Random iteration of Euclidean isometries

Abstract

We consider the statistical behaviour of independent identically distributed compositions of a finite set of Euclidean isometries of R n . We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalize immediately to Markov chains.We also give simple geometric criteria for orbits to grow linearly or sublinearly with probability one and for nondegeneracy (nonsingular covariance matrix) in the statistical limit theorems.Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.

Statistical Limit Laws for Equivariant Observations

“…In a recent paper, we applied these refined limit laws to random iterations of Euclidean isometries [1]. The appropriate mathematical model to study random iterations of Euclidean isometries is to sample with respect to a measure space Ω (rather than Ω × G) and hence the results of [4][5][6] do not immediately apply.…”

Random walks in Euclidean space

“…The Central Limit Theorem was revisited by many authors, see e.g. [14], [21], [18] and [1]. In these works the Central Limit Theorem was even generalized to cases when the distribution Y 1 has infinite second moment and the limit distribution is not Gaussian.…”