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

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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 languageEnglish
Title of host publicationRandom Matrices and Iterated Random Functions
EditorsMatthias Lowe, Gerold Alsmeyer
Number of pages27
Place of PublicationHeidelberg
Publication date2013
ISBN (Print)978-3-642-38805-7
Publication statusPublished - 2013

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