STRONG INVARIANCE PRINCIPLES FOR DEPENDENT RANDOM VARIABLES

By Wei Biao Wu. 2007

254 citations recorded by the source corpus.

We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/a4e739af46cd062e549619b2e71c0264e41171f1

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Citations in the corpus

Split invariance principles for stationary processes. (2011, methodology): “1), but, in contrast to the very restrictive condition of m-dependence, weak M-dependence holds for a huge class of stationary sequences, including those studied in Wu [49, 50] and Liu and Lin [28].”
Heteroscedasticity and Autocorrelation Robust Structural Change Detection (2013, background): “Lemma 6 is essentially an extension of Theorem 1 (i) and (iii) of Wu (2007) into nonstationary time series.”
Covariance and precision matrix estimation for high-dimensional time series (2013, background): “Write ξ jk = ˆ σ jk − σ jk . measure Case I: on the event A 1 jk , since the functional dependence for the product process it follows from the moment inequality Theorem 2.1 in Wu (2007) that where c q is a constant only depending on q .”
Structural breaks in time series (2013, background): “Recent contributions discussing the weak convergence of dependent random variables are Bradley (2009b), Dedecker et al. (2007), Wu (2007) and Aue et al. (2007).”
A bootstrap-assisted spectral test of white noise under unknown dependence (2011, background): “By Corollary 2(ii) (ii) of Wu (2007), ∑n t=1 Yt = o(n 1/2 log n) almost surely under the assumption (5).”
BANDING SAMPLE AUTOCOVARIANCE MATRICES OF STATIONARY PROCESSES (2009, unknown): “Then δα(i) = O(ri), 0 r 1, so (2.3) follows, see Theorem 5.1 in Shao and Wu (2007).”
Pointwise ergodic theorems with rate and application to limit theorems for stationary processes (2009, methodology): “Following Wu [19], we define • Sn := ∑n k=1Xk ∀n ≥ 1.”
Testing for change points in time series models and limiting theorems for NED sequences (2007, unknown): “By Proposition 1 of [37], we have ∥∥∥∥ max 1≤t≤2 |St| ∥∥∥∥ p ≤ k ∑”
Weak invariance principles for sums of dependent random functions (2013, background): “3) cannot be directly compared to the approximating martingale conditions of [27,28].”
Asymptotic normality of the principal components of functional time series (2013, background): “In particular, the notion of physical dependence of Wu [30,31] is closely related (condition (4) is replaced by a different condition).”
LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION FOR NONLINEAR PROCESSES (2007, background): “…of un+ Here $«t '%t Z is an i+i+d+ copy of $«t %t Z + The property ~9! indicates that the process $un% forgets its past exponentially fast, and it can be verified for many nonlinear time series models ~Wu and Min, 2005; Shao and Wu, 2007!+ Define the fourth cumulant spectral density f4~w1,w2 ,w3 !”
Identifying Cointegration by Eigenanalysis (2015, background): “s by contemporaneous linear transformation. A manuscript. [7] Chen,W.W.andHurvich,C.M.(2006). Semiparametric estimation of fractional cointegrating subspaces. The Annals of Statistics, 34, 2939–2979. [8] Dopico,F.M.,Moro,J.andMolera,J.M.(2000). Weyl-type relative perturbation bounds for eigensystems”
Komlós–Major–Tusnády approximation under dependence (2014, methodology): “Using martingale approximation and Skorohod embedding, Shao and Lu (1987) and Wu (2007) proved the approximation Sn = σB(n) + o(n 1/p(logn)γ) a.s.(1.3) with some σ ≥ 0, γ > 0 for some classes of stationary sequences (Xk) satisfying EX1 = 0, E|X1|p ∞ for some 2 p≤ 4.”
TESTING FOR WHITE NOISE UNDER UNKNOWN DEPENDENCE AND ITS APPLICATIONS TO DIAGNOSTIC CHECKING FOR TIME SERIES MODELS (2010, methodology): “According to Wu (2007), we have ‖P0 Zjk‖α ≤ C(δ2α(k)+ δ2α(k − j)1(k ≥ j)) if ut ∈ L2α , and δα(k) ≤ Crk for some r ∈ (0,1) provided that ut is GMC(α), α ≥ 1.”
Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples (2011, methodology): “Hence, by (41) and by using the dominated convergence theorem for discrete measures applied to the first term in the right-hand side of (44), we get that”
Asymptotic theory for stationary processes (2011, background): “Theorem 5 (Wu, 2007).”