Let N ≥ 2 be a given integer. Suppose that $df={\left(dfₙ\right)}_{n\ge 0}$ is a martingale difference sequence with values in $\ell {₁}^N$ and let ${\left(\epsilon ₙ\right)}_{n\ge 0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $\mathbb{P}(su{p}_{n\ge 0}\left|\right|{\sum }_{k=0}^n{\epsilon }_{k}d{f}_k{\left|\right|}_{\ell {₁}^N}\ge 1)\le (lnN+ln\left(3lnN\right))/(1-{\left(2lnN\right)}^{-1})su{p}_{n\ge 0}\left|\right|{\sum }_{k=0}^{n}d{f}_k{\left|\right|}_{\ell {₁}^N}$. It is shown that this result is asymptotically sharp in the sense that the least constant $C_N$ in the above estimate satisfies $li{m}_{N\to \infty }{C}_N/lnN=1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $\ell {₁}^N$.

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X=z\in X:su{p}_n\parallel {\sum }_{k=0}^n{T}^{k}z\parallel <\infty$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible Markov chains...

In [Probab. Theory Related Fields141 (2008) 543–567], the authors proved the uniqueness among the solutions of quadratic BSDEs with convex generators and unbounded terminal conditions which admit every exponential moments. In this paper, we prove that uniqueness holds among solutions which admit some given exponential moments. These exponential moments are natural as they are given by the existence theorem. Thanks to this uniqueness result we can strengthen the nonlinear Feynman–Kac formula proved...

Upper estimates are presented for the universal constant in the Katz-Petrov and Osipov inequalities which do not exceed 3.1905.

Let K be a compact, non-polar set in ℝm, m≥3 and let SKi(t)={Bi(s)+y: 0≤s≤t, y∈K} be Wiener sausages associated to independent brownian motions Bi, i=1, 2, 3 starting at 0. The expectation of volume of ⋂i=13SKi(t) with respect to product measure is obtained in terms of the equilibrium measure of K in the limit of large t.