Abstract
We study the forward and backward recursions generated by a stochastic fixed point equation (SFPE) of the form (Formula presented.), where (A, B, D) ∈ (o, ∞) x ℝ2, for both the stationary and explosive cases. In the stationary case (when 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 E[log A] > 0), we establish a central limit theorem for the forward recursion generated by the SFPE, namely the process Vn = An max{Vn-1, Dn} + Bn, where {(An, Bn, Dn) : n ∈ ℤ+} is an i.i.d. sequence of random variables. Next, we consider recursions where the driving sequence of vectors, {.An, Bn, Dn) : n ∈ ℤ+}, 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 {Vn} and the recurrence properties of an associated ξ-shifted Markov chain. We illustrate these ideas with several examples.
| Original language | English |
|---|---|
| Title of host publication | Random Matrices and Iterated Random Functions |
| Editors | Matthias Lowe, Gerold Alsmeyer |
| Number of pages | 27 |
| Volume | 63 |
| Place of Publication | Heidelberg |
| Publisher | Springer |
| Publication date | 2013 |
| Pages | 91-117 |
| ISBN (Print) | 978-3-642-38805-7 |
| Publication status | Published - 2013 |