Large Deviation Tail Estimates and Related Limit Laws for Stochastic Fixed Point Equations

Abstract

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 \(\mathbf{E}[\log \:A] < 0)\), 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 \(V _{n} = A_{n}\max \{V _{n-1},D_{n}\} + B_{n}\), where \(\{(A_{n},B_{n},D_{n}): n \in \mathbb{Z}_{+}\}\) is an i.i.d. sequence of random variables. Next, we consider recursions where the driving sequence of vectors, \(\{(A_{n},B_{n},D_{n}): n \in \mathbb{Z}_{+}\}\), 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 {V n } and the recurrence properties of an associated ξ-shifted Markov chain. We illustrate these ideas with several examples.KeywordsMarkov ChainTransition KernelTail BehaviorForward ProcessCumulant Generate FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.