H{ö}lderian weak invariance principle under Maxwell and Woodroofe condition
aa r X i v : . [ m a t h . P R ] N ov HÖLDERIAN WEAK INVARIANCE PRINCIPLE UNDER THE MAXWELLAND WOODROOFE CONDITION
DAVIDE GIRAUDO
Abstract.
We investigate the weak invariance principle in Hölder spaces under some rein-forcement of the Maxwell and Woodroofe condition. Optimality of the obtained conditionis established. Introduction and main results
Let (Ω , F , µ ) be a probability space and let T : Ω → Ω be a measure-preserving bijectiveand bi-measurable map. Let M be a sub- σ -algebra of F such that T M ⊂ M . If f : Ω → R ameasurable function, we denote S n ( T, f ) := P n − j =0 f ◦ T j and W ( n, f, T, t ) := S [ nt ] ( T, f ) + ( nt − [ nt ]) f ◦ T [ nt ] . (1.1)We shall write S n ( f ) and W ( n, f, t ) for simplicity, except when T is replaced by T .An important problem in probability theory is the understanding of the asymptotic behaviorof the process ( n − / W ( n, f, t ) , t ∈ [0 , n > . Conditions on the quantities E [ S n ( f ) | T M ] and S n ( f ) − E [ S n ( f ) | T − n M ] have been investigated. The first result in this direction was obtainedby Maxwell and Woodroofe [MW00]: if f is M -measurable and + ∞ X n =1 k E [ S n ( f ) | M ] k n / < + ∞ , (1.2)then ( n − / S n ( f )) n > converges in distribution to η N , where N is normally distributed andindependent of η . Then Volný [Vol06] proposed a method to treat the nonadapted case.Peligrad and Utev [PU05] proved the weak invariance principle under condition (1.2). Thenonadapted case was addressed in [Vol07]. Peligrad and Utev also showed that condition (1.2)is optimal among conditions on the growth of the sequence ( k E [ S n ( f ) | M ] k ) n > : if + ∞ X n =1 a n k E [ S n ( f ) | M ] k n / < ∞ (1.3)for some sequence ( a n ) n > converging to 0, the sequence ( n − / S n ( f )) n > is not necessarilystochastically bounded (Theorem 1.2. of [PU05]). Volný constructed [Vol10] an examplesatisfying (1.3) and such that the sequence (cid:16) k S n ( f ) k − S n ( f ) (cid:17) n > admits two subsequenceswhich converge weakly to two different distributions. Date : April 21, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Invariance principle, martingales, Hölder spaces, strictly stationary process.
Let us denote by H α the space of Hölder continuous functions, that is, the functions x : [0 , → R such that k x k H α := sup s 2) as well as W ( n, f, · ), we can investigate the weak convergence of the sequence ( n − / W ( n, f, · )) n > inthe the space H α , for 0 < α < / 2. The case of i.i.d. sequences and stationary martingaledifference sequences have been addressed respectively by Račkauskas and Suquet (Theorem 1of [RS03]) and Giraudo (Theorem 2.2 of [Gir16b]). In this note, we focus on conditions on thesequences ( E [ S n ( f ) | M ]) n > and ( S n ( f ) − E [ S n ( f ) | T − n M ]) n > . Theorem 1.1. Let p > and f ∈ L p . If + ∞ X k =1 k E [ S k ( f ) | Mk p k / < + ∞ , + ∞ X k =1 (cid:13)(cid:13) S k ( f ) − E [ S k ( f ) | T − k M (cid:13)(cid:13) p k / < + ∞ , (1.4) then the sequence (cid:0) n − / W ( n, f ) (cid:1) n > converges weakly to the process √ ηW in H / − /p , where W is the Brownian motion and the random variable η is independent of W and is given by η = lim n → + ∞ E (cid:2) S n ( f ) | I (cid:3) /n (where I is the σ -algebra of invariant sets and the limit is inthe L sense). Of course, if f is M -measurable, all the terms of the second series vanish