The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case

On unbounded invariant measures of stochastic dynamical systems

“…However, the main results were obtained there under much stronger assumptions, namely we assumed existence of exponential moments, that is, E[A δ + A −δ + |B| δ ] < ∞ for some δ > 0. Theorem 1.1 improves all our previous results for affine recursions and describes the asymptotic behavior of ν under optimal assumptions, that is, the weakest-known conditions implying existence of the invariant measure [2]. To our knowledge, for all the other recursions even partial results are not known.…”

“…A renewal theorem for the potential U , that is, description of its behavior at infinity, was given in [2], where the authors proved that for all h ∈ C C (Aff(R)):…”

“…This key idea has been applied in several different ways in important works on the subject, for instance, [2,11,14,17]. We use here a potential theoretic version.…”

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

“…The main result is due to Babillot et al [1], who proved that there exists a unique invariant Radon measure ν of the process {X u n }, i.e. the measure satisfying (2), but in this case it is infinite on R. Its behaviour was described by Buraczewski [6] and Brofferio et al [5].…”

The Poisson boundary of random rational affinities

“…This countable group has a natural action on the real line R and is a dense subgroup the group of real affine transformations. One can obtain interesting results concerning the behavior of the random walks on Aff(Q) using the powerful theory developed on Lie groups, when no continuity hypothesis on the measure is assumed (for instance [14], [2], [1] or [4]). Nevertheless the Poisson boundary for random walk on Aff(Q) can not be studied in such a way.…”