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Holderian weak invariance principle under a Hannan type condition
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ScienceDirect Stochastic Processes and their Applications 126 (2016) 290–311 www.elsevier.com/locate/spa
Holderian weak invariance principle under a Hannan type condition Davide Giraudo ´ Universit´e de Rouen, LMRS, Avenue de l’Universit´e, BP 12 76801 Saint-Etienne-du-Rouvray cedex, France Received 9 March 2015; received in revised form 2 September 2015; accepted 2 September 2015 Available online 10 September 2015
Abstract We investigate the invariance principle in H¨older spaces for strictly stationary martingale difference sequences. In particular, we show that the sufficient condition on the tail in the i.i.d. case does not extend to stationary ergodic martingale differences. We provide a sufficient condition on the conditional variance which guarantee the invariance principle in H¨older spaces. We then deduce a condition in the spirit of Hannan one. c 2015 Elsevier B.V. All rights reserved. ⃝
MSC: 60F05; 60F17 Keywords: Invariance principle; Martingales; Strictly stationary process
1. Introduction One of the main problems in probability theory is the understanding of the asymptotic behavi ior of Birkhoff sums Sn ( f ) := n−1 j=0 f ◦ T , where (Ω , F, µ, T ) is a dynamical system and f a map from Ω to the real line. One can consider random functions constructed from the Birkhoff sums pl
Sn ( f, t) := S[nt] ( f ) + (nt − [nt]) f ◦ T [nt]+1 ,
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spa.2015.09.001 c 2015 Elsevier B.V. All rights reserved. 0304-4149/⃝
t ∈ [0, 1]
(1.1)
D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
pl and investigate the asymptotic behavior of the sequence Sn ( f, t)
291
seen as an element of
n≥1 pl (n −1/2 (E( f 2 ))−1/2 Sn ( f ))n≥1
a function space. Donsker showed (cf. [4]) that the sequence converges in distribution in the space of continuous functions on the unit interval to a standard Brownian motion W when the sequence ( f ◦ T i )i≥0 is i.i.d. and zero mean. Then an intensive research has then been performed to extend this result to stationary weakly dependent sequences. We refer the reader to [9] for the main theorems in this direction. pl Our purpose is to investigate the weak convergence of the sequence (n −1/2 Sn ( f ))n≥1 in i H¨older spaces when ( f ◦ T )i≥0 is a strictly stationary sequence. A classical method for showing a limit theorem is to use a martingale approximation, which allows to deduce the corresponding result if it holds for martingale difference sequences provided that the approximation is good enough. To the best of our knowledge, no result about the invariance principle in H¨older space for stationary martingale difference sequences is known. 1.1. The H¨older spaces It is well known that standard Brownian motion’s paths are almost surely H¨older regular of exponent α for each α ∈ (0, 1/2), hence it is natural to consider the random function defined in (1.1) as an element of Hα [0, 1] and try to establish its weak convergence to a standard Brownian motion in this function space. Before stating the results in this direction, let us define for α ∈ (0, 1) the H¨older space Hα [0, 1] of functions x: [0, 1] → R such that sups̸=t |x(s) − x(t)| / |s − t|α is finite. The analogue of the continuity modulus in C[0, 1] is wα , defined by wα (x, δ) =
sup 0<|t−s|<δ
|x(t) − x(s)| . |t − s|α
(1.2)
We then define Hα0 [0, 1] by Hα0 [0, 1] := {x ∈ Hα [0, 1], limδ→0 wα (x, δ) = 0}. We shall essentially work with the space Hα0 [0, 1] which, endowed with ∥x∥α := wα (x, 1) + |x(0)|, is a separable Banach space (while Hα [0, 1] is not separable). Since the canonical embedding ι: Hαo [0, 1] → Hα [0, 1] is continuous, each convergence in distribution in Hαo [0, 1] also takes place in Hα [0, 1]. Let us denote by D j the set of dyadic numbers in [0, 1] of level j, that is, D0 := {0, 1} , D j := (2l − 1)2− j ; 1 ≤ l ≤ 2 j−1 , j ≥ 1. (1.3) If r ∈ D j for some j ≥ 0, we define r + := r + 2− j and r − := r − 2− j . For r ∈ D j , j ≥ 1, let Λr be the function whose graph is the polygonal path joining the points (0, 0), (r − , 0), (r, 1), (r + , 0) and (1, 0). We can decompose each x ∈ C[0, 1] as x=
λr (x)Λr =
r ∈D
+∞
λr (x)Λr ,
(1.4)
j=0 r ∈D j
and the convergence is uniform on [0, 1]. The coefficients λr (x) are given by x(r + ) + x(r − ) , 2 and λ0 (x) = x(0), λ1 (x) = x(1). λr (x) = x(r ) −
r ∈ D j , j ≥ 1,
(1.5)
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Ciesielski proved (cf. [2]) that {Λr ; r ∈ D} is a Schauder basis of Hαo [0, 1] and the norms ∥·∥α and the sequential norm defined by seq
∥x∥α := sup 2 jα max |λr (x)| , j≥0
(1.6)
r ∈D j
are equivalent. Considering the sequential norm, we can show (see Theorem 3 in [13]) that a sequence (ξn )n≥1 of random elements of Hαo vanishing at 0 is tight if and only if for each positive ε, lim lim sup µ sup 2 jα max |λr (ξn )| > ε = 0.
J →∞ n→∞
j≥J
(1.7)
r ∈D j
Notation 1.1. In the sequel, we will denote rk, j := k2− j and u k, j := [nrk, j ] (or rk and u k for short). Notice that u k+1, j − u k, j = [nrk, j + n2− j ] − u k, j ≤ 2n2− j if j ≤ log n, where log n denotes the binary logarithm of n and for a real number x, [x] is the unique integer for which [x] ≤ x < [x] + 1. Remark 1.2. Since for each x ∈ H1/2−1/ p [0, 1], each j ≥ 1 and each r ∈ D j , + x(r ) − x(r ) x(r ) − x(r − ) |λr (x)| ≤ + 2 2 + ≤ max x(r ) − x(r ) , x(r ) − x(r − ) ,
(1.8)
pl
for a function f , the sequential norm of n −1/2 Sn ( f ) does not exceed pl pl sup 2α j n −1/2 max Sn ( f, rk+1, j ) − Sn ( f, rk, j ) . 0≤k<2 j
j≥1
(1.9)
Now, we state the result obtained by Raˇckauskas and Suquet in [11]. Theorem 1.3. Let p > 2 and let ( f ◦ T j ) j≥0 be an i.i.d. centered sequence with unit variance. Then the condition lim t p µ {| f | > t} = 0
(1.10)
t→∞
pl
is equivalent to the weak convergence of the sequence (n −1/2 Sn ( f ))n≥1 to a standard Brownian o motion in the space H1/2−1/ p [0, 1]. 1.2. Some facts about the L p,∞ spaces In the rest of the paper, χ denotes the indicator function. Let p > 2. We define the L p,∞ space as the collection of functions f : Ω → R such that the quantity p
∥ f ∥ p,∞ := sup t p µ {| f | > t} < ∞.
(1.11)
t>0
This quantity is denoted like a norm, while it is not a norm (the triangle inequality may fail, for example if X = [0, 1] endowed with the Lebesgue measure, f (x) := x −1/ p and g(x) :=
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293
f (1 − x); in this case ∥ f + g∥ p,∞ ≥ 21+1/ p but ∥ f ∥ p,∞ + ∥g∥ p,∞ = 2). However, there exists a constant κ p such that for each f , ∥ f ∥ p,∞ ≤
sup
µ(A)−1+1/ p E[| f | χ A ] ≤ κ p ∥ f ∥ p,∞
(1.12)
A:µ(A)>0
and N p ( f ) := sup A:µ(A)>0 µ(A)−1+1/ p E[| f | χ A ] defines a norm. The first inequality in (1.12) can be seen from the estimate tµ {| f | > t} ≤ E [| f | χ {| f | > t}]; for the second one, we write +∞ +∞ E[| f | χ A ] = µ ({| f | > t} ∩ A) dt ≤ min {µ {| f | > t} , µ(A)} dt, (1.13) 0
0
p and we bound the integrand by min t − p ∥ f ∥ p,∞ , µ(A) . A function f satisfies (1.10) if and only if it belongs to the closure of bounded functions with respect to N p . Indeed, if f satisfies (1.10), then the sequence ( f χ | f < n|)n≥1 converges to f in L p,∞ . If N p ( f − g) < ε with g bounded, then lim sup t p µ {| f | > t} ≤ lim sup t p µ {| f − g| > t/2} ≤ 2 p ε. t→∞
(1.14)
t→∞
We now provide two technical lemmas about L p,∞ spaces. The first one will be used in the proof of the weak invariance principle for martingales, since we will have to control the tail function of the random variables involved in the construction of the truncated martingale (cf. (3.110)). The second one will provide an estimation of the L p,∞ norm of a simple function, which will be used in the proof of Theorem 2.1, since the function m is constructed as a series of simple functions. Lemma 1.4. If limt→∞ t p µ {| f | > t} limt→∞ t p µ {E[| f | | A] > t} = 0.
=
0, then for each sub-σ -algebra A, we have
Proof. For simplicity, we assume that f is non-negative. For a fixed t, the set {E[ f | A] > t} belongs to the σ -algebra A, hence tµ {E[ f | A] > t} ≤ E [E[ f | A]χ {E[ f | A] > t}] = E[ f χ {E[ f | A] > t}].
(1.15)
By definition of N p , E[ f χ {E[ f | A] > t}] ≤ N p ( f χ {E[ f | A] > t}) µ {E[ f | A] > t}1−1/ p ,
(1.16)
t p µ {E[ f | A] > t} ≤ N p ( f χ {E[ f | A] > t}) p .
(1.17)
hence
Notice that ∀s > 0,
N p ( f χ {E[ f | A] > t}) ≤ sµ {E[ f | A] > t}1/ p + N p ( f χ { f > s}) , (1.18)
hence lim sup E[ f χ {E[| f | | A] > t}] ≤ N p ( f χ { f > s}) ≤ κ p sup x p µ { f > x} . t→∞
(1.19)
x≥s
By the assumption on the function f , the right hand side goes to 0 as s goes to infinity, which concludes the proof of the lemma.
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N
i=0 ai χ (Ai ),
Lemma 1.5. Let f := a N < · · · < a0 . Then p
p
∥ f ∥ p,∞ ≤ max a j 0≤ j≤N
j
N is pairwise disjoint and 0 ≤ where the family (Ai )i=0
µ(Ai ).
(1.20)
i=0
Proof. We have the equality µ { f > t} =
N
χ(a j+1 ,a j ] (t)
j
µ(Ai ),
(1.21)
i=0
j=0
where a N +1 := 0, therefore p
t p µ { f > t} ≤ max a j 0≤ j≤N
j
µ(Ai ).
(1.22)
i=0
2. Main results The goal of the paper is to give a sharp sufficient condition on the moments of a strictly stationary martingale difference sequence which guarantees the weak invariance principle in Hαo [0, 1] for a fixed α. We first show that Theorem 1.3 does not extend to strictly stationary ergodic martingale difference sequences, that is, sequences of the form (m ◦ T i )i≥0 such that m is M measurable and E[m | T M] = 0 for some sub-σ -algebra M satisfying T M ⊂ M. An application of Kolmogorov’s continuity criterion shows that if (m ◦ T i )i≥0 is a martingale difference sequence such that m ∈ L p+δ for some positive δ and p > 2, then the partial sum pl o process (n −1/2 Sn (m))n≥1 is tight in H1/2−1/ p [0, 1] (see [6]). We provide a condition on the quadratic variance which improves the previous approach (since the previous condition can be replaced by m ∈ L p ). Then using martingale approximation we can provide a Hannan type condition which guarantees the weak invariance principle in Hαo [0, 1]. Theorem 2.1. Let p > 2 and (Ω , F, µ, T ) be a dynamical system with positive entropy. There exist a function m: Ω → R and a σ -algebra M for which T M ⊂ M such that: • the sequence (m ◦ T i )i≥0 is a martingale difference sequence with respect to the filtration (T −i M)i≥0 ; • the convergence limt→+∞ t p µ {|m| > t} = 0 takes place; pl o • the sequence (n −1/2 Sn (m))n≥1 is not tight in H1/2−1/ p [0, 1]. Theorem 2.2. Let (Ω , F, µ, T ) be a dynamical system, M a sub-σ -algebra of F such that T M ⊂ M and I the collection of sets A ∈ F such that T −1 = A. Let p > 2 and let (m ◦ T j , T −i M) be a strictly stationary martingale difference sequence. Assume that t p µ {|m| > t} → 0 and E[m 2 | T M] ∈ L p/2 . Then pl
o n −1/2 Sn (m) → η · W in distribution in H1/2−1/ p [0, 1],
(2.1)
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where the random variable η is given by η = lim E[Sn2 | I]/n in L1
(2.2)
n→∞
and η is independent of the process (Wt )t∈[0,1] . In particular, (2.1) takes place if m belongs to L p . The key point of 2.2 is an inequality in the spirit of Doob’s one, the proof of Theorem which gives n −1 E max1≤ j≤n S j (m)2 ≤ 2E[m 2 ]. It is used in order to establish tightness of pl
the sequence (n −1/2 Sn (m))n≥1 in the space C[0, 1]. Proposition 2.3. Let p > 2. There exists a constant C p depending only on p such that if (m ◦ T i )i≥1 is a martingale difference sequence, then the following inequality holds: p p/2 −1/2 pl p 2 ∥m∥ . (2.3) S (m) ≤ C + E E[m | T M] sup n p n p,∞ o n≥1
H1/2−1/ p
p,∞
Remark 2.4. As Theorem 2.1 shows, the condition limt→∞ t p µ {|m| > t} = 0 alone for pl martingale difference sequences is not sufficient to obtain the weak convergence of n −1/2 Sn (m) o in Hα [0, 1] for α = 1/2 − 1/ p. For the constructed m in Theorem 2.1, the quadratic variance is κm 2 for some constant κ and m does not belong to the L p space. By Lemma A.2 in [8], the H¨older norm of a polygonal line is reached at two vertices, hence, for a function g, g ◦ T j − g ◦ T i −1/2 pl −1/ p Sn (g − g ◦ T ) o =n max (2.4) n H1/2−1/ p 1≤i< j≤n ( j − i)1/2−1/ p ≤ 2n −1/ p max g ◦ T j . (2.5) 1≤ j≤n
As a consequence, if g belongs to
Lp,
pl then the sequence n −1/2 Sn (g − g ◦ T )
o H1/2−1/ p
n≥1
converges to 0 in probability. Therefore, we can exploit a martingale–coboundary decomposition in L p . Corollary 2.5. Let p > 2 and let f be an M-measurable function which can be written as f = m + g − g ◦ T,
(2.6)
where m, g ∈ L p and (m ◦ T i )i≥0 is a martingale difference sequence for the filtration pl o (T −i M)i≥0 . Then n −1/2 Sn ( f ) → ηW in distribution in H1/2−1/ p [0, 1], where η is given by (2.2) and independent of W . k We define for a∞function h the operators Ek (h) := E[h | T M] and Pi (h) := Ei (h)−Ei+1 (h). The condition i=0 ∥Pi ( f )∥2 was introduced by Hannan in [5] in order to deduce a central limit theorem. It actually implies the weak invariance principle (see Corollary 2 in [3]).
