Estimates of the tail of the stationary density function of certain nonlinear autoregressive processes (Q1181408)

Let the autoregressive process \(\{X_ n,n\geq 1\}\) be defined by the stochastic difference equation \(X_{n+1}=T(X_ n)+\varepsilon_{n+1},n=0,1,2,\dots,\) where \(T\) is a measurable function from \(R^ d\) to itself and \(\{\varepsilon_ n\}\) is a sequence of i.i.d. \(R^ d\)-valued random variables. The starting point \(X_ 0\in R^ d\) is arbitrary. The author studies conditions on the mapping \(T\) and the distribution of the \(\varepsilon_ n\)'s that guarantee the ergodicity of \(\{X_ n\}\), i.e., the existence of a unique stationary probability distribution for \(\{X_ n\}\). A typical set of conditions is: \(T\) is locally bounded and \[ \lim\sup_{\| x\|\to \infty}\| T(x)\|/\| x\| <1, \] the distribution of the \(\varepsilon_ n\)'s is absolutely continuous with respect to \(d\)-dimensional Lebesgue measure and the tails of the density function (assumed to be positive on the whole space) satisfy certain regularity and moment conditions. Under such conditions the author obtains sharp upper bounds for the tails of the stationary density. In practical terms, the method of proof relies on showing, by direct calculation, that the set of densities satisfying the desired tail conditions is invariant under the transition probability operator of the process. General techniques, such as the Leray-Schauder fixed point theorem, are then used to establish the existence of a stationary density (necessarily unique in this case) with the required properties. The paper under review is related to a series of influential papers by \textit{R. L. Tweedie}, [see, e.g., Stochastic Processes Appl. 3, 385-403 (1975; Zbl 0315.60035)]. Recent surveys on nonlinear autoregressive processes are the book by \textit{H. Tong} [Nonlinear time series. A dynamical system approach. (1990; Zbl 0716.62085)] and a paper by \textit{D. Tjøstheim} [Adv. Appl. Probab. 22, No. 3, 587-611 (1990; Zbl 0712.62080)].