In this paper, we investigate annealed and quenched limit theorems for random expanding dynamical systems. Making use of functional analytic techniques and more probabilistic arguments with martingales, we prove annealed versions of a central limit theorem, a large deviation principle, a local limit theorem, and an almost sure central limit theorem. We also discuss the quenched central limit th...

The introductory part will feature: rate of approximation of real numbers by rationals; theorems of Kronecker, Dirichlet, Liouville, Borel-Cantelli; connections with dynamical systems: circle rotations, hyperbolic flow in the space of lattices, geodesic flow on the modular survace, Gauss map (continued fractions); ergodicity, unique ergodicity, mixing, applications to uniform distribution of se...

k=n Ak. If E occurs, then infinitely many of Ak:s occur. Sometimes we write this as E = {An i.o.} where i.o. is to be read as ”infinitely often”, i.e., infinitely many times. E is sometimes denoted with lim supAk. We need a couple of auxiliary results (the lemmas below) of probability calculus (found, e.g., on page 3 of [1]) that are basic in the sense that they are derived directly from the Ko...

<p style='text-indent:20px;'>A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose extension to characterize the multiple occurrence on same time scale. Our results imply Logarithm Laws recurrence and hitting times, as well Poisson Limit systems which are exponentially mixing all orde...

Every Dedekind complete Riesz space X has a unique sup-completion s , which is lattice cone. This paper aims to present systematic study this cone by extending several known results the general setting, proving new and, in particular, introducing for elements of finite and infinite parts. enables us get satisfactory abstract formulation some classical setting spaces. We prove, version Borel-Can...