Review: P. Billingsley, Convergence of probability measures

By R. M. Dudley. 1971

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Subfields: Probabilistic & Bayesian Methods

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Probabilistic & Bayesian Methods

Citations in the corpus

Adversarial Example Games (2020, methodology): “By using Skorokhod’s representation theorem [7] we can show that this set is closed and thus compact (as closed subsets of compact sets are compact) using the weak convergence of measures as topology.”
Kernel-based Conditional Independence Test and Application in Causal Discovery (2011, neutral): “2 of Billingsley (1999).”
Lifelong Policy Gradient Learning of Factored Policies for Faster Training Without Forgetting (2020, methodology): “Second, (cid:107) f (cid:107) ∞ is bounded given Claim 1 and Assumption A. Finally, by Assumptions A and B, the corollary applies to the tuples H ( t ) , g ( t ) , α ( t ) [3].”
A General Family of Robust Stochastic Operators for Reinforcement Learning (2018, methodology): “Since the value-iteration sequence generated under our family of stochastic operators will be based on realizations of random variables, our theoretical results include establishing the fact that the random variables converge almost surely [5] to the same limit that produces the optimal actions.”
Gaussian Process Behaviour in Wide Deep Neural Networks (2018, background): “Convergence of the expectation of a sequence of random variables can be ensured if the sequence is uniformly integrable and the sequence converges in distribution (Billingsley, 1999).”
Inverse Reinforcement Learning in Swarm Systems (2016, background): “|V (St | π)| < V ∗ for some V ∗ ∈ R, it holds by conditional dominated convergence theorem [5] that”
Central Moment Discrepancy (CMD) for Domain-Invariant Representation Learning (2017, background): “Concerning approximation (c), the joint application of the weak law of large numbers (Billings-ley, 2008) with the continuous mapping theorem (Billingsley, 2013) proves that this approximation creates a consistent estimate.”
On the connection between symmetric $N$-player games and mean field games (2014, methodology): “By continuity of b, σ according to Assumption (A2), the almost sure continuity of τk under ⊗ λ, the extended mapping theorem [Theorem 5.5 in Billingsley (1968), page 34] applied to the relaxed controls in (4.3) (plus convergence of first moments by choice of the topology on R2), and the fact that ∈…”
Multiscale change point inference (2013, background): “(i;j)2I(c) p j i+ 1 jY i r 2log en j i+ 1 : It hence follows from Lemma 7.5 that jTc n (Y;) h n(X n;c)j max (i;j)2I(c) j q 2Tj i (Y;) p j i+ 1 Y i = o P(1): (49) Since X n ! DB, Lemma 7.7 and (Billingsley, 1968, Thm. 5.5) imply that h n(X n;c) !; h(B;c). Theorem 4.1 in (Billingsley, 1968) and (49)”
Weak convergence and empirical processes (2019, background): “Our main reference for this section would be Billingsley (2013) and also the previous editions of that book.”
Order Book Dynamics in Liquid Markets: Limit Theorems and Diffusion Approximations (2012, methodology): “), we can apply the continuous mapping theorem (Billingsley 1968, Theorem 5.1), to conclude that Let us now assume that we have deﬁned Q on [0 , τ k − 1 ] and shown that ) lies with probability 1 in the continuity set of Ψ k , so by the continuous mapping theorem As before, by continuity of the…”
Mathematics of Two-Dimensional Turbulence (2012, methodology): “2 of [Bil99] implies that D(ζ̃ ε )→ D(ζ) in P ( C(0, t0;H) ) as ε→ 0, where the process ζ is defined in the proof of Theorem 2.”
Horvitz--Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling (2009, unknown): “We have with (14), for all (s, t) ∈ [0, T ] × [0, T ], E{|Xn(t) − Xn(s)|2} ≤ C7|t − s|2β, and the sequence Xn is tight, when β > 1/2, according to Theorem 12.3 of Billingsley (1968).”
A simple test of changes in mean in the possible presence of long‐range dependence (2011, background): “Let D1⁄20; 1 be the space of functions on [0,1] which are right continuous and have left limits, endowed with the Skorokhod topology (Billingsley, 1968).”