Tail estimates for stochastic fixed point equations via nonlinear renewal theory
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Tail estimates for stochastic fixed point equations via nonlinear renewal theory. / Collamore, Jeffrey F.; Vidyashankar, Anand N.
I: Stochastic Processes and Their Applications, Bind 123, Nr. 9, 2013, s. 3378-3429.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Tail estimates for stochastic fixed point equations via nonlinear renewal theory
AU - Collamore, Jeffrey F.
AU - Vidyashankar, Anand N.
PY - 2013
Y1 - 2013
N2 - This paper presents precise large deviation estimates for solutions to stochastic fixed point equations of the type V =_D f(V), where f(v)=Av+g(v) for a random function g(v)=o(v) a.s. as v tends to infinity. Specifically, we provide an explicit characterization of the pair (C,r) in the tail estimate P(V>u)~Cu^{-r} as u tends to infinity, and also present a corresponding Lundberg-type upper bound. To this end, we introduce a novel dual change of measure on a random time interval and analyze the path properties, using nonlinear renewal theory, of the Markov chain resulting from the forward iteration of the given stochastic fixed point equation. In the process, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we also establish a new characterization of the extremal index. Finally, we provide some extensions of our methods to Markov-driven processes.
AB - This paper presents precise large deviation estimates for solutions to stochastic fixed point equations of the type V =_D f(V), where f(v)=Av+g(v) for a random function g(v)=o(v) a.s. as v tends to infinity. Specifically, we provide an explicit characterization of the pair (C,r) in the tail estimate P(V>u)~Cu^{-r} as u tends to infinity, and also present a corresponding Lundberg-type upper bound. To this end, we introduce a novel dual change of measure on a random time interval and analyze the path properties, using nonlinear renewal theory, of the Markov chain resulting from the forward iteration of the given stochastic fixed point equation. In the process, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we also establish a new characterization of the extremal index. Finally, we provide some extensions of our methods to Markov-driven processes.
U2 - 10.1016/j.spa.2013.04.015
DO - 10.1016/j.spa.2013.04.015
M3 - Journal article
VL - 123
SP - 3378
EP - 3429
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
SN - 0304-4149
IS - 9
ER -
ID: 109552199