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Theorem 2.6. Let p > 2 and let f be an M-measurable function such that i T M = 0 and E f |
(2.7)
i∈Z
∥Pi ( f )∥ p < ∞.
(2.8)
i≥0 pl
o Then n −1/2 Sn (m) → ηW in distribution in H1/2−1/ p [0, 1], where η is given by (2.2) and independent of W .
3. Proofs 3.1. Proof of Theorem 2.1 We need a result about dynamical systems of positive entropy for the construction of a counterexample. Lemma 3.1. Let (Ω , A, µ, T ) be an ergodic probability measure preserving system of positive entropy. There exist two T -invariant sub-σ -algebras B and C of A and a function g: Ω → R such that: • the σ -algebras B and C are independent; • the function g is B-measurable, takes the values −1, 0 and 1, has zero mean and the process (g ◦ T n )n∈Z is independent; • the dynamical system (Ω , C, µ, T ) is aperiodic. This is Lemma 3.8 from [7]. We consider the following four increasing sequences of positive integers (Il )l≥1 , (Jl )l≥1 , (nl )l≥1 and (L l )l≥1 . We define kl := 2 Il +Jl and impose the conditions: ∞ 1 < ∞; L l=1 l Ll lim Jl · µ |N | ≥ 41/ p = 1; l→∞ ∥g∥2
lim Jl 2−Il /2 = 0;
l→∞
lim nl
l→∞
ki = 0; n i>l i
for each l,
1/ p l−1 nl kl n i 1/ p < . L 2 Ii 2 i=1 i
(3.1) (3.2) (3.3) (3.4)
(3.5)
Here N denotes a random variable whose distribution is standard normal. Such sequences can be constructed as follows: first pick a sequence (L l )l≥1 satisfying (3.1), for example L l = l 2 . Then construct (Jl )l≥1 such that (3.2) holds. Once the sequence (Jl )l≥1 is constructed, define (Il )l≥1 satisfying (3.3). Now the sequence (kl )l≥1 is completely determined. Noticing that (3.4)
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is satisfied if the series l kl nl−1 /nl converges, we construct the sequence (nl )l≥1 by induction; once n i , i ≤ l − 1 are defined, we choose nl such that nl ≥ l 2 kl nl−1 and (3.5) holds. Using Rokhlin’s lemma, we can find for any integer l ≥ 1 a measurable set Cl ∈ C such that nl −1 −i −i the sets T Cl , i = 0, . . . , nl − 1 are pairwise disjoint and µ i=0 T Cl > 1/2. For a fixed l, we define kl, j := 2 Il +Jl − j , 0 ≤ j ≤ Jl and 1/ p kl,Jl − j−1 −1 l −1 1 J nl fl := T −i Cl χ L l j=0 kl,Jl − j i=kl,Jl − j 1/ p kl,Jl −1 nl 1 + χ T −i Cl , L l kl,Jl i=0 f :=
+∞
fl ,
(3.6)
(3.7)
m := g · f,
l=1
where g is the function obtained by Lemma 3.1. Proposition 3.2. We have the estimate ∥ fl ∥ p,∞ ≤ κ ′p L l−1 for some constant κ ′p depending only on p. As a consequence, limt→∞ t p µ {|m| > t} = 0. Proof. Notice that p 1/ p kl,Jl −1 1 n l −i χ T C l L k l,Jl l i=0
=
p,∞
Next, using Lemma 1.5 with N kl,Jl − j−1 −1 i=kl,Jl − j
:=
1 nl 1 kl,Jl · µ(Cl ) ≤ p . p L l kl,Jl Ll
Jl − 1, a j
:=
1 Ll
nl kl,Jl − j
1/ p
(3.8)
and A j
:=
T −i Cl , we obtain
p J 1/ p kl,Jl − j−1 1 l −1 nl −i χ T Cl L l j=0 kl,Jl − j i=kl,Jl − j
p,∞
≤
max
0≤ j≤Jl −1
≤
1 p Ll
1 = p Ll ≤
max
1 Ll
1/ p p j
nl kl,Jl − j nl
j kl,J −i
0≤ j≤Jl −1 kl,Jl − j i=0 j 2 Il +i max 0≤ j≤Jl −1 2 Il + j i=0
2 p, Ll
µ(A j )
(3.9)
i=0 l
nl
(3.10)
(3.11) (3.12)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
hence by (1.12), (3.8) and (3.12), ∥ fl ∥ p,∞
1 ≤ Np Ll
1/ p
nl kl,Jl
kl,Jl −1
χ
T
−i
(3.13)
Cl
i=0
1/ p kl,Jl − j−1 J l −1 1 n l + Np χ T −i Cl L l j=0 kl,Jl − j i=kl,Jl − j 1/ p kl,Jl −1 1 nl −i χ T C ≤ κp l L k l,Jl l i=0
(3.14)
(3.15)
p,∞
J 1/ p kl,Jl − j−1 1 l −1 nl −i + κp χ T C l L l j=0 kl,Jl − j i=kl,Jl − j ≤
(3.16) p,∞
1 κ p 1 + 21/ p . Ll
(3.17)
We thus define κ ′p := κ p 1 + 21/ p . We fix ε > 0; using (3.1), we can find an integer l0 such that l>l0 1/L l < ε. Since the l0 function l=1 g fl is bounded, we have, l p 0 t p +2 g fl lim sup t µ {|m| > t} ≤ lim sup t µ g fl > l>l l=1 2 t→∞ t→∞ p
p
0
p p g fl =2 l>l 0
≤
2
(3.18)
p,∞
(3.19)
p,∞ p
N p ( fl )
(3.20)
p 1 Ll l>l
(3.21)
l>l0
≤
κ ′p
0
≤ κ ′p ε p ,
(3.22)
where the second inequality comes from inequalities (1.12). Since ε is arbitrary, the proof of Proposition 3.2 is complete. We denote by M the σ -algebra generated by C and the random variables g ◦ T k , k ≤ 0. It satisfies M ⊂ T −1 M. Proposition 3.3. The sequence (m ◦ T i )i≥0 is a (stationary) martingale difference sequence with respect to the filtration (T −i M)i≥0 .
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Proof. We have to show that E[m | T M] = 0. Since the σ -algebra C is T -invariant, we have T M = σ (C ∪ σ (g ◦ T k , k ≤ −1)). This implies E[m | T M] = E[g f | T M] = f · E[g | T M].
(3.23)
Since g is centered and independent of T M, Proposition 3.3 is proved. It remains to prove that the process
pl (n −1/2 Sn (m))n≥1
is not tight in
o H1/2−1/ p [0, 1].
Proposition 3.4. Under conditions (3.2)–(3.4), there exists an integer l0 such that for l ≥ l0 1 |Su+v (g fl ) − Su (g fl )| 1 Pl := µ max ≥1 ≥ . (3.24) 1/ p 1/2−1/ p 1≤u≤nl −kl n 16 v l 1≤v≤k l
Proof. Let us fix an integer l ≥ 1. Assume that ω ∈ T −s Cl , where kl ≤ s ≤ nl − 1. Since T u ω belongs to T −(s−u) Cl we have for s − nl ≤ u ≤ s nl 1/ p 1 , if s − kl,Jl < u ≤ s; L l kl,Jl ( fl ◦ T u )(ω) = 1 (3.25) nl 1/ p , if s − kl, j−1 < u ≤ s − kl, j , and 1 ≤ j ≤ Jl ; L kl, j l 0, if s − nl ≤ u < s − kl . As a consequence, T
−s
Cl ∩
1
max
Ss−k
1/ p 1≤ j≤J l
nl
= T −s Cl ∩
max 1≤ j≤Jl
l, j−1 +1
(g fl ) − Ss−kl, j (g fl )
(kl, j−1 − 1 − kl, j )1/2−1/ p
≥1
l, j−1 +1 (g) − Ss−kl, j (g)
Ss−k
1/ p
(kl, j − 1)1/2−1/ p kl, j−1
≥ Ll
.
(3.26)
Since for kl + 1 ≤ s ≤ nl − kl and 1 ≤ j ≤ Jl , we have 1 ≤ s − kl, j ≤ nl − kl and 1 ≤ kl, j−1 − 1 − kl, j ≤ kl , the inequality Ss−k l, j−1 +1 (g f l ) − Ss−kl, j (g f l ) −s χ (T Cl ) · max 1≤ j≤Jl (kl, j−1 − 1 − kl, j )1/2−1/ p ≤ χ (T −s (Cl ))
max
1≤u≤nl −kl 1≤v≤kl
|Su+v (g fl ) − Su (g fl )| v 1/2−1/ p
(3.27)
k −1 takes place and since the sets (T −s Cl )ns=0 are pairwise disjoint, we obtain the lower bound
Pl ≥
nl −2kl
µ T
−(s+kl )
(Cl ) ∩
s=1
max
Ss+k −k l
l, j−1 +1
(g fl ) − Ss+kl −kl, j (g fl )
(kl, j−1 − 1 − kl, j )1/2−1/ p
1≤ j≤Jl
.
≥1
(3.28)
Using the fact that T is measure-preserving, this becomes Pl ≥ (nl − 2kl ) · µ T
−kl
(Cl ) ∩
max
1≤ j≤Jl
Sk −k l
l, j−1 −1
(g fl ) − Skl −kl, j (g fl )
(kl, j−1 − 1 − kl, j )1/2−1/ p
≥1
,
(3.29)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
and plugging (3.26) in the previous estimate, we get Sk −k (g) − S (g) k −k −1 l l, j l l, j−1 . (3.30) Pl ≥ (nl − 2kl )µ T −kl (Cl ) ∩ max ≥ L l 1/ p 1≤ j≤Jl (kl, j − 1)1/2−1/ p k l, j−1
The sets
max1≤ j≤Jl
Skl −kl, j−1 +1 (g)−Skl −kl, j (g) 1/ p
(kl, j −1)1/2−1/ p kl, j−1
≥ Ll
and T −kl Cl belong to the independent
sub-σ -algebras B and C respectively, hence using the fact that the sequences (g ◦ T i )i≥0 and (g ◦ T −i )i≥0 are identically distributed, we obtain Sk (g) − S (g) kl, j l, j−1 −1 Pl ≥ (nl − 2kl )µ (Cl ) µ max ≥ Ll . (3.31) 1/ p 1≤ j≤Jl (kl, j − 1)1/2−1/ p k l, j−1
By construction, we have nl · µ(Cl ) = µ kl 1 1−2 µ max Pl ≥ 1≤ j≤Jl 2 nl
nl −1 i=0
T −i Cl > 1/2, hence
l, j−1 −1 (g) − Skl, j (g) 1/ p (kl, j − 1)1/2−1/ p kl, j−1
Sk
It remains to find a lower bound for Sk (g) − S (g) k −1 l, j l, j−1 ≥ L . Pl′ := µ max l 1≤ j≤Jl (kl, j − 1)1/2−1/ p k 1/ p
≥ Ll
.
