Estimates of the tail of the stationary density function of certain nonlinear autoregressive processes

We consider the Markov chainX_n+1=T(X_n)+_n, where {_n;n1} is a ^d-valued random sequence of independent identically distributed random variables, and the functionT: ^dd is measurable and satisfies a suitable growth condition. Under certain conditions involvingT and the probability distribution of _n, we show that this Markov chain is ergodic. Moreover, we obtain sharp upper bounds for the tail of the corresponding stationary probability density function. In our proofs, we make use of the Leray-Schauder fixed-point theorem.