Foundations of Modern Probability

By Olav Kallenberg. 2021

2,431 citations recorded by the source corpus.

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Citations in the corpus

Time Series Deconfounder: Estimating Treatment Effects over Time in the Presence of Hidden Confounders (2019, methodology): “The proof for Lemma 2 uses Lemma 2.22 in (Kallenberg, 2006) (kernels and randomization): Let µ be a probability kernel from a measurable space S to a Borel space T .”
Stochastic integration in quasi-Banach spaces (2006, result): “This can be seen in the same way as in the real case (cf. [ 18, Theorem 18.10 ]).”
The Blessings of Multiple Causes (2018, background): “” The Kallenberg construction is inspired by the idea of randomization variables, Uniform[0,1] variables from which we can construct a random variable with an arbitrary distribution (Kallenberg, 1997).”
Domain Generalization by Marginal Transfer Learning (2017, background): “The disintegration theorem (see for instance Kallenberg (2002), Theorem 6.4) tells us that (under suitable regularity properties, e.g., X is a Polish space) any element P XY ∈ P X×Y can be written as a product for any integrable function h : X × Y → R .”
Error bounds for kernel-based approximations of the Koopman operator (2023, background): “In this case, the Birkhoff ergodic theorem [20, Theorem 9.6] (see also (D.1)) and its generalizations apply, and allow us to calculate expectations w.r.t. µ using long-time averages over simulation data.”
General criteria for the study of quasi-stationarity (2017, background): “The case of general one-dimensional diffusion processes [52] can be handled using our framework, although using the infinitesimal generator is more tricky [50].”
Canonical RDEs and general semimartingales as rough paths (2017, methodology): “Suppose X → X in law for a Lévy process X (see [27], Theorem 13.”
On the Equivalence between Positional Node Embeddings and Structural Graph Representations (2019, neutral): “Theorem 2 is given in the Appendix. It uses structural causal models (Galles & Pearl, 1998; Halpern, 2000) in their measure theory form via the noise outsourcing (transfer) theorem (Austin, 2008; Kallenberg, 2006) to redeﬁne Z a.s.= (A;X;), where is a G-equivariant deterministic function of A, X”
Solving inverse problems using data-driven models (2019, background): “Then there exists a regular conditional distribution of the conditional random variable (f | g) (Kallenberg 2002, Theorem 6.3).”
Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension (2020, background): “20 in [59] that the random variable Uξ has uniform distribution on [0, 1].”