In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ${\psi }_p\left(x\right)={exp\left(\right|x|}^p)-1$ with 0 < p ≤ 2) of $ma{x}_{0\le t\le \tau }\left|B_t\right|$ or $|B_{\tau }|$ to be finite, where $B={\left(B_t\right)}_{t\ge 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}<\infty$ as soon as $E\left({\tau }^k\right)=O\left(C^k{k}^k\right)$ for some constant C > 0 as k → ∞ (or equivalently $\parallel \tau {\parallel }_{{\psi }_1}<\infty$). In particular, if τ ∼ Exp(λ) or $\left|N\right(0,{\sigma }^2\left)\right|$ then the last condition is satisfied, and we obtain $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}\le K\surd E\left(\tau \right)$ with some universal constant K > 0....

In what concerns extreme values modeling, heavy tailed autoregressive processes defined with the minimum or maximum operator have proved to be good alternatives to classical linear ARMA with heavy tailed marginals (Davis and Resnick [8], Ferreira and Canto e Castro [13]). In this paper we present a complete characterization of the tail behavior of the autoregressive Pareto process known as Yeh-Arnold-Robertson Pareto(III) (Yeh et al. [32]). We shall see that it is quite similar to the first order...

Let $\Gamma$ be a group endowed with a length function $L$, and let $E$ be a linear subspace of $\mathbf{C}\Gamma$. We say that $E$ satisfies the Haagerup inequality if there exists constants $C,s\&gt;0$ such that, for any $f\in E$, the convolutor norm of $f$ on ${\ell }^2\left(\Gamma \right)$ is dominated by $C$ times the ${\ell }^2$ norm of $f(1+L{)}^s$. We show that, for $E=\mathbf{C}\Gamma$, the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on $\Gamma$. If $L$ is a word length function on a finitely generated group $\Gamma$, we show that,...