Tail behaviour of stationary distribution for Markov chains with   asymptotically zero drift
  We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function. 
Denis Denisov, Dmitry Korshunov and Vitali Wachtel
math.PR