Erratum to: An empirical central limit theorem for intermittent maps

Transcription

1 Probab. Theory Relat. Fields (2013) 155: DOI /s ERRATUM Erratum to: A empirical cetral limit theorem for itermittet maps J. Dedecker Published olie: 25 October 2011 Spriger-Verlag 2011 Erratum to: Probab. Theory Relat. Fields (2010) 148: DOI /s There are two problems i this paper. First, the defiitio of the coefficiet β 2 () is i fact too restrictive. Secodly, there is a wrog argumet i the proof of the mai result, Theorem 2.1. I this erratum, we give the correct defiitio of the coefficiet β 2 (), asitwas itroduced i Dedecker ad Prieur [1], ad we explai how to fix the proof of Theorem 2.1. The first paragraph is devoted to the defiitio of the coefficiets. I the secod paragraph, we give a slightly more geeral Rosethal-type iequality tha that give i Propositio 3.1, which will be used to fix the proof of Theorem 2.1. I the third paragraph, we explai the chages i the proof of Theorem Defiitio of the coefficiets Keepig the same otatios as i Defiitio 2.1 page 180, the term b 2 (M l, k) should be b 2 (M l, i, j) = sup P (Xi,X j ) M l ( f t f s ) P (Xi,X j )( f t f s ). (s,t) R 2 The olie versio of the origial article ca be foud uder doi: /s J. Dedecker (B) Laboratoire MAP5, UMR CNRS 8145, Uiversité Paris descartes, Sorboe Paris Cité, 45 rue des saits pères, Paris cedex 06, Frace jerome.dedecker@parisdescartes.fr

2 488 J. Dedecker ad the correct defiitio of β 2 (k) should be { } β 2 (k) = max β 1 (k), sup i> j k E((b 2 (M 0, i, j))), which is exactly the defiitio give by Dedecker ad Prieur [1]. 2 The Rosethal iequality The iequality give i Propositio 3.1 is correct, but we shall use a slightly more geeral versio. We use the covetio k i= j a i = 0if j > k, ad we use the otatio (k) + = k1 k>0. Here is the ew versio of Propositio 3.1 (ote that the previous versio ca be obtaied by takig d 1 = d 2 = =d = 0 i this ew versio). Propositio 3.1 Let X 1,...,X be real-valued radom variables i L p for some p [2, 3], with zero expectatio, ad let d 1,...,d be real umbers. Let S = X X.For1 i, let F i = σ(x 1,...,X i ). For ay 1 N, the followig iequality holds ( S p 2( p 1) where ) 1/2 ( γ i + E( X i p )+ p(p 1) γ i = 1 2 E(X 2 i ) + δ i,1 = δ i,2 = j=(i N) + +1 j j=(i N) + +1 l=(2 j i) + +1 j=(i N) + +1 (2 j i) + j=1 ( ) ) 1/p δi,1 +δ i,2 +δ i,3, i N E(X i X j ) + X j E(X i F j ) p/2, X l p 2 X j E(X i F j ) 1, X l d l p 2 E(X i X j E(X i X j ) F l ) 1, δ i,3 = 1 X j d j p 2 E(Xi 2 E(Xi 2 2 ) F j) 1. j=1 Now, the Remark 3.1 followig Propositio 3.1 should be writte as follows (ote that the idices i the defiitio of the term δ 2 of the previous versio of Remark 3.1 were wrog, ad have bee replaced by the correct idices). Remark 3.1 Assume that the X i s of Propositio 3.1 are take from a statioary sequece (X i ) i Z, ad let M i = σ(x k, k i). Let also d 1 = d 2 = = d = d i Propositio 3.1. Oe has γ i γ, δ i,1 δ 1, δ i,2 δ 2 ad δ i,3 δ 3, with

