Large deviation tail estimates and related limit laws for stochastic fixed point equations
Publikation: Bidrag til bog/antologi/rapport › Konferencebidrag i proceedings › Forskning › fagfællebedømt
Dokumenter
- SFPE-limitlawsREV.pdf
Accepteret manuskript, 234 KB, PDF-dokument
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.
\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.
Originalsprog | Engelsk |
---|---|
Titel | Random Matrices and Iterated Random Functions |
Redaktører | Matthias Lowe, Gerold Alsmeyer |
Antal sider | 27 |
Vol/bind | 63 |
Udgivelsessted | Heidelberg |
Forlag | Springer |
Publikationsdato | 2013 |
Sider | 91-117 |
ISBN (Trykt) | 978-3-642-38805-7 |
Status | Udgivet - 2013 |
Antal downloads er baseret på statistik fra Google Scholar og www.ku.dk
Ingen data tilgængelig
ID: 41838683