Karamata's Theorem Research Articles

Karamata's theorem for regularized Cauchy transforms

We prove Abelian and Tauberian theorems for regularized Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularized Cauchy transform.

Matuszewska–Orlicz indices of the Sobolev conjugate Young function

In this note we study the Matuszewska–Orlicz indices of Young and φ -functions and their conjugates. It is known, for example, that the index at zero of the inverse of a φ -function corresponds to the reciprocal of the index at infinity of the φ -function itself, and vice-versa . Likewise, the index at zero of the complementary Young function matches the Hölder conjugate of the index at infinity and the same holds for the opposite index. In this article we prove that the Matuszewska–Orlicz indices of the Sobolev conjugate Young function are equal to the Sobolev conjugate of the corresponding indices of the Young function. We then provide some examples as well which highlight the importance of an asymptotic condition in connection with Karamata theorem. Finally, we present a few examples and applications in the context of Orlicz spaces.

Rectifying an inconsistency in F(R) gravity inflation

In this letter we shall provide a formal derivation of the inflationary slow-roll indices for F(R) gravity from the power spectrum without the condition , frequently used in the literature, where is the first slow-roll index. We shall employ Karamata's theorem for regularly varying functions, and as we show, we shall derive the same expressions for the scalar and tensor spectral index of F(R) gravity, also appearing in the current literature, without the misleading and rather strong condition . The only conditions that are needed are the slow-roll assumption, and the smallness of the slow-roll indices and of their derivatives during inflation.

One-component regular variation and graphical modeling of extremes

AbstractThe problem of inferring the distribution of a random vector given that its norm is large requires modeling a homogeneous limiting density. We suggest an approach based on graphical models which is suitable for high-dimensional vectors. We introduce the notion of one-component regular variation to describe a function that is regularly varying in its first component. We extend the representation and Karamata's theorem to one-component regularly varying functions, probability distributions and densities, and explain why these results are fundamental in multivariate extreme-value theory. We then generalize the Hammersley–Clifford theorem to relate asymptotic conditional independence to a factorization of the limiting density, and use it to model multivariate tails.

On functions derived from regularly varying functions

A generalization of Karamata's theorem on integrals of regularly varying functions is proved. Using Laplace-Stieltjes transforms it is shown that any regularly varying function with exponent α (α + 1 ∉ N) is asymptotic to another regularly varying function all of whose derivations are regularly varying.

An Abel-Tauber Theorem for Laplace Transforms

An Abelian and Tauberian theorem is proved that is a refinement of Karamata's theorem for slowly varying functions. The theorem is applied to probability distributions