Real analysis and probability

By R. Ash, E. Lukács, Z. Birnbaum. 1975

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In Pure Mathematics

Citations in the corpus

On the optimal estimation of probability measures in weak and strong topologies (2013, neutral): “The result therefore follows from [11, Corollary 11.3.4] which shows that if Comparing (13) and (14), it is clear that one requires a stronger condition for weak convergence than for · k F H being just a metric. k However, these conditions can be shown to be equivalent for bounded continuous…”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “One important theoretical question that is usually considered for metrics on probability measures is (Dudley, 2002, Chapter 11): “What is the nature of the topology induced by the probability metric in relation to the usual weak topology?””
On the empirical estimation of integral probability metrics (2012, neutral): “3 of [15], gα in (2.”
GenDICE: Generalized Offline Estimation of Stationary Values (2020, neutral): “, 2006) requires ‖f‖Hk ≤ 1 where Hk denotes the RKHS with kernel k; the Dudley metric (Dudley, 2002) requires ‖f‖BL ≤ 1 where ‖f‖BL := ‖f‖∞ + ‖∇f‖2; and Wasserstein distance (Arjovsky et al.”
On Convergence of Emphatic Temporal-Difference Learning (2015, background): “Theorem 3.1 is useful here because on the separable metric space S×A×Rn+1, bounded Lipschitz continuous functions are convergence-determining for weak convergence of probability measures (Dudley, 2002, Theorem 11.3.3).”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, methodology): “Another application is in metrizing the weak convergence of probability measures on a separable metric space, where the Lévy-Prohorov distance (Dudley, 2002, Chapter 11) and dual-bounded Lipschitz distance (also called the Dudley metric) (Dudley, 2002, Chapter 11) are commonly used.”
A Model of Inductive Bias Learning (2000, background): “For more details see Dudley (1989), section 13.2.”
The Cramer Distance as a Solution to Biased Wasserstein Gradients (2017, methodology): “We will use the term probability distance to mean a symmetric divergence satisfying the relaxed triangle inequality We will ﬁrst study the p -Wasserstein metrics w p (Dudley, 2002).”
On integral probability metrics, φ-divergences and binary classification (2009, methodology): “The famous Kantorovich-Rubinstein theorem [9, Theorem 11.8.2] shows that when M is separable, the Kantorovich metric is the dual representation of the so called Wasserstein distance deﬁned as W 1 ( P , Q ) := inf µ L ( P , Q ) ρ ( x, y ) dµ ( x, y ) , where P , Q ∈ P 1 := { P : ρ ( x, y ) d P ( x )…”
On Choosing and Bounding Probability Metrics (2002, background): “Dudley (1989, p. 342) traces some of the history of these metrics. χ 2 -distance.”
Nonparametric estimation for L\'evy processes from low-frequency observations (2007, unknown): “Accordingly, we consider the following class of functions: Gk = { z 7→ w(u) ∂k ∂uk cos(uz) ∣∣∣u ∈ R} ∪ {z 7→ w(u) ∂k ∂uk sin(uz)∣∣∣u ∈ R} . An envelope function fk for Gk is given by fk = |x|k. Now we obtain from Corollary 19.35 in van der Vaart (1998) that”
Distributed Algorithms for Aggregative Games on Graphs (2016, unknown): “[10], pages 476–479), we have for any q > 0, with probability 1, lim k→∞ |Γk(i)− kpi| k 1 2 +q = 0 for all i ∈ N .”
Stochastic stability in monotone economies (2014, unknown): “See, for example, Olson (1989), Amir (2002), Gong et al. (2010), Balbus et al.”
Injective Hilbert Space Embeddings of Probability Measures (2008, methodology): “We lead to these embeddings by ﬁrst introducing the maximum mean discrepancy (MMD), which is based on the following result [Dud02, Lemma 9.3.2], related to the weak convergence of probability measures on metric spaces.”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, background): “7 in [18] and linearity of b 7→ ∫ T 0 b(Xt).”
Bisimulation Metrics for Continuous Markov Decision Processes (2011, background): “Results will be stated without proof and can be found in most classical texts on probability and analysis such as [55, 29, 20, 4].”