3 Erratum to: A empirical cetral limit theorem for itermittet maps 489 γ = 1 2 E(X 0 2 ) + δ 1 = δ 2 = k=0 E(X 0 X k ) + k=n l X 0 p 2 X k E(X k+l M k ) 1, X 0 E(X k M 0 ) p/2, X 0 d p 2 E(X k X k+l E(X k X k+l ) M 0 ) 1, k=l δ 3 = 1 X 0 d p 2 E(Xk 2 2 E(X k 2 ) M 0) 1. The proof of this ew Propositio 3.1 is almost idetical to the proof of the previous versio. The oly chages cocer the terms E(I 1 ) ad E(K 2 ). Recall that I 1 = (X 2 E(X 2 )) S p 2, ad let D k = d 1 + d 2 + +d k. Sice E((X 2 E(X 2 )) D p 2 ) = 0, we have E(I 1 ) = E((X 2 E(X 2 ))( S p 2 D p 2 )). Let Z k, j = D j + k (X i d i ), with the covetio Z 0, j = D j. The E(I 1 ) = E ( ) (X 2 E(X 2 ))( Z k, p 2 Z k 1, p 2 ). Takig the coditioal expectatio with respect to F k ad usig that x p 2 y p 2 x y p 2, we obtai that E(I 1 ) E(X 2 E(X 2 ) F k) X k d k p 2 1. (0.1) This iequality (0.1) must be used istead of the iequality (3.2) of the previous proof. I the same way, E(K 2 ) = (p 1)E k= N+1 (2k ) + Z,(2k )+ p 2 ). (X X k E(X X k ))( Z i,(2k )+ p 2 Takig the coditioal expectatio with respect to F i ad usig that x p 2 y p 2 x y p 2, we obtai that

4 490 J. Dedecker E(K 2 ) (p 1) k= N+1 (2k ) + E(X X k E(X X k ) F i ) X i d i p 2 1. (0.2) This iequality (0.2) must be used istead of the iequality (3.5) of the previous proof. Oce we have replaced (3.2) by (0.1) ad (3.5) by (0.2), the proof of the ew versio of Propositio 3.1 is exactly the same as the proof of the old versio of Propositio Correctio of the proof of Theorem 2.1 We use the same otatios as i the previous proof. Everythig is exactly idetical up to Iequality (2.14) of the previous proof. After (2.14), we proceed as follows. We ow cotrol the term E( Z (](i 1)2 L, i2 L ]) p ) with the help of the ew Propositio 3.1. LetT = 1 ()2 L <Y k i2 L ad T = T E(T ). We apply the ew Remark 3.1 to the statioary sequece (T ) k Z, by takig d = E(T ) (hece T i,0 d = T i,0 ). We obtai that E( Z (](i 1)2 L, i2 L ]) p ) = 1 ( p/2 E C T p ) ( a p/2 i + ( T i,0 p p + c i,1 +c i,2 + c i,3 )), where, for ay 1 N, a i = 1 2 Var(T i,0) + c i,1 = c i,2 = c i,3 = 1 2 l T i,0 p 2 T k=0 k=l Cov(T i,0, T ) + k=n E(T +l M k) 1, T i,0 E(T M 0) p/2, T i,0 p 2 E(T T +l E(T T +l ) M 0) 1, T i,0 p 2 E((T )2 E((T )2 ) M 0 ) 1. The term a i is the same as i the previous versio, ad ca be hadled i the same way. Hece, the iequalities (2.15) ad (2.16) of the previous versio hold true. After (2.16), we proceed as follows [usig the correct defiitio of b 2 (M l, i, j), as recalled i Paragraph 1 of this erratum, for the cotrol of the term c i,2 ].

5 Erratum to: A empirical cetral limit theorem for itermittet maps 491 For the term c i,1, sice T i,0 p 2 1 ad 2 L c i,1 2 4 N l k=0 N T 2, oe gets 2 L E( T b 1(M k, l)) (l + 1)β 1 (l). (0.3) For the term c i,2, sice T i,0 p 2 = T i,0 ad 2 L T i,0 = 1, oe gets c i,2 4 4 N 2 L E(T i,0 b 2 (M 0, k, k + l)) β 2 (k). (0.4) For the term c i,3, ote first that (T )2 E((T )2 ) = (1 2E(T ))T. Sice 1 2E(T ) 1, it follows that E((T )2 E((T )2 ) M 0 ) E(T M 0) 2b 1 (M 0, k). Hece, sice T i,0 p 2 = T i,0 ad 2 L T i,0 = 1, oe gets c i, L E(T i,0 b 1 (M 0, k)) β 1 (k). (0.5) Note that the last bouds o the right had side of (0.3), (0.4) ad (0.5) are exactly the same as the upper bouds (2.17), (2.18) ad (2.19) of the previous versio, ad so the proof of Theorem 2.1 ca be completed as previously. Ackowledgmets I wish to thak Florece Merlevède who poited out both the problem i the defiitio of β 2 (), ad the wrog argumet i the proof of Theorem 2.1. Referece 1. Dedecker, J., Prieur, C.: A empirical cetral limit theorem for depedet sequeces. Stoch. Process. Appl. 117, (2007)