(3.32)
(3.33)
l, j−1
Let us define the set Sk (g) − S (g) kl, j l, j−1 −1 ≥ L . E j := l (kl, j − 1)1/2−1/ p k 1/ p
(3.34)
l, j−1
Since the sequence (g ◦ T i )i∈Z is independent, the family (E j )1≤ j≤Jl is independent, hence Pl′ ≥ 1 −
Jl
(1 − µ(E j )).
(3.35)
j=1
We define the quantity
Ll c j := µ |N | ≥ ∥g∥2
kl, j−1 kl, j − 1
1/ p (3.36)
(we recall that N denotes a standard normally distributed random variable). By the Berry–Esseen theorem, we have for each j ∈ {1, . . . , Jl }, √ 1 2 −Il /2 1 µ(E j ) − c j ≤ ≤ 2 . (3.37) ∥g∥32 (kl, j−1 − 1)1/2 ∥g∥32
D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
301
N , Plugging the estimate (3.37) into (3.35) and noticing that for an integer N and (an )n=1 N (bn )n=1 two families of numbers in the unit interval, N N N |an − bn | , (3.38) an − bn ≤ n=1 n=1 n=1
we obtain Pl′ ≥ 1 −
Jl
(1 − µ(E j )) +
Jl
j=1
≥ 1−
Jl
j=1
(1 − c j ) −
j=1 Jl
(1 − c j ) −
Jl
(1 − c j )
(3.39)
j=1
Jl µ(E j ) − c j
(3.40)
j=1
√ 2
2−Il /2 .
(3.41)
J (1 − c j ) ≥ 1 − max 1 − c j l
(3.42)
≥ 1−
(1 − c j ) − Jl
j=1
∥g∥32
Notice that 1−
Jl
1≤ j≤Jl
j=1
and since (Il )≥ is increasing and I1 ≥ 1, we have kl, j−1 2 2 ≤ = ≤4 −1 kl, j − 1 1 − 2−Il 1 − kl, j Ll it follows by (3.36) that c j ≥ µ |N | ≥ 41/ p ∥g∥ for 1 ≤ j ≤ Jl . We thus have
(3.43)
2
Pl′
≥ 1 − 1 − µ |N | ≥ 4
1/ p
Ll ∥g∥2
Jl − Jl
√ 2 ∥g∥32
2−Il /2 .
(3.44)
Using the elementary inequality n(n − 1) 2 t 2 valid for a positive integer n and t ∈ [0, 1], we obtain √ 2 Jl2 2 −Il /2 ′ 1/ p L l 1/ p L l Pl ≥ Jl µ |N | ≥ 4 µ |N | ≥ 4 2 . − − Jl ∥g∥2 ∥g∥2 2 ∥g∥32 1 − (1 − t)n ≥ nt −
By conditions (3.3) and (3.2), there exists an integer l0′ such that if l ≥ l0′ , then 1 Sk (g) − S (g) kl, j l, j−1 −1 µ max ≥ L ≥ . l 1≤ j≤Jl (kl, j − 1)1/2−1/ p k 1/ p 4
(3.45)
(3.46)
(3.47)
l, j−1
Combining (3.32) with (3.47), we obtain for l ≥ l0′ 1 kl 1−2 . Pl ≥ 8 nl
(3.48)
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By condition (3.4), we thus get that Pl ≥ 1/16 for l ≥ l0 , where l0 ≥ l0′ and kl /nl ≤ 1/4 if l ≥ l0 . This concludes the proof of Proposition 3.4. Proposition 3.5. Under conditions (3.1)–(3.5), we have for l large enough 1 |Su+v (m) − Su (m)| 1 1 µ max ≥ ≥ . 1/ p 1/2−1/ p 1≤u≤nl −kl n 2 32 v l
(3.49)
1≤v≤kl
Since the H¨older modulus of continuity of a piecewise linear function is reached at vertices, we derive the following corollary. Corollary 3.6. If l ≥ l0 , then 1 1 1 pl kl ≥ ≥ . µ ω1/2−1/ p √ Snl (m), nl nl 2 32
(3.50)
Therefore, for each positive δ, we have 1 pl 1 1 lim sup µ ω1/2−1/ p √ Sn (m), δ ≥ ≥ , 2 32 n n→∞
(3.51)
pl
o and the process (n −1/2 Sn (m))n≥1 is not tight in H1/2−1/ p [0, 1].
Proof of Let l0 be the integer given by Proposition 3.4 and let l ≥ l0 . We define Proposition 3.5. +∞ ′′ m l′ := l−1 i=1 g f i and m l := i=l+1 g f i . We define for i ≥ 1, Ml,i :=
1 1/ p nl
max
1≤u≤nl −kl 1≤v≤kl
|Su+v (g f i ) − Su (g f i )| . v 1/2−1/ p
(3.52)
Let i be an integer such that i < l. Notice that for 1 ≤ u ≤ nl − kl and v ≤ kl , we have |Su+v (g f i ) − Su (g f i )| = U u (|Sv (g f i )|),
(3.53)
where U (h)(ω) = h (T (ω)) and since kl n i 1/ p |Sv (g f i )| ≤ v ∥g f i ∥∞ ≤ , L i 2 Ii the estimate kl n i 1/ p Ml,i ≤ 1/ p 2 Ii Li n
(3.54)
(3.55)
l
holds. Since 1 1/ p
nl
max
1≤u≤nl −kl 1≤v≤kl
Su+v (m ′ ) − Su (m ′ ) l v 1/2−1/ p
l
≤
l−1
Ml,i ,
(3.56)
i=1
we have by (3.55), 1 1/ p nl
max
1≤u≤nl −kl 1≤v≤kl
Su+v (m ′ ) − Su (m ′ ) l v 1/2−1/ p
l
≤
l−1
kl
1/ p i=1 L i n l
n 1/ p i
2 Ii
.
(3.57)
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By (3.5), the following bound takes place: Su+v (m ′ ) − Su (m ′ ) 1 1 l l max ≤ . 1/ p 1≤u≤nl −kl 1/2−1/ p 2 v nl 1≤v≤k
(3.58)
l
The following set inclusions hold: 1 Su+v (m ′′ ) − Su (m ′′ ) l l = ̸ 0 ⊂ Ml,i ̸= 0 max 1/ p 1/2−1/ p 1≤u≤nl −kl n v i>l l
(3.59)
1≤v≤kl
⊂
nl u U (g f i ) ̸= 0 .
(3.60)
i>l u=1
We thus have 1 µ n 1/ p l
max
Su+v (m ′′ ) − Su (m ′′ ) l
l
v 1/2−1/ p
1≤u≤nl −kl 1≤v≤kl
̸= 0 ≤ nl · µ {g f i ̸= 0} i>l ≤ nl
(3.61)
µ { f i ̸= 0}
(3.62)
(ki + 1)µ(Ci )
(3.63)
i>l
= nl
i>l
≤ 2nl
ki n i>l i
and by (3.4), it follows that 1 Su+v (m ′′ ) − Su (m ′′ ) 1 l l = ̸ 0 ≤ µ max . 1/2−1/ p 1/ p 1≤u≤nl −kl 32 n v l
(3.64)
(3.65)
1≤v≤kl
Accounting (3.58), we thus have 1 |Su+v (m) − Su (m)| 1 max ≥ µ n 1/ p 1≤u≤nl −kl 2 v 1/2−1/ p l 1≤v≤kl 1 Su+v (g fl + m ′′ ) − Su (g fl + m ′′ ) l l ≥µ ≥ 1 max n 1/ p 1≤u≤nl −kl v 1/2−1/ p l 1≤v≤kl 1 |Su+v (g fl ) − Su (g fl )| ≥µ max ≥ 1 n 1/ p 1≤u≤nl −kl v 1/2−1/ p l
1≤v≤kl
1 −µ n 1/ p l
max
1≤u≤nl −kl 1≤v≤kl
Su+v (m ′′ ) − Su (m ′′ ) l
v 1/2−1/ p
l
̸= 0 ,
(3.66)
hence combining Proposition 3.4 with (3.65), we obtain the conclusion of Proposition 3.5. Theorem 2.1 follows from Corollary 3.6 and Propositions 3.2 and 3.3.
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3.2. Proof of Theorem 2.2 and Proposition 2.3 Proof of Proposition 2.3. Let us fix a positive t. Recall the equivalence between ∥x∥α and seq ∥x∥α and Notation 1.1. By Remark 1.2, we have to show that for some constant C depending only on p and each integer n ≥ 1, pl pl p α j −1/2 P(n, t) := t µ sup 2 n max Sn (m, rk+1, j ) − Sn (m, rk, j ) > t 0≤k<2 j
j≥1
p/2 p ≤ C ∥m∥ p,∞ + E E[m 2 | T M] .
(3.67)
In the proof, we shall denote by C p a constant depending only on p which may change from line to line. We define pl pl α j −1/2 P1 (n, t) := µ sup 2 n max Sn (m, rk+1, j ) − Sn (m, rk, j ) > t , and (3.68) 0≤k<2 j
1≤ j≤log n
P2 (n, t) := µ
α j −1/2
sup 2 n
pl pl max Sn (m, rk+1, j ) − Sn (m, rk, j ) > t ,
0≤k<2 j
j>log n
(3.69)
hence P(n, t) ≤ t p P1 (n, t/2) + t p P2 (n, t/2).
(3.70)
We estimate P2 (n, t). For j > log n, we have the inequality rk+1, j − rk, j = (k + 1)2− j − k2− j = 2− j < 1/n,
(3.71)
hence if rk, j belongs to the interval [l/n, (l + 1)/n) for some l ∈ {0, . . . , n − 1}, then • either rk+1, j ∈ [l/n, (l + 1)/n), and in this case, pl pl Sn (m, rk+1, j ) − Sn (m, rk, j ) = m ◦ T l+1 2− j n ≤ 2− j n max U l (m) ;
(3.72)
• or rk+1, j belongs to the interval [(l + 1)/n, (l + 2)/n). The estimates pl pl pl pl Sn (m, rk+1, j ) − Sn (m, rk, j ) ≤ Sn (m, rk+1, j ) − Sn (m, (l + 1)/n) pl pl + Sn (m, (l + 1)/n) − Sn (m, rk, j ) ≤ 21− j n max U l (m)
(3.73)
1≤l≤n
1≤l≤n
hold. Considering these two cases, we obtain P2 (n, t) ≤ µ
αj
1− j −1/2
sup 2 n2
n
j>log n
max U l (m) > t
1≤l≤n
(3.74)
≤ µ 2n α−1/2 max U l (m) > t
(3.75)
≤ nµ 2n −1/ p |m| > t
(3.76)
1≤l≤n
≤
2p sup x p µ {|m| > x} . t p x>0
(3.77)
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305
Therefore, establishing inequality (3.67) reduces to find a constant C depending only on p such that p/2 p 2 p sup sup t P1 (n, t) ≤ C ∥m∥ p,∞ + E E[m | T M] . (3.78) n
t
We define u k, j := [nrk, j ] for k < 2 j and j ≥ 1 (see Notation 1.1). Notice that the inequalities pl Su k, j (m) − Sn (m, rk, j ) ≤ U u k, j +1 (m) and pl Sn (m, rk+1, j ) − Su k+1, j (m) ≤ U u k+1, j +1 (m)
(3.79) (3.80)
take place because if j ≤ log n, then u k, j ≤ nrk, j ≤ u k, j + 1 ≤ u k+1, j ≤ nrk+1, j ≤ u k+1, j + 1. Therefore, P1 (n, t) ≤ P1,1 (n, t) + P1,2 (n, t), where P1,1 (n, t) := µ max 2α j n −1/2 max Su k+1, j (m) − Su k, j (m) > t/2 , 0≤k<2 j
1≤ j≤log n
P1,2 (n, t) := µ
max
α j −1/2
1≤ j≤log n
2 n
l max U (m) > t/4 .
