Strong approximation for additive functionals of geometrically - PowerPoint PPT Presentation

Strong approximation for additive functionals of geometrically ergodic Markov chains Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM) Cincinnati Symposium on Probability Theory and Applications September 2014 Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

Strong approximation in the iid setting (1) Assume that ( X i ) i ≥ 1 is a sequence of iid centered real-valued random variables with a finite second moment σ 2 and define S n = X 1 + X 2 + · · · + X n The ASIP says that a sequence ( Z i ) i ≥ 1 of iid centered Gaussian variables may be constructed is such a way that � � � S k − σ B k � = o ( b n ) almost surely, sup 1 ≤ k ≤ n where b n = ( n log log n ) 1 / 2 (Strassen (1964)). When ( X i ) i ≥ 1 is assumed to be in addition in L p with p > 2, then we can obtain rates in the ASIP: b n = n 1 / p (see Major (1976) for p ∈ ] 2, 3 ] and Koml´ os, Major and Tusn´ ady for p > 3). Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

Strong approximation in the iid setting (2) When ( X i ) i ≥ 1 is assumed to have a finite moment generating function in a neighborhood of 0, then the famous Koml´ os-Major-Tusn´ ady theorem (1975 and 1976) says that one can construct a standard Brownian motion ( B t ) t ≥ 0 in such a way that � � | S k − σ B k | ≥ x + c log n ≤ a exp ( − bx ) sup ( 1 ) P k ≤ n where a , b and c are positive constants depending only on the law of X 1 . (1) implies in particular that � � � S k − σ B k � = O ( log n ) almost surely sup 1 ≤ k ≤ n It comes from the Erd¨ os-R´ enyi law of large numbers (1970) that this result is unimprovable. Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

Strong approximation in the multivariate iid setting Einmahl (1989) proved that we can obtain the rate O (( log n ) 2 ) in the almost sure approximation of the partial sums of iid random vectors with finite moment generating function in a neighborhood of 0 by Gaussian partial sums. Zaitsev (1998) removed the extra logarithmic factor and obtained the KMT inequality in the case of iid random vectors. What about KMT type results in the dependent setting? Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

An extension for functions of iid (Berkes, Liu and Wu (2014)) Let ( X k ) k ∈ Z be a stationary process defined as follows. Let ( ε k ) k ∈ Z be a sequence of iid r.v.’s and g : R Z → R be a measurable function such that for all k ∈ Z , X k = g ( ξ k ) with ξ k : = ( . . . , ε k − 1 , ε k ) is well defined, E ( g ( ξ k )) = 0 and � g ( ξ k ) � p < ∞ for some p > 2. Let ( ε ∗ k ) k ∈ Z an independent copy of ( ε k ) k ∈ Z . For any integer � � k ≥ 0, let ξ ∗ ξ − 1 , ε ∗ and X ∗ k = g ( ξ ∗ k = 0 , ε 1 , . . . , ε k − 1 , ε k k ) . For k ≥ 0, let δ ( k ) as introduced by Wu (2005): δ ( k ) = � X k − X ∗ k � p . Berkes, Liu and Wu (2014): The almost sure strong approximation holds with the rate o ( n 1 / p ) and σ 2 = ∑ k E ( X 0 X k ) provided that δ ( k ) = O ( k − α ) with α > 2 if p ∈ ] 2, 4 ] and α > f ( p ) if p > 4 with � f ( p ) = 1 + p 2 − 4 + ( p − 2 ) p 2 + 20 p + 4 8 p Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

What about strong approximation in the Markov setting? Let ( ξ n ) be an irreducible and aperiodic Harris recurrent Markov chain on a countably generated measurable state space ( E , B ) . Let P ( x , . ) be the transition probability. We assume that the chain is positive recurrent. Let π be its (unique) invariant probability measure. Then there exists some positive integer m , some measurable function h with values in [ 0, 1 ] with π ( h ) > 0, and some probability measure ν on E , such that P m ( x , A ) ≥ h ( x ) ν ( A ) . We assume that m = 1 The Nummelin splitting technique (1984) allows to extend the Markov chain in such a way that the extended Markov chain has a recurrent atom. This allows regeneration. Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

The Nummelin splitting technique (1) Let Q ( x , · ) be the sub-stochastic kernel defined by Q = P − h ⊗ ν The minorization condition allows to define an extended chain ( ¯ ξ n , U n ) in E × [ 0, 1 ] as follows. At time 0, U 0 is independent of ¯ ξ 0 and has the uniform distribution over [ 0, 1 ] ; for any n ∈ N , Q ( x , A ) P ( ¯ ξ n + 1 ∈ A | ¯ ξ n = x , U n = y ) = 1 y ≤ h ( x ) ν ( A ) + 1 y > h ( x ) 1 − h ( x ) : = ¯ P (( x , y ) , A ) and U n + 1 is independent of ( ¯ ξ n + 1 , ¯ ξ n , U n ) and has the uniform distribution over [ 0, 1 ] . P = ¯ ˜ P ⊗ λ ( λ is the Lebesgue measure on [ 0, 1 ] ) and ( ¯ ξ n , U n ) is an irreducible and aperiodic Harris recurrent chain, with unique invariant probability measure π ⊗ λ . Moreover ( ¯ ξ n ) is an homogenous Markov chain with transition probability P ( x , . ) . Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

