## Large deviation tail estimates and related limit laws for stochastic fixed point equations

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

### Documents

We study the forward and backward recursions generated by a stochastic fixed point equation (SFPE) of the form $V \stackrel{d}{=} A\max\{V, D\}+B$, where $(A, B, D) \in (0, \infty)\times {\mathbb R}^2$, for both the stationary and explosive cases.
In the stationary case (when ${\bf E} [\log \: A] < 0)$, we present results concerning the precise tail asymptotics for the random variable $V$ satisfying
this SFPE. In the explosive case (when ${\bf E} [\log \: A] > 0)$, we establish a central limit theorem for the forward recursion generated by the SFPE,
namely the process $V_n= A_n \max\{V_{n-1}, D_n\} +B_n$, where $\{ (A_n,B_n,D_n): n \in \pintegers \}$ is an i.i.d.\ sequence of random variables.
Next, we consider recursions
where the driving sequence of vectors, $\{(A_n, B_n, D_n): n \in \pintegers \}$, is modulated by a Markov chain in general state space.
We demonstrate an asymmetry between the forward and backward recursions and develop techniques for estimating the exceedance
probability. In the process, we establish an interesting connection between the regularity properties of $\{V_n\}$ and the recurrence properties of an associated $\xi$-shifted Markov chain. We illustrate these ideas with several examples.
Original language English Random Matrices and Iterated Random Functions Matthias Lowe, Gerold Alsmeyer 27 63 Heidelberg Springer 2013 91-117 978-3-642-38805-7 Published - 2013