1≤l≤n
(3.81)
(3.82) (3.83)
Notice that
P1,2 (n, t) ≤ µ n
α−1/2
l max U (m) > t/4
1≤l≤n
(3.84)
≤ nµ |m| > n 1/ p t/4
(3.85)
≤ 4 p t − p sup x p µ {|m| > x} ,
(3.86)
x≥0
hence (3.78) will follow from the existence of a constant C depending only on p such that p/2 p sup sup t p P1,1 (n, t) ≤ C ∥m∥ p,∞ + E E[m 2 | T M] . (3.87) n
t
We estimate P1,1 (n, t) in the following way: P1,1 (n, t) ≤
log n j=1
2 j max µ Su k+1, j (m) − Su k, j (m) > tn 1/2 2−1−α j . 0≤k<2 j
We define for 1 ≤ j ≤ log n and 0 ≤ k < 2 j the quantity P(n, j, k, t) := µ Su k+1, j (m) − Su k, j (m) > tn 1/2 2−1−α j . If ( f ◦ T j ) j≥0 is a strictly stationary sequence, we define 1/2 n Q f,n (u) := µ max f ◦ T j > u + µ U i E[ f 2 | T M] >u . i=1 1≤ j≤n
(3.88)
(3.89)
(3.90)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
The following inequality is Theorem 1 of [10]. It allows us to express the tail function of a martingale by that of the increments and the quadratic variance. Theorem 3.7. Let m be an M-measurable function such that E[m | T M] = 0. Then for each positive y and each integer n, 1 µ {|Sn (m)| > y} ≤ c(q, η) Q m,n (εq u · y)u q−1 du, (3.91) 0
where q > 0, η > 0, εq := η/q and c(q, η) := q exp(3ηeη+1 − η − 1)/η. We shall use (3.91) with q := p + 1, η = 1 and y := n 1/2 2−1−α j t in order to estimate P(n, j, k, t): 1 i P(n, j, k, t) ≤ C p µ max U (m) > n 1/2 2−1−α j tuε p+1 u p du 1≤i≤u k+1, j −u k, j
0
1/2 1 k+1, j u + Cp U i (E[m 2 | T M]) > n 1/2 2−1−α j utε p+1 u p du. (3.92) µ 0 i=u k, j +1 Exploiting the inequality u k+1, j − u k, j ≤ 2n2− j , we get from the previous bound 1 i 1/2 −1−α j P(n, j, k, t) ≤ C p µ max U (m) > n 2 tuε p+1 u p du 1≤i≤2n2− j
0
1/2 −j 1 2n2 i 2 1/2 −1−α j U (E[m | T M]) >n 2 tuε p+1 u p du. + Cp µ 0 i=1 We define for j ≤ log n, t ≥ 0 and u ∈ (0, 1), P ′ (n, j, t, u) := µ max U i (m) > n 1/2 2−1−α j tuε p+1 , 1≤i≤2n2− j
and
1/2 −j 2n2 ′′ i 2 1/2 −1−α j P (n, j, t, u) := µ U (E[m | T M]) >n 2 tuε p+1 . i=1
(3.93)
(3.94)
(3.95)
Using the fact that the random variables U i (m), 1 ≤ i ≤ 2n2− j are identically distributed, we derive the bound P ′ (n, j, t, u) ≤ 2n2− j µ |m| > n 1/2 2−1−α j tuε p+1 , (3.96) hence p
P ′ (n, j, t, u) ≤ 2n2− j (n 1/2 2−1−α j tuε p+1 )− p ∥m∥ p,∞ = 2 p+1 ε p+1 n 1− p/2 2 j (−1+ pα) t − p u − p ∥m∥ p,∞ . −p
p
(3.97)
Since α and p are linked by the relationship 1/2 − 1/ p = α, we have pα = p/2 − 1 hence 1 p P ′ (n, j, t, u)u p du ≤ C p t − p n 1− p/2 2 j ( p/2−2) ∥m∥ p,∞ . (3.98) 0
D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
307
Notice the following set of equalities: 1/2 −j 2n2 i 2 1/2 −1−α j U (E[m | T M]) > ε p+1 un 2 t i=1 −j 1 2n2 i 2 −3 2 2 2 j/ p 2 U (E[m | T M]) > 2 ε u 2 t = p+1 2n2− j i=1
(3.99)
and that n2− j ≥ 1 (because j ≤ log n), hence 1/2 −j 2n2 i 2 1/2 −1−α j U (E[m | T M]) > ε p+1 un 2 t i=1 ⊆
N ≥2
N 1 i 2 −3 2 2 2 j/ p 2 U (E[m | T M]) > 2 ε p+1 u 2 t , N i=1
(3.100)
from which it follows 1/2 −j 2n2 i 2 1/2 −1−α j U (E[m | T M]) > ε p+1 un 2 t µ i=1
N 1 i 2 −3 2 2 2 j/ p 2 ≤ µ sup U (E[m | T M]) > 2 ε p+1 u 2 t . N ≥2 N i=1
(3.101)
Combining (3.98) and (3.101), we obtain max P(n, j, k, t) ≤ C p t − p n 1− p/2 2 j ( p/2−2) ∥m∥ p,∞ p
0≤k<2 j
1
+ Cp 0
N 1 µ sup U i (E[m 2 | T M]) > 2−3 ε 2p+1 u 2 22 j/ p t 2 u p du, N ≥2 N i=1
(3.102)
hence by (3.88) and (3.89), p
P1,1 (n, t) ≤ C p t − p ∥m∥ p,∞
log n
2 j 2 j ( p/2−2) n 1− p/2
j=1
+ Cp 0
N 1 2 j µ sup U i (E[m 2 | T M]) > 2−3 ε 2p+1 u 2 22 j/ p t 2 u p du. (3.103) N N ≥2 j=1 i=1
1 log n
From the elementary bounds log n
2 j ( p/2−1) n 1− p/2 ≤ (1 − 21− p/2 )−1
(3.104)
j=1
j≥1
2 j µ |g| > 22 j/ p ≤ 2E |g| p/2 ,
for any non-negative function g,
(3.105)
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with −2 g := 23 ε −2 sup p+1 u
N ≥2
N 1 U i (E[m 2 | T M]), N i=1
u ∈ (0, 1)
(3.106)
we obtain P1,1 (n, t) ≤ C p t
−p
p ∥m∥ p,∞
+ C pt
−p
p/2 N 1 i 2 U (E[m | T M]) . sup N ≥2 N i=1
(3.107)
p/2
L1 -L∞
As the Koopman operator U is an contraction, Theorem 1 of [12] gives the existence of a constant A p such that for each h ∈ L p/2 , N 1 j U (h) ≤ A p ∥h∥ p/2 . (3.108) sup N ≥1 N j=1 p/2
Applying (3.108) with h := E[m 2 | T M], we get by (3.107) p/2 p , P1,1 (n, t) ≤ C p t − p ∥m∥ p,∞ + C p t − p E E[m 2 | T M] which establishes (3.78). This concludes the proof of Proposition 2.3.
(3.109)
Proof of Theorem 2.2. The convergence of finite dimensional distributions can be proved using theorem of [1]. Its proof works for filtrations of the form (T −i M)i≥0 where T M ⊂ M and also in the non-ergodic setting by considering the ergodic components. We deduce tightness in Theorem 2.2 from Proposition 2.3 by a truncation argument. For a fixed R, we define m R := mχ {|m| ≤ R} − E[mχ {|m| ≤ R} | T M] m ′R
and
:= mχ {|m| > R} − E[mχ {|m| > R} | T M]. T i )i≥0
(3.111)
In this way, the sequences (m R ◦ and ◦ are martingale difference sequences and m = m R + m ′R . Since |m R | ≤ 2R and (m R ◦ T i )i≥0 is a martingale difference sequence, the sequence pl o (n −1/2 Sn (m R ))n≥1 is tight in H1/2−1/ p [0, 1]. Consequently, for each positive ε, the following convergence takes place: pl αj 1/2 lim lim sup µ sup 2 max λr Sn (m R ) > εn = 0. (3.112) J →∞ n→∞
j≥J
(m ′R
(3.110)
T i )i≥0
r ∈D j
Using Proposition 2.3, we derive the following bound, valid for each ε and each R, pl lim lim sup µ sup 2α j max λr Sn (m) > εn 1/2 J →∞ n→∞
j≥J
r ∈D j
≤ C pε
−p
sup t µ {|m| χ {|m| > R} > t} + sup t µ {E[|m| χ {|m| > R} | T M] > t} p
t≥0
+ ε− p C p E
p
t≥0
E[m 2 χ {|m| > R} | T M]
p/2
.
The first term is supt≥R t p µ {|m| > t}, which goes to 0 as R goes to infinity.
(3.113)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
The second term can be bounded by supt≥R t p µ {E[|m| | T M] > t}. Indeed, if t ≥ R, we use the inclusion {E[|m| χ {|m| > R} | T M] > t} ⊂ {E[|m| | T M] > t} ,
(3.114)
and if t < R, then accounting the fact that the random variable E[|m| χ {|m| > R} | T M] is greater than R, we get E[|m| χ {|m| > R} | T M] = E[|m| χ {|m| > R} | T M]χ {E[|m| | T M] > R} ≤ E[|m| | T M]χ {E[|m| | T M] > R} ,
(3.115)
from which it follows that t p µ {E[|m| χ {|m| > R} | T M] > t} ≤ R p µ {E[|m| | T M] > R} .
(3.116)
By Lemma 1.4, the convergence lim sup t p µ {E[|m| | T M] > t} = 0
(3.117)
R→∞ t≥R
takes place. The third term of (3.113) converges to 0 as R goes to infinity by monotone convergence. This concludes the proof of tightness in Theorem 2.2. 3.3. Proof of Theorem 2.6 By (2.7), the equality f = i≥0 Pi ( f ) holds almost surely. For a fixed integer K , we define K f K := i=0 Pi ( f ). Then f K satisfies the conditions of Corollary 2.5. Indeed, we have the equalities Pi ( f ) − P0 (U i f ) = E[ f | T i M] − E[U i f | M] − E[ f | T i+1 M] + E[U i f | T M] = (I − U )E[ f | T M] − (I − U )E[ f | T i
i
i
(3.118) i+1
M]
(3.119) k and the latter term can be expressed as a coboundary noticing that (I −U i ) = (I −U ) i−1 k=0 U . K Since Pi ( f ) belongs to the L p space, we may write f K − i=0 P0 (U i f ) as (I − U )g K where K p i g K belongs to the L space. Defining m K := i=0 P0 (U ( f )), the sequence (m K ◦ T i )i≥0 is a martingale difference sequence hence for each positive ε, pl lim lim sup µ sup 2α j max λr Sn ( f K ) > εn 1/2 = 0. (3.120) J →∞ n→∞
j≥J
r ∈D j
Now, we have to show that the convergence in (3.120) holds if f K is replaced by f − f K . To this aim, we use the inclusion pl αj 1/2 sup 2 max λr Sn ( f − f K ) > εn j≥J
r ∈D j
⊆ sup 2 j≥1
αj
pl 1/2 max λr Sn ( f − f K ) > εn ,
r ∈D j
(3.121)
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hence µ sup 2
pl max λr Sn ( f − f K ) > εn 1/2
αj
r ∈D j
j≥J
pl −p 1 ≤ ε √ Sn ( f − f K ) o n H
1/2−1/ p
p
(3.122)
p,∞
p 1 pl −p = ε √ Sn Pi ( f ) n o i≥K +1 H1/2−1/ p
,
(3.123)
p,∞
from which it follows that pl 1/2 αj µ sup 2 max λr Sn ( f − f K ) > εn r ∈D j
j≥J
1 pl −p ≤ε S ( f )) √ (P i n o n H i≥K +1
1/2−1/ p
p .
(3.124)
p,∞
(U l (Pi ( f )))l≥1
Notice that for a fixed i, the sequence is a martingale difference sequence (with respect to the filtration (T −i−l M)l≥0 ). Therefore, by Proposition 2.3, we obtain 1 pl ≤ C p ∥Pi ( f )∥ p . (3.125) √ Sn (Pi ( f )) o n H 1/2−1/ p
p,∞
Plugging this estimate into (3.124), we obtain that for some constant C depending only on p, p pl −p αj 1/2 ∥Pi ( f )∥ p . (3.126) ≤ Cε µ sup 2 max λr Sn ( f − f K ) > εn r ∈D j
j≥J
i≥K +1
Combining (3.120) and (3.126), we obtain for each K : pl 1/2 αj lim lim sup µ sup 2 max λr Sn ( f ) > n ε J →∞ n→∞
j≥J
r ∈D j
p
≤ Cε
−p
∥Pi ( f )∥ p
.
(3.127)
i≥K +1
Since K is arbitrary, we conclude the proof of Theorem 2.6 thanks to assumption (2.8). Acknowledgments The author is grateful to the referee for many comments which improved the readability of the paper. The author would like to thank Dalibor Voln´y for many useful discussions which lead to the counter-example in Theorem 2.1, and also Alfredas Raˇckauskas and Charles Suquet for their encouragement.
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References [1] Patrick Billingsley, Probability and Measure, third ed., in: Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995, A Wiley-Interscience Publication. MR 1324786 (95k:60001). [2] Z. Ciesielski, On the isomorphisms of the spaces Hα and m, Bull. Acad. Pol. Sci. S´er. Sci. Math. Astron. Phys. 8 (1960) 217–222. MR 0132389 (24 #A2234). [3] J´erˆome Dedecker, Florence Merlev`ede, Dalibor Voln´y, On the weak invariance principle for non-adapted sequences under projective criteria, J. Theoret. Probab. 20 (4) (2007) 971–1004. MR 2359065 (2008g:60088). [4] Monroe D. Donsker, An invariance principle for certain probability limit theorems, Mem. Amer. Math. Soc. 1951 (6) (1951) 12. MR 0040613 (12,723a). [5] E.J. Hannan, Central limit theorems for time series regression, Z. Wahrscheinlichkeitstheor. Verwandte Geb. 26 (1973) 157–170. MR 0331683 (48 #10015). [6] G´erard Kerkyacharian, Bernard Roynette, Une d´emonstration simple des th´eor`emes de Kolmogorov, Donsker et Ito-Nisio, C. R. Acad. Sci. Paris S´er. I Math. 312 (11) (1991) 877–882. MR 1108512 (92g:60009). [7] Emmanuel Lesigne, Dalibor Voln´y, Large deviations for martingales, Stochastic Process. Appl. 96 (1) (2001) 143–159. MR 1856684 (2002k:60080). [8] J. Markeviˇci¯ut˙e, A. Raˇckauskas, Ch. Suquet, Functional central limit theorems for sums of nearly nonstationary processes, Lith. Math. J. 52 (3) (2012) 282–296. [9] Florence Merlev`ede, Magda Peligrad, Sergey Utev, Recent advances in invariance principles for stationary sequences, Probab. Surv. 3 (2006) 1–36. MR 2206313 (2007a:60025). [10] S.V. Nagaev, On probability and moment inequalities for supermartingales and martingales, in: Proceedings of the Eighth Vilnius Conference on Probability Theory and Mathematical Statistics, Part II (2002), vol. 79, 2003, pp. 35–46. MR 2021875 (2005f:60098). [11] Alfredas Raˇckauskas, Charles Suquet, Necessary and sufficient condition for the Lamperti invariance principle, Teor. ˘Imov¯ır. Mat. Stat. (68) (2003) 115–124. MR 2000642 (2004g:60050). [12] E.M. Stein, On the maximal ergodic theorem, Proc. Nat. Acad. Sci. USA 47 (1961) 1894–1897. MR 0131517 (24 #A1367). [13] Ch. Suquet, Tightness in Schauder decomposable Banach spaces, in: Proceedings of the St. Petersburg Mathematical Society, Vol. V (Providence, RI), in: Amer. Math. Soc. Transl. Ser. 2, vol. 193, Amer. Math. Soc., 1999, pp. 201–224. MR 1736910 (2000k:60009).
