Large Deviation Tail Estimates and Related Limit Laws for Stochastic Fixed Point Equations
| Autor: | Anand N. Vidyashankar, Jeffrey F. Collamore |
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| Rok vydání: | 2013 |
| Předmět: |
Limit of a function
Discrete mathematics Sequence 010102 general mathematics Space (mathematics) 01 natural sciences Fixed point equation Combinatorics 010104 statistics & probability Forward recursion 0101 mathematics Connection (algebraic framework) Random variable Mathematics Central limit theorem |
| Zdroj: | Springer Proceedings in Mathematics & Statistics ISBN: 9783642388057 |
| DOI: | 10.1007/978-3-642-38806-4_5 |
| Popis: | 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 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. |
| Databáze: | OpenAIRE |
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