and we only haveto check the convergence of the first series. Remark . If the sequence ( f ◦ T j ) j > is a martingale difference sequence with respect to thefiltration ( T − i M ), then condition (1.4) is satisfied if and only if the function f belongs to L p ,hence we recover the result of [Gir16b]. However, if the sequence ( f ◦ T j ) j > is independent,(1.4) is stronger than the sufficient condition t p µ {| f | > t } → 0. This can be explained by thefact that the key maximal inequality (2.9) does not include the quadratic variance term whichappears in the martingale inequality. In Remark 1 (after the proof of Theorem 1) in [PUW07],a version of (2.9) with this term is obtained. In our context it seems that it does not followfrom an adaptation of the proof. Remark . In [Gir16b], the conclusion of Theorem 1.1 was obtained for an M -measurable f under the condition ∞ X i =1 (cid:13)(cid:13) E (cid:2) f | T i M (cid:3) − E (cid:2) f | T i +1 M (cid:3)(cid:13)(cid:13) p < ∞ , (1.5)which holds as soon as + ∞ X k =1 (cid:13)(cid:13) E (cid:2) f ◦ T k | M (cid:3)(cid:13)(cid:13) p k /p < + ∞ , (1.6)while (1.4) holds as soon as + ∞ X k =1 (cid:13)(cid:13) E (cid:2) f ◦ T k | M (cid:3)(cid:13)(cid:13) p √ k < + ∞ . (1.7)Therefore, (1.7) gives a better sufficient condition than (1.6) if we seek for conditions relyingonly on (cid:16)(cid:13)(cid:13) E (cid:2) f ◦ T k | M (cid:3)(cid:13)(cid:13) p (cid:17) k > . ÖLDERIAN WEAK INVARIANCE PRINCIPLE UNDER THE MAXWELL AND WOODROOFE CONDITION3 However, (1.5) gives the existence of a martingale approximation in the following sense:there exists a martingale difference m ∈ L p ( M ) such that (cid:13)(cid:13)(cid:13) k W ( n, f ) − W ( n, m ) k H / − /p (cid:13)(cid:13)(cid:13) p, ∞ = o ( √ n ) . (1.8)Indeed, define for an integrable function h and a non-negative integer i , P i ( h ) := E (cid:2) h | T i M (cid:3) − E (cid:2) h | T i +1 M (cid:3) . If f satisfies (1.5), then we set m := P i > P (cid:0) U i f (cid:1) . Then for any K > f − m = P Ki =0 (cid:0) P i ( f ) − P (cid:0) U i f (cid:1)(cid:1) + P + ∞ i = K +1 (cid:0) P i ( f ) − P (cid:0) U i f (cid:1)(cid:1) holds. Since P Ki =0 (cid:0) P i ( f ) − P (cid:0) U i f (cid:1)(cid:1) may be written as ( I − U ) g K , where g K is such that t p µ {| g K | > t } → t goes to infinity, we get, by inequalities (2.4) and (2.5) of [Gir16b] thatlim sup n → + ∞ √ n (cid:13)(cid:13)(cid:13) k W ( n, f ) − W ( n, m ) k H / − /p (cid:13)(cid:13)(cid:13) p, ∞ X i > K +1 lim sup n → + ∞ √ n (cid:18)(cid:13)(cid:13)(cid:13) k W ( n, P i ( f ))) k H / − /p (cid:13)(cid:13)(cid:13) p, ∞ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W (cid:0) n, P (cid:0) U i ( f ) (cid:1)(cid:1)(cid:13)(cid:13) H / − /p (cid:13)(cid:13)(cid:13) p, ∞ (cid:19) . We conclude by Proposition 2.3 of [Gir16b].The following condition (in the spirit of Maxwell and Woodroofe’s one) is sufficient for amartingale approximation in the sense of (1.8): + ∞ X k =1 k E [ S k ( f ) | Mk p k /p < + ∞ . (1.9)Indeed, Theorem 2.3 of [CM14] gives a martingale differences sequence (cid:0) m ◦ T i (cid:1) i > such thatlim n → + ∞ n − /p k S n ( f − m ) k p = 0. Using Serfling arguments (see [Ser70]), we get that (1.9)implies lim n → + ∞ n − /p (cid:13)(cid:13)(cid:13)(cid:13) max i n | S i ( f − m ) | (cid:13)(cid:13)(cid:13)(cid:13) p = 0 . (1.10)Note that for a function h , by Lemma A.2 of [MSR12], n − / (cid:13)(cid:13)(cid:13) k W ( n, h ) k H / − /p (cid:13)(cid:13)(cid:13) p, ∞ n − /p k max j n | S j ( f ) |k p, ∞ , hence by (1.10), the martingale approximation (1.8) holds.Furthermore, using the construction given in [DV08,Dur09], in any ergodic dynamical systemof positive entropy one can construct a function satisfying condition (1.4) but not (1.5) andvice versa. Remark . For the ρ -mixing coefficient defined by ρ ( n ) = sup (cid:8) Cov( X, Y ) / ( k X k k Y k ) , X ∈ L ( σ ( f ◦ T i , i , Y ∈ L ( σ ( f ◦ T i , i > n )) (cid:9) , Lemma 1 of [PUW07] shows that for an adapted process, condition (1.4) is satisfied if the series P ∞ n =1 ρ /p (2 n ) converges. However, the conclusion of Theorem 1.1 holds if t p µ {| f | > t } → P ∞ n =1 ρ (2 n ) converges (see Theorem 2.3, [Gir16a]), which is less restrictive.It turns out that even in the adapted case, condition (1.4) is sharp among conditions on k E [ S k ( f ) | Mk p in the following sense. DAVIDE GIRAUDO Theorem 1.5. For each sequence ( a n ) n > converging to and each real number p > , thereexists a strictly stationary sequence ( f ◦ T j ) j > and a sub- σ -algebra M such that T M ⊂ M , ∞ X n =1 a n n / k E [ S n ( f ) | M ] k p < ∞ , (1.11) but the sequence (cid:0) n − / W ( n, f, t ) (cid:1) n > is not tight in H / − /p .Remark . Using the inequalities in [PUW07] in order to bound k E [ S n ( f ) | T M ] k , we cansee that the constructed f in the proof of Theorem 1.5 satisfies the classical Maxwell andWoodroofe condition (1.2) (the fact that p is strictly greater than 2 is crucial), hence the weakinvariance principle in the space of continuous functions takes place.However, it remains an open question whether condition (1.11) implies the central limittheorem or the weak invariance principle (in the space of continuous functions).2. Proofs We may observe that condition (1.4) implies by Theorem 1 of [PUW07] that the sequence( S n ( f ) / √ n ) n > is bounded in L p ; nevertheless the counter-example given in Theorem 2.6of [Gir16a] shows that we cannot deduce the weak invariance principle from this.We shall rather work with a tighness criterion. The analogue of the continuity modulus in C [0 , 1] is ω α , defined by ω α ( x, δ ) = sup < | t − s | <δ | x ( t ) − x ( s ) || t − s | α , x : [0 , → R , δ ∈ (0 , ] . Define H oα [0 , 1] := { x ∈ H α [0 , , lim δ → ω α ( x, δ ) = 0 } . We shall essentially work with the space H oα [0 , 1] which, endowed with k·k α : x ω α ( x, 1) + | x (0) | , is a separable Banach space (while H α [0 , 1] is not). Since the canonical embedding ι : H oα [0 , → H α [0 , 1] is continuous, eachconvergence in distribution in H oα [0 , 1] also takes place in H α [0 , Proposition 2.1. Let α ∈ (0 , . A sequence of processes ( ξ n ) n > with paths in H oα [0 , andsuch that ξ n (0) = 0 for each n is tight in H oα [0 , if and only if ∀ ε > , lim δ → sup n → + ∞ µ { ω α ( ξ n , δ ) > ε } = 0 . (2.1)In order to prove the weak convergence in H oα [0 , A maximal inequality. For p > 2, we define k h k p, ∞ := sup A ∈F µ ( A ) > µ ( A ) − /p E [ | h | A ] . (2.2)This norm is linked to the tail function of h by the following inequalities (see Exercice 1.1.12p. 13 in [Gra14]): (cid:18) sup t> t p µ {| h | > t } (cid:19) /p k h k p, ∞ pp − (cid:18) sup t> t p µ {| h | > t } (cid:19) /p . (2.3) ÖLDERIAN WEAK INVARIANCE PRINCIPLE UNDER THE MAXWELL AND WOODROOFE CONDITION5 As a consequence, if N is an integer and h , . . . , h n are functions, then (cid:13)(cid:13)(cid:13)(cid:13) max j N | h j | (cid:13)(cid:13)(cid:13)(cid:13) p, ∞ pp − N /p max j N k| h j |k p, ∞ . (2.4)For a positive n > 1, a function f : Ω → R and a measure-preserving map T , we define M ( n, f, T ) := max i U f )( ω ) = f ( T ω ). Definition 2.2. Let H be a closed subspace of L p . Let P be a linear operator from H to itself.We say that ( H, P ) satisfies condition ( C ) if(1) the inclusion U − H ⊂ H holds (respectively the inclusion U H ⊂ H holds);(2) P is power bounded on H , that is, for each h ∈ H , K ( P ) := sup n > sup h ∈ H \{ } k P n h k p k h k p < + ∞ ; (2.8) (3) if h ∈ H is such that P h = 0 , then the sequence ( h ◦ T i ) i > is a martingale differencesequence with respect to the filtration (cid:0) T − i M (cid:1) i > (respectively (cid:0) T − i − M (cid:1) i > );(4) P U − f = f for each f ∈ H (respectively P U f = f for each f ∈ H ). Let us give two examples of subspace H and operator P satisfying condition (C).(1) Let H be the subspace of L p which consists of M -measurable functions and P h := E [ U h | M ]. Then ( H, P ) satisfies condition (C).(2) Let H be the subspace of L p which consists of functions h such that E [ h | M ] = 0 and P h := U − h − E (cid:2) U − h | M (cid:3) . Then ( H, P ) satisfies condition (C).The goal of this subsection is to establish the following maximal inequality. Proposition 2.3. Let T : Ω → Ω be a bijective and bi-measurable measure-preserving map.Let H be a closed subspace of L p . Let r be a positive integer. For each , operator P from H to itself such that ( H, P ) satisfies condition ( C ) , each f ∈ H and each integer n satisfying r − n < r , k M ( n, f, T ) k p, ∞ C p n /p (1 + K ( P )) k f k p + K p r − X j =0 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j − X i =0 P i f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p , (2.9) where K p = 2 /p − / + 2 / (1 + K ( P )) . DAVIDE GIRAUDO If H is a closed subspace of L p and P : H → H an operator such that ( H, P ) satisfiescondition ( C ), we define for f ∈ H the quantity k f k MW( p,P ) := + ∞ X j =0 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j − X i =0 P i f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p (2.10)and the vector space MW( p, P ) := n f ∈ H | k f k MW( p,P ) < + ∞ o . (2.11)Note that MW( p, P ) endowed with k·k MW( p,P ) is a Banach space.Combining Proposition 2.3 and (2.6), we derive the following bound for the Hölderian normof the partial sum process. Corollary 2.4. Let H be a closed subspace of L p and let P be an operator from H to itselfsuch that ( H, P ) satisfies the condition ( C ) . Then there exists a constant C = C ( p, P ) suchthat for each n , and each h ∈ H , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) √ n W ( n, h ) (cid:13)(cid:13)(cid:13)(cid:13) H / − /p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p, ∞ C k h k MW( p,P ) (2.12)The proof of Proposition 2.3 is in the same spirit as the proof of Theorem 1 of [PUW07],which is done by dyadic induction. To do so, we start from the following lemma: Lemma 2.5. For each positive integer n , each function h : Ω → R and each measure-preservingmap T : Ω → Ω , the following inequality holds: M ( n, h, T ) k n (cid:12)(cid:12) h ◦ T k (cid:12)(cid:12) + 12 / − /p M (cid:16)h n i , h + h ◦ T, T (cid:17) . (2.13) Proof. First, notice that if 1 j n , then j = 2 (cid:2) j (cid:3) or j = 2 (cid:2) j (cid:3) + 1, hence (cid:12)(cid:12)(cid:12) S j ( h ) − S [ j ]( h ) (cid:12)(cid:12)(cid:12) max k n (cid:12)(cid:12) h ◦ T k (cid:12)(cid:12) . (2.14)Similarly, we have (cid:12)(cid:12)(cid:12) S i ( h ) − S [ i +22 ]( h ) (cid:12)(cid:12)(cid:12) k n (cid:12)(cid:12) h ◦ T k (cid:12)(cid:12) . (2.15)It thus follows that M ( n, h, T ) k n (cid:12)(cid:12) h ◦ T k (cid:12)(cid:12) + max i Since for j i + 4, the number of terms of the form h ◦ T q involved in S [ j ]( h ) − S [ i +22 ]( h ) isat most 2, we conclude thatmax i Now, we establish inequality (2.9) by induction on r . Proof of Proposition 2.3. We first assume that P U − = Id and U − H ⊂ H . We check thecase r = 1. Then necessarily n = 1 and the expression M ( n, f, t ) reduces to f . Since C p and K p are greater than 1, the result is a simple consequence of the triangle inequality applied to f − U − P f and U − P f .Now, assume that Proposition 2.3 holds for some r and let us show that it takes place for r + 1. We thus consider an integer n such that 2 r n < r +1 , a function f ∈ H , a measure-preserving map T : Ω → Ω bijective and bi-measurable, and a sub- σ -algebra M satisfying T M ⊂ M , a closed subspace H of L such that U − H ⊂ H and an operator P : H → H suchthat ( H, P ) satisfies condition ( C ) with P U − = Id and we have to show that (2.9) holds with r + 1 instead of r . First, using inequality M ( n, f ) M ( n, f − U − P f ) + M ( n, U − P f ) andLemma 2.5 with h := U − P f , we derive M ( n, f, T ) M (cid:0) n, f − U − P f, T (cid:1) + 6 max k n (cid:12)(cid:12) U − P f ◦ T k (cid:12)(cid:12) ++ 12 / − /p M (cid:16)h n i , ( I + U ) U − P f, T (cid:17) , (2.18)hence taking the norm k·k p, ∞ , we obtain by (2.4) that k M ( n, f, T ) k p, ∞ (cid:13)(cid:13) M ( n, f − U − P f, T ) (cid:13)(cid:13) p, ∞ + 6( n + 1) /p pp − (cid:13)(cid:13) U − P f (cid:13)(cid:13) p ++ 12 / − /p (cid:13)(cid:13)(cid:13) M (cid:16)h n i , ( I + U ) U − P f, T (cid:17)(cid:13)(cid:13)(cid:13) p, ∞ . (2.19)By inequality (2.7) and accounting the fact that 6 · ( n + 1) /p p/ ( p − C p n /p , we obtain k M ( n, f, T ) k p, ∞ C p n /p (cid:13)(cid:13) f − U − P f (cid:13)(cid:13) p + C p n /p k P f k p ++ 12 / − /p (cid:13)(cid:13)(cid:13) M (cid:16)h n i , ( I + U ) U − P f, T (cid:17)(cid:13)(cid:13)(cid:13) p, ∞ . (2.20)Since 2 r − [ n/ < r , we may apply the induction hypothesis to the integer [ n/ h := ( I + U − ) P f , T instead of T and P instead of P . This gives h n i − /p (cid:13)(cid:13)(cid:13) M (cid:16)h n i , h, T (cid:17)(cid:13)(cid:13)(cid:13) p, ∞ C p (cid:0) K (cid:0) P (cid:1)(cid:1) k h k p ++ C p f K p r − X j =0 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j − X i =0 P i (cid:0) I + U − (cid:1) P f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p , (2.21) DAVIDE GIRAUDO where f K p = 2 /p − / + 2 / (cid:0) K ( P ) (cid:1) . Notice that k h k p k P f k p , and by item 4 ofDefinition 2.2, it follows that j − X i =0 P i (cid:0) I + U − (cid:1) P f = j − X i =0 (cid:0) P i +1 f + P i f (cid:1) = j +1 − X i =0 P i f. (2.22)Accounting the inequality K (cid:0) P (cid:1) K ( P ) and f K p K p , we have h n i − /p (cid:13)(cid:13)(cid:13) M (cid:16)h n i , h, T (cid:17)(cid:13)(cid:13)(cid:13) p, ∞ K ( P )) C p k P f k p + C p K p r − X j =0 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j +1 − X i =0 P i f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p = 2 (1 + K ( P )) C p k P f k p + 2 / C p K p r X j =1 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j − X i =0 P i f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p