Introduction Maximal inequalities under uniform entropy Bernstein type maximal inequality for Harris recurrent Markov chains Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains Gabriela Cio� lek LTCI, T´ el´ ecom ParisTech, Universit´ e Paris-Saclay, 46 Rue Barrault, 75013 Paris, France gabrielaciolek@gmail.com May 11, 2017 Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Maximal inequalities under uniform entropy Bernstein type maximal inequality for Harris recurrent Markov chains Outline Introduction 1 Atomic Markov chains Harris recurrent Markov chains Nummelin’s splitting technique Maximal inequalities under uniform entropy 2 Bernstein type maximal inequality for regenerative Markov chains Hoeffding type maximal inequality for regenerative Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains 3 Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Atomic Markov chains Maximal inequalities under uniform entropy Harris recurrent Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains Nummelin’s splitting technique Atomic Markov chains Let X = ( X n ) n ∈ N be a homogeneous Markov chain on a countably generated state space ( E , E ) with transition probability Π and initial probability ν. Chain X is assumed to be ψ -irreducible and aperiodic. Regenerative Markov chain We say that the chain X is regenerative, when there exists a measurable set A such that µ ( A ) > 0 and Π( x , . ) = Π( y , . ) for all ( x , y ) ∈ A 2 Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Atomic Markov chains Maximal inequalities under uniform entropy Harris recurrent Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains Nummelin’s splitting technique Atomic Markov chains Define the sequence of regeneration times ( τ A ( j )) j ≥ 1 . Let τ A = τ A (1) = inf { n ≥ 1 : X n ∈ A } be the first time when the chain hits the regeneration set A and τ A ( j ) = inf { n > τ A ( j − 1) , X n ∈ A } for j ≥ 2 . The segments of data are of the form: B j = ( X 1+ τ A ( j ) , · · · , X τ A ( j +1) ) , j ≥ 1 and take values in the torus ∪ ∞ k =1 E k . By the strong Markov property blocks corresponding to the consecutive visitis of the chain to atom A are i.i.d. Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Atomic Markov chains Maximal inequalities under uniform entropy Harris recurrent Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains Nummelin’s splitting technique Harris recurrent Markov chains Harris recurrence Assume that X is a ψ -irreducible Markov chain. We say that X is Harris recurrent iff, starting from any point x ∈ E and any set such that ψ ( A ) > 0, we have P x ( τ A < + ∞ ) = 1 . We construct an artificial regeneration set via Nummelin technique. Small set We say that a set S ∈ E is small if there exists a parameter δ > 0 , a positive probability measure Φ supported by S and an integer m ∈ N ∗ such that ∀ x ∈ S , A ∈ E Π m ( x , A ) ≥ δ Φ( A ) , (1) where Π m denotes the m -th iterate of the transition probability Π . Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Atomic Markov chains Maximal inequalities under uniform entropy Harris recurrent Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains Nummelin’s splitting technique Nummelin’s splitting technique Let ( Y n ) n ∈ N be a sequence of independent r.v.’s with parameter δ. We construct the bivriate chain X M = ( X n , Y n ) n ∈ N with a joint distribution P ν, M . The construction relies on the mixture representation of Π on S , namely Π( x , A ) = δ Φ( A ) + (1 − δ ) Π( x , A ) − δ Φ( A ) . It can be retrieved by the following 1 − δ randomization of the transition probability Π each time the chain X visits the set S . If X n ∈ S and if Y n = 1 (which happens with probability δ ∈ ]0 , 1[), then X n +1 is distributed according to the probability measure Φ, if Y n = 0 (that happens with probability 1 − δ ), then X n +1 is distributed according to the probability measure (1 − δ ) − 1 (Π( X n , · ) − δ Φ( · )) . ˆ S = S × { 1 } is an atom for the split chain. Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Atomic Markov chains Maximal inequalities under uniform entropy Harris recurrent Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains Nummelin’s splitting technique Nummelin’s splitting technique Figure: Regeneration block construction for AR(1) model. Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Bernstein type maximal inequality for regenerative Markov chains Maximal inequalities under uniform entropy Hoeffding type maximal inequality for regenerative Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains We introduce the following notation for partial sums of the regeneration cycles f ( B i ) = � τ A ( j +1) i =1+ τ A ( j ) f ( X i ) . In the following, we assume that the mean inter-renewal time α = E A [ τ A ] < ∞ and write l n = � n i =1 I { X i ∈ A } for the total number of consecutive visits of the chain to the atom A . The regenerative approach is based on the following decomposition of the sum � n i =1 f ( X i ) : ⌊ n α ⌋ n � � f ( X i ) = f ( B i ) + ∆ n , i =1 i =1 where l n 2 τ A n ∆ n = 1 f ( X i ) + 1 f ( B i ) + 1 � � � f ( X i ) , n n n i =1 i = l n 1 i = τ A ( l n − 1) �� n �� n � − 1 , l n − 1 � � − 1 , l n − 1 � where l n 1 = min , l n 2 = max and α α � τ A � 1 σ 2 ( f ) = � { f ( X i ) − µ ( f ) } 2 E A ( τ A ) E A i =1 is the asymptotic variance. Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Bernstein type maximal inequality for regenerative Markov chains Maximal inequalities under uniform entropy Hoeffding type maximal inequality for regenerative Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains In empirical processes theory for processes indexed by class of functions, it is important to assess the complexity of considered classes. The information about entropy of F helps us to inspect how large our class is. Covering and uniform entropy number The covering number N p ( ǫ, Q , F ) is the minimal number of balls { g : � g − f � L p ( Q ) < ǫ } of radius ǫ needed to cover the set F . The centers of the balls need not to belong to F , but they should have finite norms. The entropy (without bracketing) is the logarithm of the covering number. We define uniform entropy number as N p ( ǫ, F ) = sup Q N p ( ǫ, Q , F ) , where the supremum is taken over all discrete probability measures Q . Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
Introduction Bernstein type maximal inequality for regenerative Markov chains Maximal inequalities under uniform entropy Hoeffding type maximal inequality for regenerative Markov chains Bernstein type maximal inequality for Harris recurrent Markov chains We impose the following conditions on the chain: A1. (Bernstein’s block moment condition) There exists a positive constant M such that for any p ≥ 2 and for every f ∈ F E A | f ( B 1 ) | p ≤ 1 2 p ! σ 2 ( f ) M p − 2 . (2) A2. (Block length moment assumption) There exists a positive constant N such that for any p ≥ 2 E A | τ A | p ≤ N p . (3) A3. (Non-regenerative block exponential moment assumption) There exists λ 0 > 0 �� τ A � � � ��� � such that for every f ∈ F we have E ν exp i =1 f ( X i ) < ∞ . λ 0 A4. (Exponential block moment assumption) There exists λ 1 > 0 such that for every f ∈ F we have E A [exp [ λ 1 | f ( B 1 ) | ]] < ∞ . A5. (uniform entropy number condition) N 2 ( ǫ, F ) < ∞ . Gabriela Cio� lek Sharp Bernstein and Hoeffding type inequalities for regenerative Markov chains
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