Martingale Limit Theory and Its Application

By Peter Hall, E. Lukács, Z. Birnbaum, Chris C. Heyde. 1980

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Adaptivity and Computation-Statistics Tradeoffs for Kernel and Distance based High Dimensional Two Sample Testing (2015, methodology): “This seems to have ﬁrst been proved by Bai and Saranadasa [1996] for T BS using a martingale central limit theorem (see Hall and Heyde [2014]).”
Testing for Homogeneity with Kernel Fisher Discriminant Analysis (2007, methodology): “We use the central limit theorem (CLT) for triangular array of martingale diﬀerence (Hall and Heyde, 1980, Theorem 3.2).”
Basic properties of strong mixing conditions. A survey and some open questions (2005, background): “o full justice to it. Here are a few references on important topics not covered in this survey. For the approximation of mixing sequences by martingale diﬀerences, see e.g. the book by Hall and Heyde [80]. For the direct approximation of mixing random variables by independent ones, see e.g. [43, Cha”
Rate of convergence in probability to the Marchenko-Pastur law (2004, background): “…1 p trR 1 vp Ej (5)1 j2jtrR E trRj þ jE trRj pv Ej (5)1 j2: (4:32) Furthermore, 1 vp Ej (5)1 j2jtrR E trRj C vn3 EjtrR E trRj3: By Burkholder’s inequality for martingales (see Hall and Heyde 1980, p. 24), we obtain EjtrR E trRj3 C ffiffiffipp Xp k¼1 Ejªk j3, (4:33) with ªk defined in (4.13).”
Sieve inference on possibly misspecified semi-nonparametric time series models (2014, background): “Assumption 3.4 is a very mild one, and can be easily verified by applying any existing triangular array CLT for weakly dependent data (see, e.g., Hall and Heyde, 1980).”
Forecasting with Factor-Augmented Regression: A Frequentist Model Averaging Approach, Second Version (2013, unknown): “This result is similar to the theory of Hansen (2008) for forecast combination without estimated factors. With the generated regressors, the in-sample squared error LT (w) and theMallows criterion CT (w) are both inflated. However, Theorem 1 shows that the in-sample squared error and the Mallows cri”
Empirical Processes Based on Pseudo-observations 11: The Multivariate Case (2004, background): “Now (diTq,., Gi)lLi5n is a martingale difference and it follows from Rosenthal's inequality (e.g. Hall and Heyde 1980), that for some positive constant c. 404 Kilani Ghoudi and Bruno RBmillard End of the proof of Lemma 7.1 Set Gi = A.”
EMPIRICAL LIMITS FOR TIME SERIES ECONOMETRIC MODELS (2003, background): “…(in view of A1), it is reasonable to assume that t≤n tt // + B n is 'small' compared with B n , or, for each vector h, t≤n h tt dd h + h B n h = ooh B n hhh w. ploberger and p. c. b. phillips This requirement is a standard assumption in asymptotic theory, c.f. Hall and Heyde (1980, Ch. 6.).”
Asymptotically Unbiased Inference for a Dynamic Panel Model with Fixed Effects when Both N and T are Large (2002, unknown): “Using Lemma 10 along with the well-known relation between cumulants and central moments, we obtain E ·3 ε0DT ε− σ(2) 2 (−T + 1) ́2 ̧ = O ¡ T 2 ¢ , from which (16) follows.”
LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION FOR NONLINEAR PROCESSES (2007, background): “When d0 = 1, it follows from Theorem 1 in Hannan (1973). ♦ Proof of Proposition 3.”
Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process (2005, unknown): “Since ρ > 1, it follows from Chow’s theorem (see [11]) that Mn n→∞ −→ M∞ a.”