and we infer (cid:13)(cid:13)(cid:13) M (cid:16)h n i , h, T (cid:17)(cid:13)(cid:13)(cid:13) p, ∞ (cid:16) n (cid:17) /p (cid:16) K ( P )) − K p √ (cid:17) C p k P f k p + n /p / − /p C p K p r X j =0 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j − X i =0 P i f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p . (2.23)Pluggling this into (2.20), we derive k M ( n, f, T ) k p, ∞ C p n /p (1 + K ( P )) k f k p + n /p C p K p r X j =0 − j/ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) j − X i =0 P i f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ++ C p n /p (cid:16) − /p (1 + K ( P )) − / − /p K p (cid:17) k P f k p . (2.24)The definition of K p implies that 2 /p − / − √ K ( P )) − K p = 0, hence (2.9) is established.This concludes the proof of Proposition 2.3 in the case P U − = Id.When P U = Id and U H ⊂ H we do the same proof, but replacing each occurrence of U − by U . This ends the proof of Proposition 2.3. (cid:3) Proof of Theorem 1.1. Since the convergence of the finite dimensional distributions iscontained in the main result of [Vol07], the only difficulty in proving Theorem 1.1 is to establishtightness. To this aim, we shall proceed as in the proof of Theorem 5.3 in [Cun14]. Proposition 2.6. Let T be a measure preserving map, H a closed subspace of L p ( p > ) andlet P be an operator from H to itself such that ( H, P ) satisfies condition ( C ) . Assume that h is an element of H such that k h k MW( p,P ) < + ∞ Then the sequence ( n − / W ( n, h )) n > is tight in H / − /p .Proof. Let us define V n := P n − i =0 P i . Using k V n V k k p K ( P ) min n k k V n k p , n k V k k p o , wederive that for each f ∈ MW( p, P ), k V n f k MW( p,P ) n K ( P ) k V n f k p n/ + X k > n +1 (cid:13)(cid:13)(cid:13) V k f (cid:13)(cid:13)(cid:13) p k/ (2.25) ÖLDERIAN WEAK INVARIANCE PRINCIPLE UNDER THE MAXWELL AND WOODROOFE CONDITION9 which goes to 0 as n goes to infinity. If m > n is such that 2 n m < n +1 ,then k V m f k MW( p,P ) m K ( P ) m n X k =0 k V k f k MW( p,P ) K ( P ) m n X k =0 k ε k , (2.26)where ( ε k ) k > is a sequence converging to 0. This entails that the operator P is mean-ergodicon MW( p, P ). Furthermore, since P has no non trivial fixed points on the Banach space (cid:16) MW( p, P ) , k·k MW( p,P ) (cid:17) , we derive by Theorem 1.3 p.73 of [Kre85] that the subspace ( I − P )MW( p, P ) is dense in MW( p, P ) for the topology induced by the norm k·k MW( p,P ) .Let h ∈ H be such that k h k MW( p,P ) < + ∞ and x > 0. We can find f ∈ ( I − P )MW( p, P )such that k h − f k MW( p,P ) < x . Consequently, using Corollary 2.4, we derive that for eachpositive ε and δ , µ (cid:26) ω / − /p (cid:18) √ n W ( n, h ) , δ (cid:19) > ε (cid:27) ε − p x + µ (cid:26) ω / − /p (cid:18) √ n W ( n, f ) , δ (cid:19) > ε (cid:27) . (2.27)Now, since the function f belongs to ( I − P )MW( p, P ), we can find f ′ ∈ MW( p, P ) suchthat f = f ′ − P f ′ . If P U − = Id, then we write f = f ′ − U − P f ′ + ( U − − I ) f ′ and if P U = Id, then f = f ′ − U P f ′ + ( U − I ) f ′ . In other words, f admits a martingale-coboundarydecomposition in L p (since f ′ belongs to L p ). Consequently, by Corollary 2.5 of [Gir16b], thesequence ( n − / W ( n, f )) n > is tight in H / − /p . By Proposition 2.1 and (2.27), we derivethat for each positive ε and x ,lim δ → lim sup n → + ∞ µ (cid:26) ω / − /p (cid:18) √ n W ( n, h ) , δ (cid:19) > ε (cid:27) ε − p x. (2.28)Since x is arbitrary we conclude