ScienceDirect Stochastic Processes and their Applications 126 (2016) 290–311 www.elsevier.com/locate/spa
Holderian weak invariance principle under a Hannan type condition Davide Giraudo ´ Universit´e de Rouen, LMRS, Avenue de l’Universit´e, BP 12 76801 Saint-Etienne-du-Rouvray cedex, France Received 9 March 2015; received in revised form 2 September 2015; accepted 2 September 2015 Available online 10 September 2015
Abstract We investigate the invariance principle in H¨older spaces for strictly stationary martingale difference sequences. In particular, we show that the sufficient condition on the tail in the i.i.d. case does not extend to stationary ergodic martingale differences. We provide a sufficient condition on the conditional variance which guarantee the invariance principle in H¨older spaces. We then deduce a condition in the spirit of Hannan one. c 2015 Elsevier B.V. All rights reserved. ⃝
MSC: 60F05; 60F17 Keywords: Invariance principle; Martingales; Strictly stationary process
1. Introduction One of the main problems in probability theory is the understanding of the asymptotic behavi ior of Birkhoff sums Sn ( f ) := n−1 j=0 f ◦ T , where (Ω , F, µ, T ) is a dynamical system and f a map from Ω to the real line. One can consider random functions constructed from the Birkhoff sums pl
Sn ( f, t) := S[nt] ( f ) + (nt − [nt]) f ◦ T [nt]+1 ,
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spa.2015.09.001 c 2015 Elsevier B.V. All rights reserved. 0304-4149/⃝
t ∈ [0, 1]
(1.1)
D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
pl and investigate the asymptotic behavior of the sequence Sn ( f, t)
291
seen as an element of
n≥1 pl (n −1/2 (E( f 2 ))−1/2 Sn ( f ))n≥1
a function space. Donsker showed (cf. [4]) that the sequence converges in distribution in the space of continuous functions on the unit interval to a standard Brownian motion W when the sequence ( f ◦ T i )i≥0 is i.i.d. and zero mean. Then an intensive research has then been performed to extend this result to stationary weakly dependent sequences. We refer the reader to [9] for the main theorems in this direction. pl Our purpose is to investigate the weak convergence of the sequence (n −1/2 Sn ( f ))n≥1 in i H¨older spaces when ( f ◦ T )i≥0 is a strictly stationary sequence. A classical method for showing a limit theorem is to use a martingale approximation, which allows to deduce the corresponding result if it holds for martingale difference sequences provided that the approximation is good enough. To the best of our knowledge, no result about the invariance principle in H¨older space for stationary martingale difference sequences is known. 1.1. The H¨older spaces It is well known that standard Brownian motion’s paths are almost surely H¨older regular of exponent α for each α ∈ (0, 1/2), hence it is natural to consider the random function defined in (1.1) as an element of Hα [0, 1] and try to establish its weak convergence to a standard Brownian motion in this function space. Before stating the results in this direction, let us define for α ∈ (0, 1) the H¨older space Hα [0, 1] of functions x: [0, 1] → R such that sups̸=t |x(s) − x(t)| / |s − t|α is finite. The analogue of the continuity modulus in C[0, 1] is wα , defined by wα (x, δ) =
sup 0<|t−s|<δ
|x(t) − x(s)| . |t − s|α
(1.2)
We then define Hα0 [0, 1] by Hα0 [0, 1] := {x ∈ Hα [0, 1], limδ→0 wα (x, δ) = 0}. We shall essentially work with the space Hα0 [0, 1] which, endowed with ∥x∥α := wα (x, 1) + |x(0)|, is a separable Banach space (while Hα [0, 1] is not separable). Since the canonical embedding ι: Hαo [0, 1] → Hα [0, 1] is continuous, each convergence in distribution in Hαo [0, 1] also takes place in Hα [0, 1]. Let us denote by D j the set of dyadic numbers in [0, 1] of level j, that is, D0 := {0, 1} , D j := (2l − 1)2− j ; 1 ≤ l ≤ 2 j−1 , j ≥ 1. (1.3) If r ∈ D j for some j ≥ 0, we define r + := r + 2− j and r − := r − 2− j . For r ∈ D j , j ≥ 1, let Λr be the function whose graph is the polygonal path joining the points (0, 0), (r − , 0), (r, 1), (r + , 0) and (1, 0). We can decompose each x ∈ C[0, 1] as x=
λr (x)Λr =
r ∈D
+∞
λr (x)Λr ,
(1.4)
j=0 r ∈D j
and the convergence is uniform on [0, 1]. The coefficients λr (x) are given by x(r + ) + x(r − ) , 2 and λ0 (x) = x(0), λ1 (x) = x(1). λr (x) = x(r ) −
r ∈ D j , j ≥ 1,
(1.5)
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Ciesielski proved (cf. [2]) that {Λr ; r ∈ D} is a Schauder basis of Hαo [0, 1] and the norms ∥·∥α and the sequential norm defined by seq
∥x∥α := sup 2 jα max |λr (x)| , j≥0
(1.6)
r ∈D j
are equivalent. Considering the sequential norm, we can show (see Theorem 3 in [13]) that a sequence (ξn )n≥1 of random elements of Hαo vanishing at 0 is tight if and only if for each positive ε, lim lim sup µ sup 2 jα max |λr (ξn )| > ε = 0.
J →∞ n→∞
j≥J
(1.7)
r ∈D j
Notation 1.1. In the sequel, we will denote rk, j := k2− j and u k, j := [nrk, j ] (or rk and u k for short). Notice that u k+1, j − u k, j = [nrk, j + n2− j ] − u k, j ≤ 2n2− j if j ≤ log n, where log n denotes the binary logarithm of n and for a real number x, [x] is the unique integer for which [x] ≤ x < [x] + 1. Remark 1.2. Since for each x ∈ H1/2−1/ p [0, 1], each j ≥ 1 and each r ∈ D j , + x(r ) − x(r ) x(r ) − x(r − ) |λr (x)| ≤ + 2 2 + ≤ max x(r ) − x(r ) , x(r ) − x(r − ) ,
(1.8)
pl
for a function f , the sequential norm of n −1/2 Sn ( f ) does not exceed pl pl sup 2α j n −1/2 max Sn ( f, rk+1, j ) − Sn ( f, rk, j ) . 0≤k<2 j
j≥1
(1.9)
Now, we state the result obtained by Raˇckauskas and Suquet in [11]. Theorem 1.3. Let p > 2 and let ( f ◦ T j ) j≥0 be an i.i.d. centered sequence with unit variance. Then the condition lim t p µ {| f | > t} = 0
(1.10)
t→∞
pl
is equivalent to the weak convergence of the sequence (n −1/2 Sn ( f ))n≥1 to a standard Brownian o motion in the space H1/2−1/ p [0, 1]. 1.2. Some facts about the L p,∞ spaces In the rest of the paper, χ denotes the indicator function. Let p > 2. We define the L p,∞ space as the collection of functions f : Ω → R such that the quantity p
∥ f ∥ p,∞ := sup t p µ {| f | > t} < ∞.
(1.11)
t>0
This quantity is denoted like a norm, while it is not a norm (the triangle inequality may fail, for example if X = [0, 1] endowed with the Lebesgue measure, f (x) := x −1/ p and g(x) :=
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293
f (1 − x); in this case ∥ f + g∥ p,∞ ≥ 21+1/ p but ∥ f ∥ p,∞ + ∥g∥ p,∞ = 2). However, there exists a constant κ p such that for each f , ∥ f ∥ p,∞ ≤
sup
µ(A)−1+1/ p E[| f | χ A ] ≤ κ p ∥ f ∥ p,∞
(1.12)
A:µ(A)>0
and N p ( f ) := sup A:µ(A)>0 µ(A)−1+1/ p E[| f | χ A ] defines a norm. The first inequality in (1.12) can be seen from the estimate tµ {| f | > t} ≤ E [| f | χ {| f | > t}]; for the second one, we write +∞ +∞ E[| f | χ A ] = µ ({| f | > t} ∩ A) dt ≤ min {µ {| f | > t} , µ(A)} dt, (1.13) 0
0
p and we bound the integrand by min t − p ∥ f ∥ p,∞ , µ(A) . A function f satisfies (1.10) if and only if it belongs to the closure of bounded functions with respect to N p . Indeed, if f satisfies (1.10), then the sequence ( f χ | f < n|)n≥1 converges to f in L p,∞ . If N p ( f − g) < ε with g bounded, then lim sup t p µ {| f | > t} ≤ lim sup t p µ {| f − g| > t/2} ≤ 2 p ε. t→∞
(1.14)
t→∞
We now provide two technical lemmas about L p,∞ spaces. The first one will be used in the proof of the weak invariance principle for martingales, since we will have to control the tail function of the random variables involved in the construction of the truncated martingale (cf. (3.110)). The second one will provide an estimation of the L p,∞ norm of a simple function, which will be used in the proof of Theorem 2.1, since the function m is constructed as a series of simple functions. Lemma 1.4. If limt→∞ t p µ {| f | > t} limt→∞ t p µ {E[| f | | A] > t} = 0.
=
0, then for each sub-σ -algebra A, we have
Proof. For simplicity, we assume that f is non-negative. For a fixed t, the set {E[ f | A] > t} belongs to the σ -algebra A, hence tµ {E[ f | A] > t} ≤ E [E[ f | A]χ {E[ f | A] > t}] = E[ f χ {E[ f | A] > t}].
(1.15)
By definition of N p , E[ f χ {E[ f | A] > t}] ≤ N p ( f χ {E[ f | A] > t}) µ {E[ f | A] > t}1−1/ p ,
(1.16)
t p µ {E[ f | A] > t} ≤ N p ( f χ {E[ f | A] > t}) p .
(1.17)
hence
Notice that ∀s > 0,
N p ( f χ {E[ f | A] > t}) ≤ sµ {E[ f | A] > t}1/ p + N p ( f χ { f > s}) , (1.18)
hence lim sup E[ f χ {E[| f | | A] > t}] ≤ N p ( f χ { f > s}) ≤ κ p sup x p µ { f > x} . t→∞
(1.19)
x≥s
By the assumption on the function f , the right hand side goes to 0 as s goes to infinity, which concludes the proof of the lemma.
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N
i=0 ai χ (Ai ),
Lemma 1.5. Let f := a N < · · · < a0 . Then p
p
∥ f ∥ p,∞ ≤ max a j 0≤ j≤N
j
N is pairwise disjoint and 0 ≤ where the family (Ai )i=0
µ(Ai ).
(1.20)
i=0
Proof. We have the equality µ { f > t} =
N
χ(a j+1 ,a j ] (t)
j
µ(Ai ),
(1.21)
i=0
j=0
where a N +1 := 0, therefore p
t p µ { f > t} ≤ max a j 0≤ j≤N
j
µ(Ai ).
