION.

By M. Rosenblatt. 1956

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Citations in the corpus

Fairness in Criminal Justice Risk Assessments: The State of the Art (2017, background): “The IID requirement can be relaxed somewhat (Rosenblatt 1956; Wu 2005).”
Basic properties of strong mixing conditions. A survey and some open questions (2005, neutral): “Random elds”
Bias and Generalization in Deep Generative Models: An Empirical Study (2018, methodology): “In all of our experiments, the support of p(z) does not exceed 500, so we accurately approximate p(z) [25] with a reasonably sized dataset (100k-1M examples in our experiments).”
A Thinned Block Bootstrap Variance Estimation Procedure for Inhomogeneous Spatial Point Patterns (2007, unknown): “We do this by using the model-free strong mixing coefficient (Rosenblatt 1956), defined as α(p; k) ≡ sup{|P(A1 ∩ A2) − P(A1)P (A2)| : A1 ∈F(E1),A2 ∈F(E2), E2 = E1 + s, |E1| = |E2| ≤ p,d(E1,E2) ≥ k } .”
On the functional central limit theorem for stationary processes (2000, unknown): “The strong mixing coefficient of Rosenblatt [22] is defined by .”
On Conditional Density Estimation (2003, background): “The a-mixing condition (ROSENBLATT, 1956) is satisfied if there exists a sequence of nonnegative numbers called mixing coefficients (a(k)) such VVS, 2003 that lim kfi 1 a(k) ¼ 0 and for any A in F n1 ¼ rfðX1; Y1Þ; . . . ; ðXn; YnÞg and any set B in F1nþk ¼ rfðXnþk; YnþkÞ; . . .g, we have…”
An empirical central limit theorem for dependent sequences (2007, unknown): “In 1979, Yoshihara obtained the same result for α-mixing sequences in the sense of Rosenblatt (1956) under the condition α(n) = O(n−a) for some a > 3, and ∗Laboratoire de Statistique Théorique et Appliquée, Université Paris 6, Site Chevaleret, 13 rue Clisson, 75013 Paris, France.”
Nonparametric Estimation of Expected Shortfall (2007, unknown): “The α -mixing coefﬁcient introduced by Rosenblatt (1956) is The series is said to be α -mixing if lim k →∞ α ( k ) = 0.”
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.”
Nonparametric two‐step regression estimation when regressors and error are dependent (2000, unknown): “…(a), (d)–(g) hold and that in addition, (h) {ηt, νt,Xt} is a stationary sequence, (i) For some p > 2, E(|ν1|p) ∞, (j) {ηt, νt,Xt} is strong mixing (Rosenblatt, 1956) with mixing numbers αt for which∑∞t=1 α(p−2)/pt < ∞, (k) There is a sequence {ζn}, such that for some N > 0, supn>N…”
Advances on asymptotic normality in non-parametric functional time series analysis (2009, methodology): “We assume that the n pairs ( X i , Y i ) are α-mixing with the usual definition of α-mixing coefficients introduced by Rosenblatt 16 and given by Rio 12 , [ 10 , p.”
Recent advances in invariance principles for stationary sequences (2006, background): “Starting with the seminal works by Rosenblatt [62] and Ibragimov [27], the central limit theorem, maximal inequalities and the invariance principle were thoroughly analyzed by many mathematicians.”
On Additive Conditional Quantiles With High-Dimensional Covariates (2003, methodology): “ASYMPTOTIC BEHAVIOR In this section we derive the asymptotic behavior of the average quantile estimator under -mixing (see, e.g., Rosenblatt 1956).”
Entry and return times distribution (2013, unknown): “(5) Uniform mixing [18,19]: sup n sup U,V ∣∣∣∣ 1 k k ∑”
Split invariance principles for stationary processes. (2011, methodology): “In the case when {Yk, k ∈ Z} allows a Wiener–Rosenblatt representation Yk = f(εk, εk−1, . . .), k ∈ Z,(6) with an i.i.d. sequence {εk, k ∈ Z}, weak M-dependence is very close to Wu’s physical dependence condition in [49], except that we allow a larger freedom in choosing the approximating random var”
EFFICIENT SEMIPARAMETRIC ESTIMATION OF THE FAMA-FRENCH MODEL AND EXTENSIONS (2012, unknown): “The processes {Ut} is called strongly mixing (Rosenblatt (1956)) if sup 1≤t sup A∈F t−∞ B∈F ∞t+k |Pr(A∩B)− Pr(A)Pr(B)| ≡ α(k)→ 0 as k→ ∞ We make the following assumptions.”
How Generative Adversarial Networks and Their Variants Work (2017, background): “Boundary equilibrium GAN (BEGAN) [10] uses the fact that pixelwise loss distribution follows a normal distribution by CLT.”