  Given a sequence of i.i.d. random functions
$\Psi_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated
function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$
and $X_{n}^{x}:=\Psi_{n-1}(X_{n-1}^{x})$ for $x\in\mathbb{R}$ and
$n\in\mathbb{N}$. Under the two basic assumptions that the $\Psi_{n}$ are a.s.
continuous at any point in $\mathbb{R}$ and asymptotically linear at the
"endpoints" $\pm\infty$, we study the tail behavior of the stationary laws of
such Markov chains by means of Markov renewal theory. Our approach provides an
extension of Goldie's implicit renewal theory and can also be viewed as an
adaptation of Kesten's work on products of random matrices to one-dimensional
function systems as described. Our results have applications in quite different
areas of applied probability like queuing theory, econometrics, mathematical
finance and population dynamics. Our results have applications in quite
different areas of applied probability like queuing theory, econometrics,
mathematical finance and population dynamics, e.g. ARCH models and random
logistic transforms.
