The Riesz transforms of a positive singular measure $\nu \in M\left({\mathbf{R}}^n\right)$ satisfy the weak type inequality$$m\left[\sum _{j=1}^n|R_j\nu |\>\lambda \right]\ge \frac{C\parallel \nu \parallel }{\lambda },\lambda \>0$$where $m$ denotes Lebesgue measure and $C$ is a positive constant only depending on $m$.

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $\parallel Y{\parallel }_W\le \left(2(\alpha +1)²\right)/(2\alpha +1)\parallel X{\parallel }_{L^{\infty }}$. Here W is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

For a sequence of dependent random variables ${\left(X_k\right)}_{k\in \mathbb{N}}$ we consider a large class of summability methods defined by R. Jajte in [jaj] as follows: For a pair of real-valued nonnegative functions g,h: ℝ⁺ → ℝ⁺ we define a sequence of “weighted averages” $1/g\left(n\right){\sum }_{k=1}^n\left(X_k\right)/h\left(k\right)$, where g and h satisfy some mild conditions. We investigate the almost sure behavior of such transformations. We also take a close look at the connection between the method of summation (that is the pair of functions (g,h)) and the coefficients that measure...

Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small when 1 is a simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤1/2; otherwise, it is called large).

P. Samek and D. Volný, in the paper ``Uniqueness of a martingale-coboundary decomposition of a stationary processes" (1992), showed the uniqueness of martingale-coboundary decomposition of strictly stationary processes. The original proof is given by reducing the problem to the ergodic case. In this note we give another proof without such reduction.

Some duality results and some inequalities are proved for two-parameter Vilenkin martingales, for Fourier backwards martingales and for Vilenkin and Fourier coefficients.

L. C. G. Rogers has given an elementary proof of the fundamental theorem of asset pricing in the case of finite discrete time, due originally to Dalang, Morton and Willinger. The purpose of this paper is to give an even simpler proof of this important theorem without using the existence of regular conditional distribution, in contrast to Rogers' proof.

Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered. Their dual spaces are characterized and an inequality due to Stein and Lepingle is extended.

We consider markets with proportional transaction costs and shortsale restrictions. We give necessary and sufficient conditions for the absence of arbitrage and also estimate the super-replication price.

This paper is mainly devoted to establishing an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. More precisely, let $(\Omega ,ℱ,\mathbb{P})$ be a probability space and $p(·):\Omega \to (0,\infty )$ be a $ℱ$-measurable function such that $0<{inf}_{x\in \Omega }p\left(x\right)\le {sup}_{x\in \Omega }p\left(x\right)<\infty$. It is proved that a predictable martingale Hardy space ${𝒫}_{p(·)}$ has an atomic decomposition by some key observations and new techniques. As an application, we obtain the boundedness of fractional integrals on the predictable martingale Hardy space with...