Regeneration Define now the set C in E × [ 0, 1 ] by C = { ( x , y ) ∈ E × [ 0, 1 ] such that y ≤ h ( x ) } . For any ( x , y ) in C , P ( ¯ ξ n + 1 ∈ A | ¯ ξ n = x , U n = y ) = ν ( A ) . Since π ⊗ λ ( C ) = π ( h ) > 0, the set C is an atom of the extended chain, and it can be proven that this atom is recurrent. Let T 0 = inf { n ≥ 1 : U n ≤ h ( ¯ ξ n ) } and T k = inf { n > T k − 1 : U n ≤ h ( ¯ ξ n ) } , and the return times ( τ k ) k > 0 by τ k = T k − T k − 1 . Note that T 0 is a.s. finite and the return times τ k are iid and integrable. Let S n ( f ) = ∑ n k = 1 f ( ¯ ξ k ) . The random vectors ( τ k , S T k ( f ) − S T k − 1 ( f )) k > 0 are iid and their common law is the law of ( τ 1 , S T 1 ( f ) − S T 0 ( f )) under P C . Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

o (1995): If the r.v’s S T k ( | f | ) − S T k − 1 ( | f | ) have a Cs´ aki and Cs¨ org¨ finite moment of order p for some p in ] 2, 4 ] and if E ( τ p / 2 ) < ∞ , k then one can construct a standard Wiener process ( W t ) t ≥ 0 such that S n ( f ) − n π ( f ) − σ ( f ) W n = O ( a n ) a.s. . with a n = n 1 / p ( log n ) 1 / 2 ( log log n ) α and σ 2 ( f ) = lim n 1 n Var S n ( f ) . The above result holds for any bounded function f only if the return times have a finite moment of order p . The proof is based on the regeneration properties of the chain, on the Skorohod embedding and on an application of the results of KMT (1975) to the partial sums of the iid random variables S T k + 1 ( f ) − S T k ( f ) , k > 0. Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

On the proof of Cs´ aki and Cs¨ org¨ o For any i ≥ 1, let X i = ∑ T i ℓ = T i − 1 + 1 f ( ¯ ξ ℓ ) . Since the ( X i ) i > 0 are iid, if E | X 1 | 2 + δ < ∞ , there exists a standard Brownian motion ( W ( t )) t > 0 such that k � � � � = o ( n 1 / ( 2 + δ ) ) ∑ sup X i − σ ( f ) W ( k ) a . s . k ≤ n i = 1 Let ρ ( n ) = max { k : T k ≤ n } . If E | τ 1 | q < ∞ for some 1 ≤ q ≤ 2, then n E ( τ 1 ) + O ( n 1 / q ( log log n ) α ) ρ ( n ) = a . s . � � ρ ( n ) � ∑ � = o ( n 1 / ( 2 + δ ) ) a.s. i = 1 X i − S n ( f ) � � = O � � n n 1 / ( 2 q ) ( log n ) 1 / 2 ( log log n ) α ) ρ ( n )) − W ( W a.s. E ( τ 1 ) With this method, no way to do better than O ( n 1 / ( 2 q ) ( log n ) 1 / 2 ) (1 ≤ q ≤ 2 ) even if f is bounded and τ 1 has exponential moment. Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

Link between the moments of return times and the coefficients of absolute regularity For positive measures µ and ν , let � µ − ν � denote the total variation of µ − ν Set � E � P n ( x , . ) − π � d π ( x ) . β n = The coefficients β n are called absolute regularity (or β -mixing) coefficients of the chain. Bolthausen (1980-1982): for any p > 1, E ( τ p 1 ) = E C ( T p 0 ) < ∞ if and only if ∑ n p − 2 β n < ∞ . n > 0 Hence, according to the strong approximation result of M.-Rio (2012), if f is bounded and E ( τ p 1 ) for some p in ] 2, 3 [ , then the strong approximation result holds with the rate o ( n 1 / p ( log n ) ( p − 2 ) / ( 2 p ) ) . Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)

Main result: M. Rio (2014) Assume that β n = O ( ρ n ) for some real ρ with 0 < ρ < 1, If f is bounded and such that π ( f ) = 0 then there exists a standard Wiener process ( W t ) t ≥ 0 and positive constants a , b and c depending on f and on the transition probability P ( x , · ) such that, for any positive real x and any integer n ≥ 2, � � � � � ≥ c log n + x � S k ( f ) − σ ( f ) W k sup ≤ a exp ( − bx ) . P π k ≤ n where σ 2 ( f ) = π ( f 2 ) + 2 ∑ n > 0 π ( fP n f ) > 0. � � � S k ( g ) − σ ( f ) W k � = O ( log n ) a.s. Therefore sup k ≤ n Florence Merlev` ede Joint work with E. Rio Universit´ e Paris-Est-Marne-La-Vall´ ee (UPEM)