(1.22)
i=0
2. Main results The goal of the paper is to give a sharp sufficient condition on the moments of a strictly stationary martingale difference sequence which guarantees the weak invariance principle in Hαo [0, 1] for a fixed α. We first show that Theorem 1.3 does not extend to strictly stationary ergodic martingale difference sequences, that is, sequences of the form (m ◦ T i )i≥0 such that m is M measurable and E[m | T M] = 0 for some sub-σ -algebra M satisfying T M ⊂ M. An application of Kolmogorov’s continuity criterion shows that if (m ◦ T i )i≥0 is a martingale difference sequence such that m ∈ L p+δ for some positive δ and p > 2, then the partial sum pl o process (n −1/2 Sn (m))n≥1 is tight in H1/2−1/ p [0, 1] (see [6]). We provide a condition on the quadratic variance which improves the previous approach (since the previous condition can be replaced by m ∈ L p ). Then using martingale approximation we can provide a Hannan type condition which guarantees the weak invariance principle in Hαo [0, 1]. Theorem 2.1. Let p > 2 and (Ω , F, µ, T ) be a dynamical system with positive entropy. There exist a function m: Ω → R and a σ -algebra M for which T M ⊂ M such that: • the sequence (m ◦ T i )i≥0 is a martingale difference sequence with respect to the filtration (T −i M)i≥0 ; • the convergence limt→+∞ t p µ {|m| > t} = 0 takes place; pl o • the sequence (n −1/2 Sn (m))n≥1 is not tight in H1/2−1/ p [0, 1]. Theorem 2.2. Let (Ω , F, µ, T ) be a dynamical system, M a sub-σ -algebra of F such that T M ⊂ M and I the collection of sets A ∈ F such that T −1 = A. Let p > 2 and let (m ◦ T j , T −i M) be a strictly stationary martingale difference sequence. Assume that t p µ {|m| > t} → 0 and E[m 2 | T M] ∈ L p/2 . Then pl
o n −1/2 Sn (m) → η · W in distribution in H1/2−1/ p [0, 1],
(2.1)
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where the random variable η is given by η = lim E[Sn2 | I]/n in L1
(2.2)
n→∞
and η is independent of the process (Wt )t∈[0,1] . In particular, (2.1) takes place if m belongs to L p . The key point of 2.2 is an inequality in the spirit of Doob’s one, the proof of Theorem which gives n −1 E max1≤ j≤n S j (m)2 ≤ 2E[m 2 ]. It is used in order to establish tightness of pl
the sequence (n −1/2 Sn (m))n≥1 in the space C[0, 1]. Proposition 2.3. Let p > 2. There exists a constant C p depending only on p such that if (m ◦ T i )i≥1 is a martingale difference sequence, then the following inequality holds: p p/2 −1/2 pl p 2 ∥m∥ . (2.3) S (m) ≤ C + E E[m | T M] sup n p n p,∞ o n≥1
H1/2−1/ p
p,∞
Remark 2.4. As Theorem 2.1 shows, the condition limt→∞ t p µ {|m| > t} = 0 alone for pl martingale difference sequences is not sufficient to obtain the weak convergence of n −1/2 Sn (m) o in Hα [0, 1] for α = 1/2 − 1/ p. For the constructed m in Theorem 2.1, the quadratic variance is κm 2 for some constant κ and m does not belong to the L p space. By Lemma A.2 in [8], the H¨older norm of a polygonal line is reached at two vertices, hence, for a function g, g ◦ T j − g ◦ T i −1/2 pl −1/ p Sn (g − g ◦ T ) o =n max (2.4) n H1/2−1/ p 1≤i< j≤n ( j − i)1/2−1/ p ≤ 2n −1/ p max g ◦ T j . (2.5) 1≤ j≤n
As a consequence, if g belongs to
Lp,
pl then the sequence n −1/2 Sn (g − g ◦ T )
o H1/2−1/ p
n≥1
converges to 0 in probability. Therefore, we can exploit a martingale–coboundary decomposition in L p . Corollary 2.5. Let p > 2 and let f be an M-measurable function which can be written as f = m + g − g ◦ T,
(2.6)
where m, g ∈ L p and (m ◦ T i )i≥0 is a martingale difference sequence for the filtration pl o (T −i M)i≥0 . Then n −1/2 Sn ( f ) → ηW in distribution in H1/2−1/ p [0, 1], where η is given by (2.2) and independent of W . k We define for a∞function h the operators Ek (h) := E[h | T M] and Pi (h) := Ei (h)−Ei+1 (h). The condition i=0 ∥Pi ( f )∥2 was introduced by Hannan in [5] in order to deduce a central limit theorem. It actually implies the weak invariance principle (see Corollary 2 in [3]).
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Theorem 2.6. Let p > 2 and let f be an M-measurable function such that i T M = 0 and E f |
(2.7)
i∈Z
∥Pi ( f )∥ p < ∞.
(2.8)
i≥0 pl
o Then n −1/2 Sn (m) → ηW in distribution in H1/2−1/ p [0, 1], where η is given by (2.2) and independent of W .
3. Proofs 3.1. Proof of Theorem 2.1 We need a result about dynamical systems of positive entropy for the construction of a counterexample. Lemma 3.1. Let (Ω , A, µ, T ) be an ergodic probability measure preserving system of positive entropy. There exist two T -invariant sub-σ -algebras B and C of A and a function g: Ω → R such that: • the σ -algebras B and C are independent; • the function g is B-measurable, takes the values −1, 0 and 1, has zero mean and the process (g ◦ T n )n∈Z is independent; • the dynamical system (Ω , C, µ, T ) is aperiodic. This is Lemma 3.8 from [7]. We consider the following four increasing sequences of positive integers (Il )l≥1 , (Jl )l≥1 , (nl )l≥1 and (L l )l≥1 . We define kl := 2 Il +Jl and impose the conditions: ∞ 1 < ∞; L l=1 l Ll lim Jl · µ |N | ≥ 41/ p = 1; l→∞ ∥g∥2
lim Jl 2−Il /2 = 0;
l→∞
lim nl
l→∞
ki = 0; n i>l i
for each l,
1/ p l−1 nl kl n i 1/ p < . L 2 Ii 2 i=1 i
(3.1) (3.2) (3.3) (3.4)
(3.5)
Here N denotes a random variable whose distribution is standard normal. Such sequences can be constructed as follows: first pick a sequence (L l )l≥1 satisfying (3.1), for example L l = l 2 . Then construct (Jl )l≥1 such that (3.2) holds. Once the sequence (Jl )l≥1 is constructed, define (Il )l≥1 satisfying (3.3). Now the sequence (kl )l≥1 is completely determined. Noticing that (3.4)
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is satisfied if the series l kl nl−1 /nl converges, we construct the sequence (nl )l≥1 by induction; once n i , i ≤ l − 1 are defined, we choose nl such that nl ≥ l 2 kl nl−1 and (3.5) holds. Using Rokhlin’s lemma, we can find for any integer l ≥ 1 a measurable set Cl ∈ C such that nl −1 −i −i the sets T Cl , i = 0, . . . , nl − 1 are pairwise disjoint and µ i=0 T Cl > 1/2. For a fixed l, we define kl, j := 2 Il +Jl − j , 0 ≤ j ≤ Jl and 1/ p kl,Jl − j−1 −1 l −1 1 J nl fl := T −i Cl χ L l j=0 kl,Jl − j i=kl,Jl − j 1/ p kl,Jl −1 nl 1 + χ T −i Cl , L l kl,Jl i=0 f :=
+∞
fl ,
(3.6)
(3.7)
m := g · f,
l=1
where g is the function obtained by Lemma 3.1. Proposition 3.2. We have the estimate ∥ fl ∥ p,∞ ≤ κ ′p L l−1 for some constant κ ′p depending only on p. As a consequence, limt→∞ t p µ {|m| > t} = 0. Proof. Notice that p 1/ p kl,Jl −1 1 n l −i χ T C l L k l,Jl l i=0
=
p,∞
Next, using Lemma 1.5 with N kl,Jl − j−1 −1 i=kl,Jl − j
:=
1 nl 1 kl,Jl · µ(Cl ) ≤ p . p L l kl,Jl Ll
Jl − 1, a j
:=
1 Ll
nl kl,Jl − j
1/ p
(3.8)
and A j
:=
T −i Cl , we obtain
p J 1/ p kl,Jl − j−1 1 l −1 nl −i χ T Cl L l j=0 kl,Jl − j i=kl,Jl − j
p,∞
≤
max
0≤ j≤Jl −1
≤
1 p Ll
1 = p Ll ≤
max
1 Ll
1/ p p j
nl kl,Jl − j nl
j kl,J −i
0≤ j≤Jl −1 kl,Jl − j i=0 j 2 Il +i max 0≤ j≤Jl −1 2 Il + j i=0
2 p, Ll
µ(A j )
(3.9)
i=0 l
nl
(3.10)
(3.11) (3.12)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
hence by (1.12), (3.8) and (3.12), ∥ fl ∥ p,∞
1 ≤ Np Ll
1/ p
nl kl,Jl
kl,Jl −1
χ
T
−i
(3.13)
Cl
i=0
1/ p kl,Jl − j−1 J l −1 1 n l + Np χ T −i Cl L l j=0 kl,Jl − j i=kl,Jl − j 1/ p kl,Jl −1 1 nl −i χ T C ≤ κp l L k l,Jl l i=0
(3.14)
(3.15)
p,∞
J 1/ p kl,Jl − j−1 1 l −1 nl −i + κp χ T C l L l j=0 kl,Jl − j i=kl,Jl − j ≤
(3.16) p,∞
1 κ p 1 + 21/ p . Ll
(3.17)
We thus define κ ′p := κ p 1 + 21/ p . We fix ε > 0; using (3.1), we can find an integer l0 such that l>l0 1/L l < ε. Since the l0 function l=1 g fl is bounded, we have, l p 0 t p +2 g fl lim sup t µ {|m| > t} ≤ lim sup t µ g fl > l>l l=1 2 t→∞ t→∞ p
p
0
p p g fl =2 l>l 0
≤
2
(3.18)
p,∞
(3.19)
p,∞ p
N p ( fl )
(3.20)
p 1 Ll l>l
(3.21)
l>l0
≤
κ ′p
0
≤ κ ′p ε p ,
(3.22)
where the second inequality comes from inequalities (1.12). Since ε is arbitrary, the proof of Proposition 3.2 is complete. We denote by M the σ -algebra generated by C and the random variables g ◦ T k , k ≤ 0. It satisfies M ⊂ T −1 M. Proposition 3.3. The sequence (m ◦ T i )i≥0 is a (stationary) martingale difference sequence with respect to the filtration (T −i M)i≥0 .
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Proof. We have to show that E[m | T M] = 0. Since the σ -algebra C is T -invariant, we have T M = σ (C ∪ σ (g ◦ T k , k ≤ −1)). This implies E[m | T M] = E[g f | T M] = f · E[g | T M].
(3.23)
Since g is centered and independent of T M, Proposition 3.3 is proved. It remains to prove that the process
pl (n −1/2 Sn (m))n≥1
is not tight in
o H1/2−1/ p [0, 1].
Proposition 3.4. Under conditions (3.2)–(3.4), there exists an integer l0 such that for l ≥ l0 1 |Su+v (g fl ) − Su (g fl )| 1 Pl := µ max ≥1 ≥ . (3.24) 1/ p 1/2−1/ p 1≤u≤nl −kl n 16 v l 1≤v≤k l
Proof. Let us fix an integer l ≥ 1. Assume that ω ∈ T −s Cl , where kl ≤ s ≤ nl − 1. Since T u ω belongs to T −(s−u) Cl we have for s − nl ≤ u ≤ s nl 1/ p 1 , if s − kl,Jl < u ≤ s; L l kl,Jl ( fl ◦ T u )(ω) = 1 (3.25) nl 1/ p , if s − kl, j−1 < u ≤ s − kl, j , and 1 ≤ j ≤ Jl ; L kl, j l 0, if s − nl ≤ u < s − kl . As a consequence, T
−s
Cl ∩
1
max
Ss−k
1/ p 1≤ j≤J l
nl
= T −s Cl ∩
max 1≤ j≤Jl
l, j−1 +1
(g fl ) − Ss−kl, j (g fl )
(kl, j−1 − 1 − kl, j )1/2−1/ p
≥1
l, j−1 +1 (g) − Ss−kl, j (g)
Ss−k
1/ p
(kl, j − 1)1/2−1/ p kl, j−1
≥ Ll
.
(3.26)
Since for kl + 1 ≤ s ≤ nl − kl and 1 ≤ j ≤ Jl , we have 1 ≤ s − kl, j ≤ nl − kl and 1 ≤ kl, j−1 − 1 − kl, j ≤ kl , the inequality Ss−k l, j−1 +1 (g f l ) − Ss−kl, j (g f l ) −s χ (T Cl ) · max 1≤ j≤Jl (kl, j−1 − 1 − kl, j )1/2−1/ p ≤ χ (T −s (Cl ))
max
1≤u≤nl −kl 1≤v≤kl
|Su+v (g fl ) − Su (g fl )| v 1/2−1/ p
(3.27)
k −1 takes place and since the sets (T −s Cl )ns=0 are pairwise disjoint, we obtain the lower bound
Pl ≥
nl −2kl
µ T
−(s+kl )
(Cl ) ∩
s=1
max
Ss+k −k l
l, j−1 +1
(g fl ) − Ss+kl −kl, j (g fl )
(kl, j−1 − 1 − kl, j )1/2−1/ p
1≤ j≤Jl
.
