TY - GEN

T1 - Large deviation tail estimates and related limit laws for stochastic fixed point equations

AU - Collamore, Jeffrey F.

AU - Vidyashankar, Anand N.

PY - 2013

Y1 - 2013

N2 - 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 $V$ satisfyingthis 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 recursionswhere 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.

AB - 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 $V$ satisfyingthis 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 recursionswhere 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.

M3 - Article in proceedings

SN - 978-3-642-38805-7

VL - 63

SP - 91

EP - 117

BT - Random Matrices and Iterated Random Functions

A2 - Lowe, Matthias

A2 - Alsmeyer, Gerold

PB - Springer

CY - Heidelberg

ER -