Sharp Tail Bounds for Functionals of Markov Chains
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Niveau: Supérieur
Sharp Tail Bounds for Functionals of Markov Chains – Journees de Probabilite - Lille 2008 Stephan Clemenc¸on Telecom ParisTech - LTCI UMR CNRS/Telecom ParisTech No 5141 Joint work with Patrice Bertail (Modal'x - Universite Paris X) Stephan Clemenc¸on (LTCI) Tail bounds for Markov chains 05/09/2008 1 / 21
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Sharp Tail Bounds for Functionals of Markov Chains – Journees de Probabilite - Lille 2008 Stephan Clemenc¸on Telecom ParisTech - LTCI UMR CNRS/Telecom ParisTech No 5141 Joint work with Patrice Bertail (Modal'x - Universite Paris X) Stephan Clemenc¸on (LTCI) Tail bounds for Markov chains 05/09/2008 1 / 21
- ltci umr
- mx n?2
- journees de probabilite - lille
- markov chain theory
- instantaneous functionals
- markov chains
Sharp Tail Bounds for Functionals of Markov Chains – Journe´esdeProbabilit´e-Lille2008
21
JointworkwithPatriceBertail(Modal’x-Universit´eParisX)
Ste´phanCle´men¸con
Telecom ParisTech -LTCI UMR CNRS/Telecom ParisTech No 5141
2
A little Markov chain theory Markov with regeneration times General Harris Markov chains
1
Overall purpose
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Probabilistic study based on renewal theory Limit behavior First order results
forMarkovchains0/5902/00282/1
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The regenerative method Sharper results Tail bounds through the regenerative approach
whereH(x) = (1 +x) log(1 +x)−x. (Fuk & Nagaev, ’71)Relaxing the boundedness constraint: ∀M,x>0, P{XYi≥x} ≤ +exp )nP{Y1>M}, n−Mnσ22MH(nMσ2xM i=1
withσ2M=E[Yi2I{|Yi| ≤M}]. Our goal:extension to instantaneous functionals of a Markov chain, i.e. Yi=f(Xi), by means of theregenerative method.
(Bennett, ’62)LetY1, . . . ,Yn such that r.v.’sbe i.i.d.E[Yi] = 0, E[Yi2] =σ2<∞and|Yi| ≤Ma.s. . P{i=nX1Yi≥x} ≤exp−Mnσ22H(Mnσx2),
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Overall purpose
80/3/902tSe´hpcon¸LTn(Clanme´ednuorofsT)ICblia