≥1
(3.28)
Using the fact that T is measure-preserving, this becomes Pl ≥ (nl − 2kl ) · µ T
−kl
(Cl ) ∩
max
1≤ j≤Jl
Sk −k l
l, j−1 −1
(g fl ) − Skl −kl, j (g fl )
(kl, j−1 − 1 − kl, j )1/2−1/ p
≥1
,
(3.29)
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and plugging (3.26) in the previous estimate, we get Sk −k (g) − S (g) k −k −1 l l, j l l, j−1 . (3.30) Pl ≥ (nl − 2kl )µ T −kl (Cl ) ∩ max ≥ L l 1/ p 1≤ j≤Jl (kl, j − 1)1/2−1/ p k l, j−1
The sets
max1≤ j≤Jl
Skl −kl, j−1 +1 (g)−Skl −kl, j (g) 1/ p
(kl, j −1)1/2−1/ p kl, j−1
≥ Ll
and T −kl Cl belong to the independent
sub-σ -algebras B and C respectively, hence using the fact that the sequences (g ◦ T i )i≥0 and (g ◦ T −i )i≥0 are identically distributed, we obtain Sk (g) − S (g) kl, j l, j−1 −1 Pl ≥ (nl − 2kl )µ (Cl ) µ max ≥ Ll . (3.31) 1/ p 1≤ j≤Jl (kl, j − 1)1/2−1/ p k l, j−1
By construction, we have nl · µ(Cl ) = µ kl 1 1−2 µ max Pl ≥ 1≤ j≤Jl 2 nl
nl −1 i=0
T −i Cl > 1/2, hence
l, j−1 −1 (g) − Skl, j (g) 1/ p (kl, j − 1)1/2−1/ p kl, j−1
Sk
It remains to find a lower bound for Sk (g) − S (g) k −1 l, j l, j−1 ≥ L . Pl′ := µ max l 1≤ j≤Jl (kl, j − 1)1/2−1/ p k 1/ p
≥ Ll
.
(3.32)
(3.33)
l, j−1
Let us define the set Sk (g) − S (g) kl, j l, j−1 −1 ≥ L . E j := l (kl, j − 1)1/2−1/ p k 1/ p
(3.34)
l, j−1
Since the sequence (g ◦ T i )i∈Z is independent, the family (E j )1≤ j≤Jl is independent, hence Pl′ ≥ 1 −
Jl
(1 − µ(E j )).
(3.35)
j=1
We define the quantity
Ll c j := µ |N | ≥ ∥g∥2
kl, j−1 kl, j − 1
1/ p (3.36)
(we recall that N denotes a standard normally distributed random variable). By the Berry–Esseen theorem, we have for each j ∈ {1, . . . , Jl }, √ 1 2 −Il /2 1 µ(E j ) − c j ≤ ≤ 2 . (3.37) ∥g∥32 (kl, j−1 − 1)1/2 ∥g∥32
D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
301
N , Plugging the estimate (3.37) into (3.35) and noticing that for an integer N and (an )n=1 N (bn )n=1 two families of numbers in the unit interval, N N N |an − bn | , (3.38) an − bn ≤ n=1 n=1 n=1
we obtain Pl′ ≥ 1 −
Jl
(1 − µ(E j )) +
Jl
j=1
≥ 1−
Jl
j=1
(1 − c j ) −
j=1 Jl
(1 − c j ) −
Jl
(1 − c j )
(3.39)
j=1
Jl µ(E j ) − c j
(3.40)
j=1
√ 2
2−Il /2 .
(3.41)
J (1 − c j ) ≥ 1 − max 1 − c j l
(3.42)
≥ 1−
(1 − c j ) − Jl
j=1
∥g∥32
Notice that 1−
Jl
1≤ j≤Jl
j=1
and since (Il )≥ is increasing and I1 ≥ 1, we have kl, j−1 2 2 ≤ = ≤4 −1 kl, j − 1 1 − 2−Il 1 − kl, j Ll it follows by (3.36) that c j ≥ µ |N | ≥ 41/ p ∥g∥ for 1 ≤ j ≤ Jl . We thus have
(3.43)
2
Pl′
≥ 1 − 1 − µ |N | ≥ 4
1/ p
Ll ∥g∥2
Jl − Jl
√ 2 ∥g∥32
2−Il /2 .
(3.44)
Using the elementary inequality n(n − 1) 2 t 2 valid for a positive integer n and t ∈ [0, 1], we obtain √ 2 Jl2 2 −Il /2 ′ 1/ p L l 1/ p L l Pl ≥ Jl µ |N | ≥ 4 µ |N | ≥ 4 2 . − − Jl ∥g∥2 ∥g∥2 2 ∥g∥32 1 − (1 − t)n ≥ nt −
By conditions (3.3) and (3.2), there exists an integer l0′ such that if l ≥ l0′ , then 1 Sk (g) − S (g) kl, j l, j−1 −1 µ max ≥ L ≥ . l 1≤ j≤Jl (kl, j − 1)1/2−1/ p k 1/ p 4
(3.45)
(3.46)
(3.47)
l, j−1
Combining (3.32) with (3.47), we obtain for l ≥ l0′ 1 kl 1−2 . Pl ≥ 8 nl
(3.48)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
By condition (3.4), we thus get that Pl ≥ 1/16 for l ≥ l0 , where l0 ≥ l0′ and kl /nl ≤ 1/4 if l ≥ l0 . This concludes the proof of Proposition 3.4. Proposition 3.5. Under conditions (3.1)–(3.5), we have for l large enough 1 |Su+v (m) − Su (m)| 1 1 µ max ≥ ≥ . 1/ p 1/2−1/ p 1≤u≤nl −kl n 2 32 v l
(3.49)
1≤v≤kl
Since the H¨older modulus of continuity of a piecewise linear function is reached at vertices, we derive the following corollary. Corollary 3.6. If l ≥ l0 , then 1 1 1 pl kl ≥ ≥ . µ ω1/2−1/ p √ Snl (m), nl nl 2 32
(3.50)
Therefore, for each positive δ, we have 1 pl 1 1 lim sup µ ω1/2−1/ p √ Sn (m), δ ≥ ≥ , 2 32 n n→∞
(3.51)
pl
o and the process (n −1/2 Sn (m))n≥1 is not tight in H1/2−1/ p [0, 1].
Proof of Let l0 be the integer given by Proposition 3.4 and let l ≥ l0 . We define Proposition 3.5. +∞ ′′ m l′ := l−1 i=1 g f i and m l := i=l+1 g f i . We define for i ≥ 1, Ml,i :=
1 1/ p nl
max
1≤u≤nl −kl 1≤v≤kl
|Su+v (g f i ) − Su (g f i )| . v 1/2−1/ p
(3.52)
Let i be an integer such that i < l. Notice that for 1 ≤ u ≤ nl − kl and v ≤ kl , we have |Su+v (g f i ) − Su (g f i )| = U u (|Sv (g f i )|),
(3.53)
where U (h)(ω) = h (T (ω)) and since kl n i 1/ p |Sv (g f i )| ≤ v ∥g f i ∥∞ ≤ , L i 2 Ii the estimate kl n i 1/ p Ml,i ≤ 1/ p 2 Ii Li n
(3.54)
(3.55)
l
holds. Since 1 1/ p
nl
max
1≤u≤nl −kl 1≤v≤kl
Su+v (m ′ ) − Su (m ′ ) l v 1/2−1/ p
l
≤
l−1
Ml,i ,
(3.56)
i=1
we have by (3.55), 1 1/ p nl
max
1≤u≤nl −kl 1≤v≤kl
Su+v (m ′ ) − Su (m ′ ) l v 1/2−1/ p
l
≤
l−1
kl
1/ p i=1 L i n l
n 1/ p i
2 Ii
.
(3.57)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
By (3.5), the following bound takes place: Su+v (m ′ ) − Su (m ′ ) 1 1 l l max ≤ . 1/ p 1≤u≤nl −kl 1/2−1/ p 2 v nl 1≤v≤k
(3.58)
l
The following set inclusions hold: 1 Su+v (m ′′ ) − Su (m ′′ ) l l = ̸ 0 ⊂ Ml,i ̸= 0 max 1/ p 1/2−1/ p 1≤u≤nl −kl n v i>l l
(3.59)
1≤v≤kl
⊂
nl u U (g f i ) ̸= 0 .
(3.60)
i>l u=1
We thus have 1 µ n 1/ p l
max
Su+v (m ′′ ) − Su (m ′′ ) l
l
v 1/2−1/ p
1≤u≤nl −kl 1≤v≤kl
̸= 0 ≤ nl · µ {g f i ̸= 0} i>l ≤ nl
(3.61)
µ { f i ̸= 0}
(3.62)
(ki + 1)µ(Ci )
(3.63)
i>l
= nl
i>l
≤ 2nl
ki n i>l i
and by (3.4), it follows that 1 Su+v (m ′′ ) − Su (m ′′ ) 1 l l = ̸ 0 ≤ µ max . 1/2−1/ p 1/ p 1≤u≤nl −kl 32 n v l
(3.64)
(3.65)
1≤v≤kl
Accounting (3.58), we thus have 1 |Su+v (m) − Su (m)| 1 max ≥ µ n 1/ p 1≤u≤nl −kl 2 v 1/2−1/ p l 1≤v≤kl 1 Su+v (g fl + m ′′ ) − Su (g fl + m ′′ ) l l ≥µ ≥ 1 max n 1/ p 1≤u≤nl −kl v 1/2−1/ p l 1≤v≤kl 1 |Su+v (g fl ) − Su (g fl )| ≥µ max ≥ 1 n 1/ p 1≤u≤nl −kl v 1/2−1/ p l
1≤v≤kl
1 −µ n 1/ p l
max
1≤u≤nl −kl 1≤v≤kl
Su+v (m ′′ ) − Su (m ′′ ) l
v 1/2−1/ p
l
̸= 0 ,
(3.66)
hence combining Proposition 3.4 with (3.65), we obtain the conclusion of Proposition 3.5. Theorem 2.1 follows from Corollary 3.6 and Propositions 3.2 and 3.3.
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3.2. Proof of Theorem 2.2 and Proposition 2.3 Proof of Proposition 2.3. Let us fix a positive t. Recall the equivalence between ∥x∥α and seq ∥x∥α and Notation 1.1. By Remark 1.2, we have to show that for some constant C depending only on p and each integer n ≥ 1, pl pl p α j −1/2 P(n, t) := t µ sup 2 n max Sn (m, rk+1, j ) − Sn (m, rk, j ) > t 0≤k<2 j
j≥1
p/2 p ≤ C ∥m∥ p,∞ + E E[m 2 | T M] .
(3.67)
In the proof, we shall denote by C p a constant depending only on p which may change from line to line. We define pl pl α j −1/2 P1 (n, t) := µ sup 2 n max Sn (m, rk+1, j ) − Sn (m, rk, j ) > t , and (3.68) 0≤k<2 j
1≤ j≤log n
P2 (n, t) := µ
α j −1/2
sup 2 n
pl pl max Sn (m, rk+1, j ) − Sn (m, rk, j ) > t ,
0≤k<2 j
j>log n
(3.69)
hence P(n, t) ≤ t p P1 (n, t/2) + t p P2 (n, t/2).
(3.70)
We estimate P2 (n, t). For j > log n, we have the inequality rk+1, j − rk, j = (k + 1)2− j − k2− j = 2− j < 1/n,
(3.71)
hence if rk, j belongs to the interval [l/n, (l + 1)/n) for some l ∈ {0, . . . , n − 1}, then • either rk+1, j ∈ [l/n, (l + 1)/n), and in this case, pl pl Sn (m, rk+1, j ) − Sn (m, rk, j ) = m ◦ T l+1 2− j n ≤ 2− j n max U l (m) ;
(3.72)
• or rk+1, j belongs to the interval [(l + 1)/n, (l + 2)/n). The estimates pl pl pl pl Sn (m, rk+1, j ) − Sn (m, rk, j ) ≤ Sn (m, rk+1, j ) − Sn (m, (l + 1)/n) pl pl + Sn (m, (l + 1)/n) − Sn (m, rk, j ) ≤ 21− j n max U l (m)
(3.73)
1≤l≤n
1≤l≤n
hold. Considering these two cases, we obtain P2 (n, t) ≤ µ
αj
1− j −1/2
sup 2 n2
n
j>log n
max U l (m) > t
1≤l≤n
(3.74)
≤ µ 2n α−1/2 max U l (m) > t
(3.75)
≤ nµ 2n −1/ p |m| > t
(3.76)
1≤l≤n
≤
2p sup x p µ {|m| > x} . t p x>0
(3.77)
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305
Therefore, establishing inequality (3.67) reduces to find a constant C depending only on p such that p/2 p 2 p sup sup t P1 (n, t) ≤ C ∥m∥ p,∞ + E E[m | T M] . (3.78) n
t
We define u k, j := [nrk, j ] for k < 2 j and j ≥ 1 (see Notation 1.1). Notice that the inequalities pl Su k, j (m) − Sn (m, rk, j ) ≤ U u k, j +1 (m) and pl Sn (m, rk+1, j ) − Su k+1, j (m) ≤ U u k+1, j +1 (m)
(3.79) (3.80)
take place because if j ≤ log n, then u k, j ≤ nrk, j ≤ u k, j + 1 ≤ u k+1, j ≤ nrk+1, j ≤ u k+1, j + 1. Therefore, P1 (n, t) ≤ P1,1 (n, t) + P1,2 (n, t), where P1,1 (n, t) := µ max 2α j n −1/2 max Su k+1, j (m) − Su k, j (m) > t/2 , 0≤k<2 j
1≤ j≤log n
P1,2 (n, t) := µ
max
α j −1/2
1≤ j≤log n
2 n
l max U (m) > t/4 .
