Rates in almost sure invariance principle for quickly mixing dynamical systems

For a large class of quickly mixing dynamical systems, we prove that the error in the almost sure approximation with a Brownian motion is of order [Formula: see text] with [Formula: see text]. Specifically, we consider nonuniformly expanding maps with exponential and stretched exponential decay of correlations, with one-dimensional Hölder continuous observables.

Almost sure mixing rates for non-uniformly expanding maps

We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that if the volume measure of the set of points failing the non-uniform expansion or the slow recurrence to the critical set, at a certain times has a (stretched) exponential decay for almost all random orbits, then the decay of correlations along random orbits is stretched exponential, up to some waiting time. As applications we obtain almost sure stretched exponential decay of correlations along random orbits for Viana maps, as for a class of non-uniformly expanding local diffeomorphisms and a family of unimodal maps.

Asymptotic Properties of a Random Graph with Duplications

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number cd > 0 almost surely as the number of steps goes to ∞, and cd ~ (eπ)1/2d1/4e-2√d holds as d → ∞.

Asymptotic Properties of a Random Graph with Duplications

We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degreedtends to some positive numbercd> 0 almost surely as the number of steps goes to ∞, andcd~ (eπ)1/2d1/4e-2√dholds asd→ ∞.