Une condition suffisante de recurrence pour des chaines de Markov sur la droite

Abstract

A direct, elementary, proof is given to the following result: "Let P(x,dy) a transition probability on the real line. Assume that:

• i) for f continuous and bounded, Pf is continuous.

• ii) P is irreductible, with respect to open sets.

• iii) for some constant K, P(x, ]−∞, −K[)=O for x close enough to +∞ and P(x, ]+K, +∞[)=O for x close enough to −∞.

• iv) for x outside a compact set, ∫y P(x,dy)=x

Then the Markov chain associated to P is topologically recurrent on the line" A similar result is given on ℤ.