Large deviation tail estimates and related limit laws for stochastic fixed point equations
Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › fagfællebedømt
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Large deviation tail estimates and related limit laws for stochastic fixed point equations. / Collamore, Jeffrey F.; Vidyashankar, Anand N.
Random Matrices and Iterated Random Functions. red. / Matthias Lowe; Gerold Alsmeyer. Bind 63 Heidelberg : Springer, 2013. s. 91-117.Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › fagfællebedømt
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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 -
ID: 41838683