Reflection positivity on real intervals

We study functions f: (a, b) → R on open intervals in R with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel f(x+y2) is positive definite. We call f negative definite if, for every h\u3e 0 , the function e-hf is positive definite. Our first main result is a Lévy–Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For (a, b) = (0 , ∞) it generalizes classical results by Bernstein and Horn. On a symmetric interval (- a, a) , we call f reflection positive if it is positive definite and, in addition, the kernel f(x-y2) is positive definite. We likewise define reflection negative functions and obtain a Lévy–Khintchine formula for reflection negative functions on all of R. Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in R

Riesz transform characterization of H^1 spaces associated with certain Laguerre expansions

For alpha>0 we consider the system l_k^{(alpha-1)/2}(x) of the Laguerre functions which are eigenfunctions of the differential operator Lf =-\frac{d^2}{dx^2}f-\frac{alpha}{x}\frac{d}{dx}f+x^2 f. We define an atomic Hardy space H^1_{at}(X), which is a subspace of L^1((0,infty), x^alpha dx). Then we prove that the space H^1_{at}(X) is also characterized by the Riesz transform Rf=\sqrt{\pi}\frac{d}{dx}L^{-1/2}f in the sense that f\in H^1_{at}(X) if and only if f,Rf \in L^1((0,infty),x^alpha dx)

FPT is Characterized by Useful Obstruction Sets

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions.Comment: Extended abstract with appendix, as accepted to WG 201

Local proper scoring rules of order two

Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if it encourages truthful reporting. It is local of order $k$ if the score depends on the predictive density only through its value and the values of its derivatives of order up to $k$ at the realizing event. Complementing fundamental recent work by Parry, Dawid and Lauritzen, we characterize the local proper scoring rules of order 2 relative to a broad class of Lebesgue densities on the real line, using a different approach. In a data example, we use local and nonlocal proper scoring rules to assess statistically postprocessed ensemble weather forecasts.Comment: Published in at http://dx.doi.org/10.1214/12-AOS973 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Translation Invariant Kernels and Screw Functions

We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Bochner's theorem for translation invariant positive definite kernels on the real line (Rudin, 1962)