Sharp Tail Bounds for Functionals of Markov Chains

Stephan Clemenc¸On

Description

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

• ltci umr


• mx n?2


• journees de probabilite - lille


• markov chain theory


• instantaneous functionals


• markov chains

Sujets

• Harris
• Markov chain

Informations

• Publié par: zauj
• Nombre de lectures: 7
• Langue: English

Extrait

e´hptSn(LTn¸co´emeanClrofsdnuobliaT)IC/005nsaichovrkMa209/1/08

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

l´nChaepon¸cenemaT)ICTL(sdnuobliS´t

2

A little Markov chain theory Markov with regeneration times General Harris Markov chains

1

Overall purpose

3

Probabilistic study based on renewal theory Limit behavior First order results

forMarkovchains0/5902/00282/1

4

The regenerative method Sharper results Tail bounds through the regenerative approach

05/0ainsovchMark

whereH(x) = (1 +x) log(1 +x)−x. (Fuk & Nagaev, ’71)Relaxing the boundedness constraint: ∀M,x>0, P{ X Y i ≥x} ≤ +exp )nP{Y 1 >M}, n  − M nσ 2 2 M H( n M σ 2 x M  i=1

withσ 2 M =E[Y i 2 I{|Y i | ≤M}]. Our goal:extension to instantaneous functionals of a Markov chain, i.e. Y i =f(X i ), by means of theregenerative method.

(Bennett, ’62)LetY 1 , . . . ,Y n  such that r.v.’sbe i.i.d.E[Y i ] = 0, E[Y i 2 ] =σ 2 <∞and|Y i | ≤Ma.s. . P{ i= n X 1 Y i ≥x} ≤exp  − M nσ 2 2 H( M nσ x 2 )  ,

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Overall purpose

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