Linear processes and functions of linear processes

which holds as soon as P τ = n = On −1−p2 log n −p2−ε , where P is the probability of the chain starting from 0, and τ = inf{n 0, X n = 0}. Now P τ = n = 1 − a n Π n −1 i=1 a i for n ≥ 2. Consequently, if a i = 1 − p 2i 1 + 1 + ε log i for i large enough , the condition 4.2 is satisfied and the conclusion of Theorem 2.1 holds. Remark 4.2. If f is bounded and K f 6= 0, the central limit theorem may fail to hold for S n = P n i=1 f Y i − E f Y i . We refer to the Example 2, page 321, given by Davydov 1973, where S n properly normalized converges to a stable law with exponent strictly less than 2. Proof of Proposition 4.1. Let B p F be the set of F -measurable random variables such that kZk p ≤ 1. We first notice that kEX 2 k |F − σ 2 k p 2 = sup Z ∈B p p−2 F CovZ, X 2 k . Applying Rio’s covariance inequality 1993, we get that kEX 2 k |F − σ 2 k p 2 ≤ 2 Z α 1 k Q p udu 2 p , which shows that the convergence of the second series in 4.1 implies 2.2. Now, from Fréchet 1957, we have that kEX 2 k |F − σ 2 k 1,Φ,p = sup E1 ∨ |Z| p −2 | EX 2 k |F − σ 2 | , Z F -measurable, Z ∼ N 0, 1 . Hence, setting ǫ k = signEX 2 k |F − σ 2 , kEX 2 k |F − σ 2 k 1,Φ,p = sup Covǫ k 1 ∨ |Z| p −2 , X 2 k , Z F -measurable, Z ∼ N 0, 1 . Applying again Rio’s covariance inequality 1993, we get that kEX 2 k |F − σ 2 k 1,Φ,p ≤ C Z α 1 k 1 ∨ logu −1 p−22 Q 2 udu , which shows that the convergence of the first series in 4.1 implies 2.1.

4.2 Linear processes and functions of linear processes

In what follows we say that the series P i ∈Z a i converges if the two series P i ≥0 a i and P i a i converge. Theorem 4.1. Let a i i ∈Z be a sequence of real numbers in ℓ 2 such that P i ∈Z a i converges to some real A. Let ǫ i i ∈Z be a stationary sequence of martingale differences in L p for p ∈]2, 3]. Let X k = P j ∈Z a j ǫ k − j , and σ 2 n = n −1 E S 2 n . Let b = a −A and b j = a j for j 6= 0. Let A n = P j ∈Z P n k=1 b k − j 2 . If A n = on, then σ 2 n converges to σ 2 = A 2 E ǫ 2 . If moreover ∞ X n=1 1 n 2 −p2 E 1 n n X j=1 ǫ j 2 F − Eǫ 2 p 2 ∞ , 4.3 then we have 985 1. If A n = O1, then ζ 1 P n −12 S n , G σ 2 = On −12 logn, for p = 3, 2. If A n = On r+2−pr , then ζ r P n −12 S n , G σ 2 = On 1 −p2 , for r ∈ [p − 2, 1] and p 6= 3, 3. If A n = On 3 −p , then ζ r P n −12 S n , G σ 2 = On 1 −p2 , for r ∈]1, 2], 4. If A n = On 3 −p , then ζ r P n −12 S n , G σ 2 n = On 1 −p2 , for r ∈]2, p]. Remark 4.3. If the condition given by Heyde 1975 holds, that is ∞ X n=1 X k ≥n a k 2 ∞ and ∞ X n=1 X k ≤−n a k 2 ∞ , 4.4 then A n = O1, so that it satisfies all the conditions of items 1-4. Remark 4.4. Under the additional assumption P i ∈Z |a i | ∞, one has the bound A n ≤ 4B n , where B n = n X k=1 X j ≥k |a j | 2 + X j ≤−k |a j | 2 . 4.5 Proof of Theorem 4.1. We start with the following decomposition: S n = A n X j=1 ǫ j + ∞ X j= −∞ n X k=1 b k − j ǫ j . 4.6 Let R n = P ∞ j= −∞ P n k=1 b k − j ǫ j . Since kR n k 2 2 = A n kǫ k 2 2 and since |σ n − σ| ≤ n −12 kR n k 2 , the fact that A n = on implies that σ n converges to σ. We now give an upper bound for kR n k p . From Burkholder’s inequality, there exists a constant C such that kR n k p ≤ C n ∞ X j= −∞ n X k=1 b k − j 2 ǫ 2 j p 2 o 1 2 ≤ Ckǫ k p p A n . 