Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q) \in GL(d, \R) \times \R^d$, we consider the random difference equation (RDE) $$R_{n}=M_{n}R_{n-1}+Q_{n},$$ $n\ge 1$, and assume the existence of $\kappa >0$ such that \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^\kappa})^{\frac{1}{n}} = 1 . We prove, under suitable assumptions, that the sequence $S_n = R_1 + ... + R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $\kappa$. As a by-product, we show that the unique stationary solution $R$ of the RDE is regularly varying with index $\kappa$, and give a precise description of its tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .Comment: 15 page

A new view of electrochemistry at highly oriented pyrolytic graphite

Major new insights on electrochemical processes at graphite electrodes are reported, following extensive investigations of two of the most studied redox couples, Fe(CN)64–/3– and Ru(NH3)63+/2+. Experiments have been carried out on five different grades of highly oriented pyrolytic graphite (HOPG) that vary in step-edge height and surface coverage. Significantly, the same electrochemical characteristic is observed on all surfaces, independent of surface quality: initial cyclic voltammetry (CV) is close to reversible on freshly cleaved surfaces (>400 measurements for Fe(CN)64–/3– and >100 for Ru(NH3)63+/2+), in marked contrast to previous studies that have found very slow electron transfer (ET) kinetics, with an interpretation that ET only occurs at step edges. Significantly, high spatial resolution electrochemical imaging with scanning electrochemical cell microscopy, on the highest quality mechanically cleaved HOPG, demonstrates definitively that the pristine basal surface supports fast ET, and that ET is not confined to step edges. However, the history of the HOPG surface strongly influences the electrochemical behavior. Thus, Fe(CN)64–/3– shows markedly diminished ET kinetics with either extended exposure of the HOPG surface to the ambient environment or repeated CV measurements. In situ atomic force microscopy (AFM) reveals that the deterioration in apparent ET kinetics is coupled with the deposition of material on the HOPG electrode, while conducting-AFM highlights that, after cleaving, the local surface conductivity of HOPG deteriorates significantly with time. These observations and new insights are not only important for graphite, but have significant implications for electrochemistry at related carbon materials such as graphene and carbon nanotubes