Probability Theory: A Comprehensive Course

By A. Klenke. 2008

973 citations recorded by the source corpus.

Subfields: Management Science and Operations Research, Statistics and Probability

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Citations in the corpus

Wasserstein Fair Classification (2019, background): “The brackets 〈·, ·〉 denote the trace See Klenke (2013) for a formal definition of almost sure convergence of random variables.”
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients (2010, neutral): “…step is then to the bounds lim sup E for all p ∈ [1 , ∞ ) for the dominating stochastic processes (see Lemma 3.5). follow These nicely from Doob’s submartingale inequality (see, e.g., Theorem 11.2 in Klenke [19]), from uniform boundedness of σ ( ∈ ≥ 15) and (16) shows that sup for all p ∈ [1 , ∞ ).”
Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems (2015, background): “For more details and properties of conditional 95 expectation and regular conditional distributions see [19], [20] and [2], [3].”
Generalized Energy Based Models (2020, methodology): “Thus (Klenke, 2008, Theorem 6.28) applies, and it follows that ∇θ ∫ exp(−E(Gθ0(z)))dη(z)= ∫ ∇xE(Gθ0(z))∇θGθ0(z)exp(−E(Gθ0(z)))dη(z).”
The role of the information set for forecasting—with applications to risk management (2014, background): “The metric space (Θ, d) is separable; see Klenke (2008), page 252.”
Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation (2019, background): “8 Measure-Theoretic Details The content of this section draws from [14, 15].”
Loss of regularity for Kolmogorov equations (2012, background): “Moreover, Lemma 22.2 in Klenke [40] yields P[W1 > y] ≥ e − y 2 2 y √ 2π(1+y−2) ≥ e − y 2 2 y √ 8π ≥ e−4y2 for all y ∈ [1,∞).”
Statistical Solutions of Hyperbolic Conservation Laws: Foundations (2016, background): “Thus, any function m 2 : D 2 → R satisfying the properties of being a covariance function (see e.g. unique [37, Section 21.2]) corresponds to a Gaussian measure µ ∈ P ( L p ( D )), and vice versa . for For instance, Brownian motion is obtained by letting m 2 ( t, s ) = min( t, s ) t, s > 0.”
On the exact region determined by Kendall's τ and Spearman's ρ (2015, background): “For more details and properties of regular conditional distributions and disintegration we refer to [8, 9].”