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Tail probability and singularity of Laplace-Stieltjes transform of a Pareto type random variable

Kenji Nakagawa (2015)

Applications of Mathematics

We give a sufficient condition for a non-negative random variable X to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution...

The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if f ( s ) = ∑ n = 1 ∞ a ₙ n - s with | | f | | ∞ : = s u p ℜ s > 0 | f ( s ) | < ∞ , then ∑ n = 1 ∞ | a ₙ | n - 2 ≤ | | f | | ∞ and even slightly better, and ∑ n = 1 ∞ | a ₙ | n - 1 / 2 ≤ C | | f | | ∞ , C being an absolute constant.

The Bohr-Pál theorem and the Sobolev space W ₂ 1 / 2

Vladimir Lebedev (2015)

Studia Mathematica

The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space W ₂ 1 / 2 ( ) . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if α < 1/2,...

The Fourier integral for a certain class of distributions

Josef Matušů (1991)

Applications of Mathematics

The aim of this paper is to derive by elementary means a theorem on the representation of certain distributions in the form of a Fourier integral. The approach chosen was found suitable especially for students of post-graduate courses at technical universities, where it is in some situations necessary to restrict a little the extent of the mathematical theory when concentrating on a technical problem.

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