\[ X_{n}=\Psi_{n}\circ...\circ\Psi_{1}(X_{0}) \]
for $n\ge 1$ , where $\Psi_{1},\Psi_{2},...$ are iid random Lipschitz functions on $\X$ with Lipschitz constants $L(\Psi_{1}),L(\Psi_{2}),...$ Let $\pi$ denote the unique stationary distribution of $(X_{n})_{n\ge 0}$ and $x_{0}\in\X$ an arbitrary reference point.
Assuming $\Prob_{\pi}(d(x_{0},X_{0})>r)>0$ for all $r>0$ , we will provide bounds for the lower and upper tail index $\vth_{*}$ and $\vth^{*}$ of $d(x_{0},X_{0})$ in equilibrium (under $\Prob_{\pi}$ ), defined by
\vth_{*}:=-\limsup_{x\to\infty}\frac{\log\Prob(X>x)}{\log x}\quad\text{and}\quad\vth^{*}:=-\liminf_{x\to\infty}\frac{\log\Prob(X>x)}{\log x}.
This will be done by providing lower and upper bounds for $d(x_{0},X_{n})$ under $\Prob_{\pi}$ in terms of rather simple IFS on $\R_{\ge}$ and the use of Goldie's implicit renewal theorem. Special attention is paid to the particularly relevant case when $\X=\R$ . The method is illustrated by some examples including the well-known AR (1) model with ARCH(1) errors which has been studied earlier in some detail by Borkovec and Klüppelberg.
References
Alsmeyer , G.: On the stationary tail index of iterated random Lipschitz functions. Stoch. Proc. Appl. (Online first), DOI: http://dx.doi.org /10.1016/j.spa.2015.08.004
Borkovec , M., Klüppelberg , C.: The tail of the stationary distribution of an autoregressive process with ARCH}(1) errors. Ann. Appl. Probab. 11(4), 1220-1241 (2001)
Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl . Probab . 1(1), 126--166 (1991)