1≤l≤n
(3.81)
(3.82) (3.83)
Notice that
P1,2 (n, t) ≤ µ n
α−1/2
l max U (m) > t/4
1≤l≤n
(3.84)
≤ nµ |m| > n 1/ p t/4
(3.85)
≤ 4 p t − p sup x p µ {|m| > x} ,
(3.86)
x≥0
hence (3.78) will follow from the existence of a constant C depending only on p such that p/2 p sup sup t p P1,1 (n, t) ≤ C ∥m∥ p,∞ + E E[m 2 | T M] . (3.87) n
t
We estimate P1,1 (n, t) in the following way: P1,1 (n, t) ≤
log n j=1
2 j max µ Su k+1, j (m) − Su k, j (m) > tn 1/2 2−1−α j . 0≤k<2 j
We define for 1 ≤ j ≤ log n and 0 ≤ k < 2 j the quantity P(n, j, k, t) := µ Su k+1, j (m) − Su k, j (m) > tn 1/2 2−1−α j . If ( f ◦ T j ) j≥0 is a strictly stationary sequence, we define 1/2 n Q f,n (u) := µ max f ◦ T j > u + µ U i E[ f 2 | T M] >u . i=1 1≤ j≤n
(3.88)
(3.89)
(3.90)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
The following inequality is Theorem 1 of [10]. It allows us to express the tail function of a martingale by that of the increments and the quadratic variance. Theorem 3.7. Let m be an M-measurable function such that E[m | T M] = 0. Then for each positive y and each integer n, 1 µ {|Sn (m)| > y} ≤ c(q, η) Q m,n (εq u · y)u q−1 du, (3.91) 0
where q > 0, η > 0, εq := η/q and c(q, η) := q exp(3ηeη+1 − η − 1)/η. We shall use (3.91) with q := p + 1, η = 1 and y := n 1/2 2−1−α j t in order to estimate P(n, j, k, t): 1 i P(n, j, k, t) ≤ C p µ max U (m) > n 1/2 2−1−α j tuε p+1 u p du 1≤i≤u k+1, j −u k, j
0
1/2 1 k+1, j u + Cp U i (E[m 2 | T M]) > n 1/2 2−1−α j utε p+1 u p du. (3.92) µ 0 i=u k, j +1 Exploiting the inequality u k+1, j − u k, j ≤ 2n2− j , we get from the previous bound 1 i 1/2 −1−α j P(n, j, k, t) ≤ C p µ max U (m) > n 2 tuε p+1 u p du 1≤i≤2n2− j
0
1/2 −j 1 2n2 i 2 1/2 −1−α j U (E[m | T M]) >n 2 tuε p+1 u p du. + Cp µ 0 i=1 We define for j ≤ log n, t ≥ 0 and u ∈ (0, 1), P ′ (n, j, t, u) := µ max U i (m) > n 1/2 2−1−α j tuε p+1 , 1≤i≤2n2− j
and
1/2 −j 2n2 ′′ i 2 1/2 −1−α j P (n, j, t, u) := µ U (E[m | T M]) >n 2 tuε p+1 . i=1
(3.93)
(3.94)
(3.95)
Using the fact that the random variables U i (m), 1 ≤ i ≤ 2n2− j are identically distributed, we derive the bound P ′ (n, j, t, u) ≤ 2n2− j µ |m| > n 1/2 2−1−α j tuε p+1 , (3.96) hence p
P ′ (n, j, t, u) ≤ 2n2− j (n 1/2 2−1−α j tuε p+1 )− p ∥m∥ p,∞ = 2 p+1 ε p+1 n 1− p/2 2 j (−1+ pα) t − p u − p ∥m∥ p,∞ . −p
p
(3.97)
Since α and p are linked by the relationship 1/2 − 1/ p = α, we have pα = p/2 − 1 hence 1 p P ′ (n, j, t, u)u p du ≤ C p t − p n 1− p/2 2 j ( p/2−2) ∥m∥ p,∞ . (3.98) 0
D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
307
Notice the following set of equalities: 1/2 −j 2n2 i 2 1/2 −1−α j U (E[m | T M]) > ε p+1 un 2 t i=1 −j 1 2n2 i 2 −3 2 2 2 j/ p 2 U (E[m | T M]) > 2 ε u 2 t = p+1 2n2− j i=1
(3.99)
and that n2− j ≥ 1 (because j ≤ log n), hence 1/2 −j 2n2 i 2 1/2 −1−α j U (E[m | T M]) > ε p+1 un 2 t i=1 ⊆
N ≥2
N 1 i 2 −3 2 2 2 j/ p 2 U (E[m | T M]) > 2 ε p+1 u 2 t , N i=1
(3.100)
from which it follows 1/2 −j 2n2 i 2 1/2 −1−α j U (E[m | T M]) > ε p+1 un 2 t µ i=1
N 1 i 2 −3 2 2 2 j/ p 2 ≤ µ sup U (E[m | T M]) > 2 ε p+1 u 2 t . N ≥2 N i=1
(3.101)
Combining (3.98) and (3.101), we obtain max P(n, j, k, t) ≤ C p t − p n 1− p/2 2 j ( p/2−2) ∥m∥ p,∞ p
0≤k<2 j
1
+ Cp 0
N 1 µ sup U i (E[m 2 | T M]) > 2−3 ε 2p+1 u 2 22 j/ p t 2 u p du, N ≥2 N i=1
(3.102)
hence by (3.88) and (3.89), p
P1,1 (n, t) ≤ C p t − p ∥m∥ p,∞
log n
2 j 2 j ( p/2−2) n 1− p/2
j=1
+ Cp 0
N 1 2 j µ sup U i (E[m 2 | T M]) > 2−3 ε 2p+1 u 2 22 j/ p t 2 u p du. (3.103) N N ≥2 j=1 i=1
1 log n
From the elementary bounds log n
2 j ( p/2−1) n 1− p/2 ≤ (1 − 21− p/2 )−1
(3.104)
j=1
j≥1
2 j µ |g| > 22 j/ p ≤ 2E |g| p/2 ,
for any non-negative function g,
(3.105)
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with −2 g := 23 ε −2 sup p+1 u
N ≥2
N 1 U i (E[m 2 | T M]), N i=1
u ∈ (0, 1)
(3.106)
we obtain P1,1 (n, t) ≤ C p t
−p
p ∥m∥ p,∞
+ C pt
−p
p/2 N 1 i 2 U (E[m | T M]) . sup N ≥2 N i=1
(3.107)
p/2
L1 -L∞
As the Koopman operator U is an contraction, Theorem 1 of [12] gives the existence of a constant A p such that for each h ∈ L p/2 , N 1 j U (h) ≤ A p ∥h∥ p/2 . (3.108) sup N ≥1 N j=1 p/2
Applying (3.108) with h := E[m 2 | T M], we get by (3.107) p/2 p , P1,1 (n, t) ≤ C p t − p ∥m∥ p,∞ + C p t − p E E[m 2 | T M] which establishes (3.78). This concludes the proof of Proposition 2.3.
(3.109)
Proof of Theorem 2.2. The convergence of finite dimensional distributions can be proved using theorem of [1]. Its proof works for filtrations of the form (T −i M)i≥0 where T M ⊂ M and also in the non-ergodic setting by considering the ergodic components. We deduce tightness in Theorem 2.2 from Proposition 2.3 by a truncation argument. For a fixed R, we define m R := mχ {|m| ≤ R} − E[mχ {|m| ≤ R} | T M] m ′R
and
:= mχ {|m| > R} − E[mχ {|m| > R} | T M]. T i )i≥0
(3.111)
In this way, the sequences (m R ◦ and ◦ are martingale difference sequences and m = m R + m ′R . Since |m R | ≤ 2R and (m R ◦ T i )i≥0 is a martingale difference sequence, the sequence pl o (n −1/2 Sn (m R ))n≥1 is tight in H1/2−1/ p [0, 1]. Consequently, for each positive ε, the following convergence takes place: pl αj 1/2 lim lim sup µ sup 2 max λr Sn (m R ) > εn = 0. (3.112) J →∞ n→∞
j≥J
(m ′R
(3.110)
T i )i≥0
r ∈D j
Using Proposition 2.3, we derive the following bound, valid for each ε and each R, pl lim lim sup µ sup 2α j max λr Sn (m) > εn 1/2 J →∞ n→∞
j≥J
r ∈D j
≤ C pε
−p
sup t µ {|m| χ {|m| > R} > t} + sup t µ {E[|m| χ {|m| > R} | T M] > t} p
t≥0
+ ε− p C p E
p
t≥0
E[m 2 χ {|m| > R} | T M]
p/2
.
The first term is supt≥R t p µ {|m| > t}, which goes to 0 as R goes to infinity.
(3.113)
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The second term can be bounded by supt≥R t p µ {E[|m| | T M] > t}. Indeed, if t ≥ R, we use the inclusion {E[|m| χ {|m| > R} | T M] > t} ⊂ {E[|m| | T M] > t} ,
(3.114)
and if t < R, then accounting the fact that the random variable E[|m| χ {|m| > R} | T M] is greater than R, we get E[|m| χ {|m| > R} | T M] = E[|m| χ {|m| > R} | T M]χ {E[|m| | T M] > R} ≤ E[|m| | T M]χ {E[|m| | T M] > R} ,
(3.115)
from which it follows that t p µ {E[|m| χ {|m| > R} | T M] > t} ≤ R p µ {E[|m| | T M] > R} .
(3.116)
By Lemma 1.4, the convergence lim sup t p µ {E[|m| | T M] > t} = 0
(3.117)
R→∞ t≥R
takes place. The third term of (3.113) converges to 0 as R goes to infinity by monotone convergence. This concludes the proof of tightness in Theorem 2.2. 3.3. Proof of Theorem 2.6 By (2.7), the equality f = i≥0 Pi ( f ) holds almost surely. For a fixed integer K , we define K f K := i=0 Pi ( f ). Then f K satisfies the conditions of Corollary 2.5. Indeed, we have the equalities Pi ( f ) − P0 (U i f ) = E[ f | T i M] − E[U i f | M] − E[ f | T i+1 M] + E[U i f | T M] = (I − U )E[ f | T M] − (I − U )E[ f | T i
i
i
(3.118) i+1
M]
(3.119) k and the latter term can be expressed as a coboundary noticing that (I −U i ) = (I −U ) i−1 k=0 U . K Since Pi ( f ) belongs to the L p space, we may write f K − i=0 P0 (U i f ) as (I − U )g K where K p i g K belongs to the L space. Defining m K := i=0 P0 (U ( f )), the sequence (m K ◦ T i )i≥0 is a martingale difference sequence hence for each positive ε, pl lim lim sup µ sup 2α j max λr Sn ( f K ) > εn 1/2 = 0. (3.120) J →∞ n→∞
j≥J
r ∈D j
Now, we have to show that the convergence in (3.120) holds if f K is replaced by f − f K . To this aim, we use the inclusion pl αj 1/2 sup 2 max λr Sn ( f − f K ) > εn j≥J
r ∈D j
⊆ sup 2 j≥1
αj
pl 1/2 max λr Sn ( f − f K ) > εn ,
r ∈D j
(3.121)
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D. Giraudo / Stochastic Processes and their Applications 126 (2016) 290–311
hence µ sup 2
pl max λr Sn ( f − f K ) > εn 1/2
αj
r ∈D j
j≥J
pl −p 1 ≤ ε √ Sn ( f − f K ) o n H
1/2−1/ p
p
(3.122)
p,∞
p 1 pl −p = ε √ Sn Pi ( f ) n o i≥K +1 H1/2−1/ p
,
(3.123)
p,∞
from which it follows that pl 1/2 αj µ sup 2 max λr Sn ( f − f K ) > εn r ∈D j
j≥J
1 pl −p ≤ε S ( f )) √ (P i n o n H i≥K +1
1/2−1/ p
p .
(3.124)
p,∞
(U l (Pi ( f )))l≥1
Notice that for a fixed i, the sequence is a martingale difference sequence (with respect to the filtration (T −i−l M)l≥0 ). Therefore, by Proposition 2.3, we obtain 1 pl ≤ C p ∥Pi ( f )∥ p . (3.125) √ Sn (Pi ( f )) o n H 1/2−1/ p
p,∞
Plugging this estimate into (3.124), we obtain that for some constant C depending only on p, p pl −p αj 1/2 ∥Pi ( f )∥ p . (3.126) ≤ Cε µ sup 2 max λr Sn ( f − f K ) > εn r ∈D j
j≥J
i≥K +1
Combining (3.120) and (3.126), we obtain for each K : pl 1/2 αj lim lim sup µ sup 2 max λr Sn ( f ) > n ε J →∞ n→∞
j≥J
r ∈D j
p
≤ Cε
−p
∥Pi ( f )∥ p
.
(3.127)
i≥K +1
Since K is arbitrary, we conclude the proof of Theorem 2.6 thanks to assumption (2.8). Acknowledgments The author is grateful to the referee for many comments which improved the readability of the paper. The author would like to thank Dalibor Voln´y for many useful discussions which lead to the counter-example in Theorem 2.1, and also Alfredas Raˇckauskas and Charles Suquet for their encouragement.
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