Convergence of Probability Measures

By P. M. Lee, P. Billingsley. 1970

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Citations in the corpus

On Convergence of Emphatic Temporal-Difference Learning (2015, neutral): “(A.22) This in turn implies that the sequence {‖W̃t,K‖, t ≥ 0} is uniformly integrable (see e.g., Billingsley, 1968, p. 32); i.e., sup t≥0 E [ ‖W̃t,K‖ · 1 ( ‖W̃t,K‖ ≥ a )] → 0 as a→ +∞, (where 1 ( ‖W̃t,K‖ ≥ a ) is the indicator for the event ‖W̃t,K‖ ≥ a).”
Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model (2010, methodology): “The processWa−1 is the corresponding fractional Brownian motion (fBM) of type II, and =⇒ is used for convergence in distribution as a process on a function space (Cp or Dp), see Billingsley (1968) or Kallenberg (2001).”
Distribution free goodness-of-fit tests for linear processes (2006, methodology): “ght continuous functions on [0,π] with left-hand 4 M. A. DELGADO, J. HIDALGO AND C. VELASCO side limits, the cadla`g space. The functional space D[0,π] is endowed with the Skorohod metric (see, e.g., [3]) and convergence in distribution in the corresponding topology will be denoted by “⇒”. Under sui”
Bootstrap Standard Error Estimates for Linear Regression (2005, methodology): “(4) t β̃ ∗ 2n = √ n ( β̃ ∗ 2n − β̂2n ) √ C ∗ n,22 .”
Optimal mass transportation and Mather theory (2004, unknown): “[35, 16, 9].”
Asymptotics of empirical processes of long memory moving averages with infinite variance (2001, background): “…normalized process n1=2(F̂n − F) converges weakly in the space D( R) with the sup-topology, which we denote by ⇒D( R) in the sequel, to a Gaussian process Z(x); x ∈ R; with zero mean and covariance E[Z(x)Z(y)]=F(x∧y)−F(x)F(y) (see e.g. Billingsley, 1968; Doukhan et al., 1995; Shao and Yu, 1996).”
Novel scaling limits for critical inhomogeneous random graphs (2009, unknown): “Again we use that convergence of the process for t ≥ 0 follows when the process converges for t ∈ [0 , T ] for all T > 0 (see [9, Lemma 3, page 173]).”