4.7 According to Remark 2.1, since 4.3 holds, the two conditions 2.1 and 2.2 of Theorem 2.1 are satisfied by the martingale M n = A P n k=1 ǫ k . To conclude the proof, we use Lemma 5.2 given in Section 5.2, with the upper bound 4.7. ƒ Proof of Remarks 4.3 and 4.4. To prove Remark 4.3, note first that A n = n X j=1 − j X l= −∞ a l + ∞ X l=n+1 − j a l 2 + ∞ X i=1 n+i −1 X l=i a l 2 + ∞ X i=1 −i X l= −i−n+1 a l 2 . It follows easily that A n = O1 under 4.4. To prove the bound 4.5, note first that A n ≤ 3B n + ∞ X i=n+1 n+i −1 X l=i |a l | 2 + ∞ X i=n+1 −i X l= −i−n+1 |a l | 2 . 986 Let T i = P ∞ l=i |a l | and Q i = P −i l= −∞ |a l |. We have that ∞ X i=n+1 n+i −1 X l=i |a l | 2 ≤ T n+1 ∞ X i=n+1 T i − T n+i ≤ nT 2 n+1 ∞ X i=n+1 −i X l= −i−n+1 |a l | 2 ≤ Q n+1 ∞ X i=n+1 Q i − Q n+i ≤ nQ 2 n+1 . Since nT 2 n+1 + Q 2 n+1 ≤ B n , 4.5 follows. ƒ In the next result, we shall focus on functions of real-valued linear processes X k = h X i ∈Z a i ǫ k −i − E h X i ∈Z a i ǫ k −i , 4.8 where ǫ i i ∈Z is a sequence of iid random variables. Denote by w h ., M the modulus of continuity of the function h on the interval [ −M, M], that is w h t, M = sup{|hx − h y|, |x − y| ≤ t, |x| ≤ M, | y| ≤ M} . Theorem 4.2. Let a i i ∈Z be a sequence of real numbers in ℓ 2 and ǫ i i ∈Z be a sequence of iid random variables in L 2 . Let X k be defined as in 4.8 and σ 2 n = n −1 E S 2 n . Assume that h is γ-Hölder on any compact set, with w h t, M ≤ C t γ M α , for some C 0, γ ∈]0, 1] and α ≥ 0. If for some p ∈]2, 3], E |ǫ | 2 ∨α+γp ∞ and X i ≥1 i p 2−1 X | j|≥i a 2 j γ2 ∞, 4.9 then the series P k ∈Z CovX , X k converges to some nonnegative σ 2 , and 1. ζ 1 P n −12 S n , G σ 2 = On −12 log n, for p = 3, 2. ζ r P n −12 S n , G σ 2 = On 1 −p2 for r ∈ [p − 2, 2] and r, p 6= 1, 3, 3. ζ r P n −12 S n , G σ 2 n = On 1 −p2 for r ∈]2, p]. Proof of Theorem 4.2. Theorem 4.2 is a consequence of the following proposition: Proposition 4.2. Let a i i ∈Z , ǫ i i ∈Z and X i i ∈Z be as in Theorem 4.2. Let ǫ ′ i i ∈Z be an independent copy of ǫ i i ∈Z . Let V = P i ∈Z a i ǫ −i and M 1,i = |V | ∨ X j i a j ǫ − j + X j ≥i a j ǫ ′ − j and M 2,i = |V | ∨ X j i a j ǫ ′ − j + X j ≥i a j ǫ − j . If for some p ∈]2, 3], X i ≥1 i p 2−1 w h X j ≥i a j ǫ − j , M 1,i p ∞ and X i ≥1 i p 2−1 w h X j −i a j ǫ − j , M 2, −i p ∞, 4.10 then the conclusions of Theorem 4.2 hold. 987 To prove Theorem 4.2, it remains to check 4.10. We only check the first condition. Since w h t, M ≤ C t γ M α and the random variables ǫ i are iid, we have w h X j ≥i a j ǫ − j , M 1,i p ≤ C X j ≥i a j ǫ − j γ |V | α p + C X j ≥i a j ǫ − j γ p k|V | α k p , so that w h X j ≥i a j ǫ − j , M 1,i p ≤ C 2 α X j ≥i a j ǫ − j α+γ p + X j ≥i a j ǫ − j γ p k|V | α k p + 2 α X j i a j ǫ − j α p . From Burkholder’s inequality, for any β 0, X j ≥i a j ǫ − j β p = X j ≥i a j ǫ − j β β p ≤ K X j ≥i a 2 j β2 kǫ k β 2 ∨β p . Applying this inequality with β = γ or β = α + γ, we infer that the first part of 4.10 holds under 4.9. The second part can be handled in the same way. ƒ Proof of Proposition 4.2. Let F i = σǫ k , k ≤ i. We shall first prove that the condition 3.2 of Theorem 3.1 holds. We write kES 2 n |F − ES 2 n k p 2 ≤ 2 n X i=1 n −i X k=0 kEX i X k+i |F − EX i X k+i k p 2 ≤ 4 n X i=1 n X k=i kEX i X k+i |F k p 2 + 2 n X i=1 i X k=1 kEX i X k+i |F − EX i X k+i k p 2 . We first control the second term. Let ǫ ′ be an independent copy of ǫ, and denote by E ǫ · the conditional expectation with respect to ǫ. Define Y i = X j i a j ǫ i − j , Y ′ i = X j i a j ǫ ′ i − j , Z i = X j ≥i a j ǫ i − j , and Z ′ i = X j ≥i a j ǫ ′ i − j . Taking F ℓ = σǫ i , i ≤ ℓ, and setting h = h − Eh P i ∈Z a i ǫ i , we have kEX i X k+i |F − EX i X k+i k p 2 = E ǫ h Y ′ i + Z i h Y ′ k+i + Z k+i − E ǫ h Y ′ i + Z ′ i h Y ′ k+i + Z ′ k+i p 2 . Applying first the triangle inequality, and next Hölder’s inequality, we get that kEX i X k+i |F − EX i X k+i k p 2 ≤ kh Y ′ k+i + Z k+i k p kh Y ′ i + Z i − h Y ′ i + Z ′ i k p + kh Y ′ i + Z ′ i k p kh Y ′ k+i + Z k+i − h Y ′ k+i + Z ′ k+i k p . 988 Let m 1,i = |Y ′ i + Z i | ∨ |Y ′ i + Z ′ i |. Since w h t, M = w h t, M , it follows that kEX i X k+i |F − EX i X k+i k p 2 ≤ kh Y ′ k+i + Z k+i k p w h X j ≥i a j ǫ i − j − ǫ ′ i − j , m 1,i p + kh Y ′ i + Z ′ i k p w h X j ≥k+i a j ǫ k+i − j − ǫ ′ k+i − j , m 1,k+i p . By subadditivity, we obtain that w h X j ≥i a j ǫ i − j − ǫ ′ i − j , m 1,i p ≤ w h X j ≥i a j ǫ i − j , m 1,i p + w h X j ≥i a j ǫ ′ i − j , m 1,i p . Since the three couples P j ≥i a j ǫ i − j , m 1,i , P j ≥i a j ǫ ′ i − j , m 1,i and P j ≥i a j ǫ − j , M 1,i are identically distributed, it follows that w h X j ≥i a j ǫ i − j − ǫ ′ i − j , m 1,i p ≤ 2 w h X j ≥i a j ǫ − j , M 1,i p . In the same way w h X j ≥k+i a j ǫ k+i − j − ǫ ′ k+i − j , m 1,k+i p ≤ 2 w h X j ≥k+i a j ǫ − j , M 1,k+i p . Consequently X n ≥1 1 n 3 −p2 n X i=1 i X k=1 kEX i X k+i |F − EX i X k+i k p 2 ∞ provided that the first condition in 4.10 holds. We turn now to the control of P n i=1 P n k=i kEX i X k+i |F k p 2 . We first write that kEX i X k+i |F k p 2 ≤ kE X i − EX i |F i+[k 2] X k+i |F k p 2 + kE EX i |F i+[k 2] X k+i |F k p 2 ≤ kX k p kX i − EX i |F i+[k 2] k p + kX k p kEX k+i |F i+[k 2] k p . Let bk = k − [k2]. Since kEX k+i |F i+[k 2] k p = kEX bk |F k p , we have that kEX k+i |F i+[k 2] k p = E ǫ h X j bk a j ǫ ′ bk − j + X j ≥bk a j ǫ bk − j − h X j bk a j ǫ ′ bk − j + X j ≥bk a j ǫ ′ bk − j p . Using the same arguments as before, we get that kEX k+i |F i+[k 2] k p = kEX bk |F k p ≤ 2 w h X j ≥bk a j ǫ − j , M 1,bk p . 4.11 In the same way, X i − EX i |F i+[k 2] p = E ǫ h X j −[k2] a j ǫ i − j + X j ≥−[k2] a j ǫ i − j − h X j −[k2] a j ǫ ′ i − j + X j ≥−[k2] a j ǫ i − j p . 989 Let m 2,i,k = X j ∈Z a i ǫ i − j ∨ X j −[k2] a j ǫ ′ i − j + X j ≥−[k2] a j ǫ i − j . Using again the subbadditivity of t → w h t, M , and the fact that P j −[k2] a j ǫ i − j , m 2,i,k , P j −[k2] a j ǫ ′ i − j , m 2,i,k and P j −[k2] a j ǫ − j , M 2, −[k2] are identically distributed, we obtain that X i − EX i |F i+[k 2] p = X −[k2] − EX −[k2] |F p ≤ 2 w h X j −[k2] a j ǫ − j , M 2, −[k2] p . 4.12 Consequently X n ≥1 1 n 3 −p2 n X i=1 n X k=i kEX i X k+i |F k p 2 ∞ provided that 4.10 holds. This completes the proof of 3.2. Using the bounds 4.11 and 4.12 taking bk = n in 4.11 and [k 2] = n in 4.12, we see that the condition 3.1 of Theorem 3.1 and also the condition 3.4 of Theorem 3.2 in the case p = 3 holds under 4.10. ƒ

4.3 Functions of