# Random Walks on Reductive Groups by Yves Benoist, Jean-François Quint (auth.)

By Yves Benoist, Jean-François Quint (auth.)

The classical idea of *Random Walks* describes the asymptotic habit of sums of self reliant identically dispensed random actual variables. This ebook explains the generalization of this conception to items of autonomous identically disbursed random matrices with actual coefficients.

Under the idea that the motion of the matrices is semisimple – or, equivalently, that the Zariski closure of the crowd generated by way of those matrices is *reductive* - and lower than appropriate second assumptions, it truly is proven that the norm of the goods of such random matrices satisfies a few classical probabilistic laws.

This booklet contains worthy historical past at the thought of reductive algebraic teams, likelihood thought and operator thought, thereby supplying a contemporary advent to the topic.

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**Example text**

3 Ergodicity of Markov Operators We again let (X, X ) be a standard Borel space, P be a Markov chain on (X, X ) and ν be a P -invariant probability measure. We shall give equivalent definitions for ergodicity. First let us describe the functions which are ν-almost surely P -invariant. 6 Let (X, X ) be a standard Borel space, P be a Markov operator on X and ν be a P -invariant probability measure. Then, every ν-almost surely P invariant bounded Borel function ϕ is equal ν-almost everywhere to a P -invariant bounded Borel function ψ .

2) Hence, in particular, Eν (ψ ◦ T n | Xn ) = Eν (ψ | X0 ) ◦ T n . 3) (a) If ψ is a bounded Borel function on Ω, we let ϕ denote the bounded Borel function on X given by, for every x in X, ϕ(x) = Ω ψ(ω) dPx (ω). In other words, ϕ(x) is the expected value of the function ψ for the trajectories of the Markov chain starting at x. The map ψ → ϕ is onto and, we have, for ν-almost any ω in Ω, Eν (ψ | X0 )(ω) = ϕ(ω0 ) and Eν (ψ ◦ T | X0 )(ω) = P ϕ(ω0 ). Thus, we get Eν (ψ) = ν(ϕ) and Eν (ψ ◦ T ) = ν(P ϕ), whence the result.

Bn , . ) ∈ B. We now construct the forward dynamical system on B × X. We equip B × X with the σ -algebra B ⊗ X of Borel subsets and we introduce the skew-product transformation T X : (b, x) → (T b, b1 x). We identify the σ -algebra X of Borel subsets of X with the sub-σ -algebra of Borel subsets of B × X which do not depend on the first coordinate. For any x in X, set Pμ,x = μ ∗ δx . One easily check that this defines a Markov–Feller operator Pμ on X. We explain now how the forward dynamical system on B × X is related to the forward dynamical system (Ω, T ) of the Markov operator P = Pμ that we introduced in Sect.