We determine the distributional behavior of products of free (in the sense of Voiculescu) identically distributed random variables. Analogies and differences with the classical theory of independent random variables are then discussed.
We consider transient random walks in random environment on with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
We describe the limit measures for some class of deformations of the free convolution, introduced by A. D. Krystek and Ł. J. Wojakowski. In particular, we provide a counterexample to a conjecture from their paper.
Observations are made on a point process in in a window of volume . The observation, or ‘score’ at a point , here denoted , is a function of the points within a random distance of . When the input is a Poisson or binomial point process, the large limit theory for the total score , when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input ....
Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable functionf. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence...
Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.
The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this note, the exact distributions of , and are derived when and are independent normal and logistic random variables. The normal and logistic distributions have been two of the most popular models for measurement errors in engineering.
By direct calculus we identify explicitly the lipschitzian norm of the solution of the Poisson equation in terms of various norms of g, where is a Sturm–Liouville operator or generator of a non-singular diffusion in an interval. This allows us to obtain the best constant in the L1-Poincaré inequality (a little stronger than the Cheeger isoperimetric inequality) and some sharp transportation–information inequalities and concentration inequalities for empirical means. We conclude with several illustrative...