Stochastic processes with sample paths in reproducing kernel Hilbert spaces

By Milan Lukić, Jay H. Beder. 2001

133 citations recorded by the source corpus.

Subfields: Applied Mathematics, Finance, Mathematical Physics

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Citations in the corpus

Assessing Nonlinear Granger Causality from Multivariate Time Series (2008, background): “An RKHS (HY ,k Y ) is separable, when the topological space Y is separable 444 X. Sun and kY is continuous on Y×Y [ 16 ].”
On the Use of Reproducing Kernel Hilbert Spaces in Functional Classification (2015, background): “(a) Note that, except for trivial cases, the trajectories x of the process X ( t ) are not included, with probability one, in H ( K ); see, e.g., (Berlinet and Thomas-Agnan, 2004, p. 66) and Luki´c and Beder (2001) for details.”
Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences (2018, methodology): “We then review Driscoll’s theorem [Driscoll, 1973, Lukić and Beder, 2001], which provides a necessary and sufficient condition for a GP sample path to lie in a given RKHS (which can be different from the RKHS associated with the GP covariance kernel) in Section 4.2.”
Convergence properties of the expected improvement algorithm with fixed mean and covariance functions (2007, unknown): “From the probabilistic point of view, however, the event { ξ ( · ) ∈ H} almost never happens according to Driscoll’s theorem (Lukic and Beder, 2001). paper states The second result of this that the result of Theorem 6 also holds P -almost surely in Ω .”
A new kernel-based approach for linear system identification (2010, background): “For known y and γ, the minimum variance estimate of f is given by f̂ = arg min g∈H⊕B n∑ i=1 (yi − Lui [g]) 2 + γ‖P[g]‖2H (4) Remark 2 The above proposition states the duality between Gaussian processes and RKHS [48,26], which will be further exploited in the sequel.”
Separability of reproducing kernel spaces (2015, background): “For example, Steinwart and Christmann’s [33, Thm. 7.22] oracle inequality for SVMs, Christmann and Steinwart [8, Thms. 7 & 12], [7], [6], Steinwart and Christmann [34], [35], De Vito, Rosasco and Toigo [9], Hable and Christmann [18, Thm. 3.2], Lukić and Beder [25], Steinwart [32] and Vovk [38, Thm.”