Nonlinearity 16 (2003) 977-987
Markus Ådahl, Ian Melbourne and Matthew Nicol
Abstract
We consider the statistical behaviour of i.i.d. compositions
of a finite set of Euclidean isometries of Rn$.
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 generalise 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.