the proof of (2.6) by using again Proposition 2.1. (cid:3) Proof of Theorem 1.1. Writing f = E [ f | M ] + f − E [ f | M ], the proof reduces (as men-tioned in the begining of the section) to establish tightness in H o / − /p [0 , 1] of the sequences( W n ) n > := (cid:0) n − / W ( n, E [ f | M ]) (cid:1) n > and ( W ′ n ) n > := (cid:0) n − / W ( n, f − E [ f | M ]) (cid:1) n > . • Tightness of ( W n ) n > . We define P ( f ) := E [ U f | M ] and H := { f ∈ L p , f is M -measurable } . (2.29)Then ( H, P ) satisfies condition ( C ). Since n − X i =0 P i ( E [ f | M ]) = E [ S n ( f ) | M ] , (2.30)the convergence of the first series in (1.4) is equivalent to f ∈ MW( p, P ) (byLemma 2.7 of [PU05]). By Proposition 2.6, we derive that the sequence ( W n ) n > is tight in H o / − /p [0 , • Tightness of ( W ′ n ) n > . We define P ( f ) := U − f − E (cid:2) U − f | M (cid:3) and H := { f ∈ L p , E [ f | M ] = 0 } . (2.31)Since for each f ∈ H and each k > (cid:13)(cid:13) P k f (cid:13)(cid:13) p k f k p , ( H, P ) satisfies condi-tion ( C ) (see the proof of Proposition 2 in [Vol07] for the other conditions). Since P ( E [ f | M ]) = 0, we have n X i =1 P i ( f − E [ f | M ]) = n X i =1 P i f = U − n (cid:0) S n ( f ) − E (cid:2) S n ( f ) | T − n M (cid:3)(cid:1) , (2.32)hence the convergence of the second series in (1.4) implies that f belongs to MW( p, P )(by Lemma 37 of [MP13]). By Proposition 2.6, we derive that the sequence ( W ′ n ) n > is tight in H o / − /p [0 , (cid:3) Proof of Theorem 1.5. We take a similar construction as in the proof of Proposition 1of [PUW07]. We consider a non-negative sequence ( a n ) n > , and a sequence ( u k ) k > of realnumbers such that u = 1 , u = 2 , u p/ k + 1 < u k +1 for k > a t k − for t > u k . (2.33)Notice that since p > 2, the conditions (2.33) are more restrictive than that of the proof ofProposition 1 of [PUW07]. If i = u j for some j > 1, then we define p i := cj/u p/ j and p i = 0otherwise. Let ( Y k ) k > be a discrete time Markov chain with the state space Z + and transitionmatrix given by p k,k − = 1 for k > p ,j − := p j , j > 1. We shall also consider a randomvariable τ which takes its values among non-negative integers, and whose distribution is givenby µ ( τ = j ) = p j . Then the stationary distribution exists and is given by π j = π ∞ X i = j +1 p i , j > , where π = 1 / E [ τ ] . (2.34)We start from the stationary distribution ( π j ) j > and we take g ( x ) := x =0 − π , where π = µ { Y = 0 } . We then define f ◦ T j = X j := g ( Y j ).It is already checked in [PUW07] that the sequence ( X j ) j > satisfies (1.11), where M = σ ( X k , k j ) and S n = P nj =1 X j . To conclude the proof, it remains to check that the sequence (cid:0) n − / W ( n, f, T ) (cid:1) n > is not tight in H o / − /p , which will be done by disproving (2.1) for aparticular choice of ε . To this aim, we define T = 0 , T k = min { t > T k − | Y t = 0 } , τ k = T k − T k − , k > . (2.35)Then ( τ k ) k > is an independent sequence and each τ k is distributed as τ and S T k = k X j =1 (1 − π τ j ) = k − π T k . (2.36)Let us fix some integer K greater than E [ τ ]. Let δ ∈ (0 , 1) be fixed and n an integer suchthat 1 /n < δ . Then the inequality1( nK ) /p max i The author would like to thank an anonymous referee for many valu-able comments which improved the presentation of the paper and led to a shorter proof ofTheorem 1.1. 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