Spectral gap properties and limit theorems for some random walks and dynamical systems

Abstract

We give a description of some limit theorems and the corresponding proofs for various transfer operators. Our examples are closely related with random walks on homogeneous spaces. The results are obtained using spectral gap methods in Hölder spaces or Hilbert spaces. We describe also their geometrical setting and the basic corresponding properties. In particular we focus on precise large deviations for products of random matrices, Fréchet's law for affine random walks and local limit theorems for Euclidean motion groups or nilmanifolds.

Expansions in the Local and the Central Limit Theorems for Dynamical Systems

“…Then the assumptions (α)[r ] for ψ = ξ = 1 for all r and Assumption (β)[ p] hold for all p > 0. See [27]. Hence, the first result Theorem 7.1 can be further generalized.…”

“…This ensures (α) with B j = ( ( j) i0 (ξ ))(x) up to decrease if necessary the value of δ to get the second bound. Moreover (β) [ p] for all p > 0 is also proved in [27]…”

“…As the last example, we describe briefly how our results apply to random matrix products, and more generally, to random walks on split semisimple Lie groups. The ideas we use are from [6,27,38]. We refer the readers to those references and the references therein for the historical development of the subject as well as for the complete statements of the results we use and their proofs.…”

“…which is the ergodic sum of φ(g, x) = log g•x x . A local limit theorem has been established in [27] under the following assumptions.…”

Precise large deviation asymptotics for products of random matrices

“…As a special case of (1.6) with l = 0 and ψ compactly supported we obtain Theorem 3.3 of Guivarc'h [20]. With l = 0, ψ the indicator function of the interval [0, ∞) and ϕ = r s , we get the main result in [8].…”