On the rate of convergence in the martingale central limit theorem

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1 Bernoulli 192), 2013, DOI: /12-BEJ417 arxiv: v2 [math.pr] 21 Mar 2013 On the rate of convergence in the martingale central limit theorem JEAN-CHRISTOPHE MOURRAT Ecole olytechnique fédérale de Lausanne, institut de mathématiques, station 8, 1015 Lausanne, Switzerland Consider a discrete-time martingale, and let V 2 be its normalized quadratic variation. As V 2 aroaches 1, and rovided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale aroaches the Gaussian distribution. For any 1, Ann. Probab ) ) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say A +B, where u to a constant, A = V 2 1 /2+1). Here we discuss the otimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, Ann. Probab ) ) sketched a strategy to rove otimality for =1. Here we extend this strategy to any 1, thereby justifying the otimality of the term A. As a necessary ste, we also rovide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that imroves on the term B, generalizing another result of Ann. Probab ) ). Keywords: central limit theorem; martingale; rate of convergence 1. Introduction Let X=X 1,...,X n ) be a sequence of square-integrable random variables such that for any i, X i satisfies E[X i F i 1 ]=0, where F i is the σ-algebra generated by X 1,...,X i ). In other words, X is a square-integrable martingale difference sequence. Following the notation of [1], we write M n for the set of all such sequences of length n, and introduce s 2 X) = E[Xi 2 ], V 2 X) =s 2 X) SX)= X i. E[Xi F 2 i 1 ], This is an electronic rerint of the original article ublished by the ISI/BS in Bernoulli, 2013, Vol. 19, No. 2, This rerint differs from the original in agination and tyograhic detail c 2013 ISI/BS

2 2 J.-C. Mourrat V 2 X) can be called the normalized quadratic variation of X. Let X n ) n N be such that for any n, X n M n. It is well known see, e.g., [2], Section 7.7.a) that if V 2 X n ) rob.) n ) and some Lindeberg condition is satisfied, then the rescaled sum SX n )/sx n ) converges in distribution to a standard Gaussian random variable, that is, t R, P[SX n )/sx n ) t] Φt), 1.2) n + where Φt)=2π) 1/2 t e x2 /2 dx. We are interested in bounds on the seed of convergence in this central limit theorem. Several results have been obtained under a variety of additional assumtions. One natural way to strengthen the convergence in robability 1.1) is to change it for a convergence in L for some [1,+ ]. Indeed, quantitative estimates in terms of V 2 1 seem articularly convenient when the aim is to aly the result to ractical situations. We write and DX)=su P[SX)/sX) t] Φt) t R X = max 1 i n X i [1,+ ]). [4] roved the following result. Theorem 1.1 [4]). Let [1,+ ). There exists a constant C >0 such that for any n 1 and any X M n, DX) C V 2 X) 1 +s 2 X) In [7], Theorem 1.1 is generalized to the following. X i 2) 1/2+1). 1.3) Theorem 1.2 [7]). Let [1,+ ] and [1,+ ). There exists C, >0 such that for any n 1 and any X M n, DX) C, [ V 2 X) 1 /2+1) + s 2 X) X i 2 2 ) 1/2 +1)]. 1.4) Here /2+1) = 1/2 for = +. In fact, a stronger, nonuniform bound is given; see [7], Theorem 2.2 or, equivalently, [8]), for details.

3 Rate of convergence in the martingale central limit theorem 3 The main question addressed here concerns the otimality of the term V 2 X) 1 /2+1) aearing in the right-hand side of 1.3) or 1.4). About this, [4] constructed a sequence of elements X n M n such that sx n ) n, DX n ) log 1/2 n), V 2 X) 1 s 2 X) X 2 2 s 2 X) n X i 2 2 log 2+1)/2 n), wherewewritea n b n ifthereexistsc >0,suchthata n /C b n Ca n forallsufficiently large n. This examle demonstrates that the exonent 1/2 + 1) aearing on the outer bracket of the right-hand side of 1.3) cannot be imroved. But because the two terms of the right-hand side of 1.3) are of the same order, no conclusions can be drawn about the otimality of the term V 2 X) 1 /2+1) alone. Most imortantly, it is rather disaointing that in the examle, X 2 2 and n X i 2 2 are of the same order, if the tyical martingales that one is interested in have increments of roughly the same order. Using a similar construction, but also imosing the condition that V 2 X)=1 a.s., [7], Examle 2.4, roved the otimality of the exonent 1/2 +1) aearing in the second term of the sum in the right-hand side of 1.4). However, the author did not discuss the otimality of the first term V 2 X) 1 /2+1). For 1 2, Theorem 1.1 was in fact already roved by [6]. In [5], Section 3.6, the authors could show only that the bound on DX) can be no better than V 2 X) 1 1/2 1. The roof of Theorem 1.1 given by [4] is insired by a method introduced by [1], who roved the following results. Theorem 1.3 [1]). Let γ 0,+ ). There exists a constant C γ >0 such that for any n 2 and any X M n satisfying X γ and V 2 X)=1 a.s., DX) C γ nlogn) s 3 X). Tyically, sx) is of order n when X M n. Under such circumstances, Theorem 1.3 thus gives a bound of order logn)/ n. Moreover, [1] rovided an examle of a sequence of elements X n M n satisfying the conditions of Theorem 1.3, such that s 2 X n )=n and for which lim su nlog 1 n)dx n )>0, n + and so the result is otimal. Relaxing the condition that V 2 X)=1 a.s., [1] then showed the following result. Corollary 1.4 [1]). Let γ 0,+ ). There exists a constant C γ >0 such that for any n 2 and any X M n satisfying X γ, DX) C γ [ nlogn) s 3 X) +min V 2 X) 1 1/3 1, V 2 X) 1 1/2 ) ]. 1.5)

4 4 J.-C. Mourrat See [7], Theorem 3.2, for a nonuniform version of this result. A strategy was sketched by [1] to rove that the bound V 2 X) 1 1/3 1 is indeed otimal, even on the restricted class considered by Corollary 1.4 of martingales with bounded increments. This examle rovides a satisfactory answer to our question of otimality for =1. The aim of the resent aer is to generalize Corollary 1.4 and the otimality result to any [1,+ ). We begin by roving the following general result. Theorem 1.5. Let [1,+ ) and γ 0,+ ). There exists a constant C,γ >0 such that for any n 2 and any X M n satisfying X γ, [ ] nlogn) DX) C,γ s 3 X) + V 2 X) 1 +s 2 X)) 1/2+1). 1.6) Notethat,somewhatsurrisingly,the term s 2 X) n X i 2 2 aearingininequality 1.3) is no longer resent in 1.5), and is changed for the smaller s 2 X) in 1.6). Finally, we justify the otimality of the term V 2 X) 1 /2+1) aearing in the right-hand side of 1.6). Theorem 1.6. Let [1,+ ) and α 1/2,1). There exists a sequence of elements X n M n such that X n 2, sx n ) n, V 2 X n ) 1 /2+1) =On α 1)/2 ), limsu n + n 1 α)/2 DX n )>0. Our strategy for roving Theorem 1.6 builds on the aroach sketched by [1] for the case where =1. Interestingly, Theorem 1.5 is used in the roof of Theorem 1.6. The question of otimality of the term V 2 X) 1 /2+1), now settled by Theorem 1.6, arises naturally in the roblem of showing a quantitative central limit theorem for the random walk among random conductances on Z d [9]. There, the random walk is aroximated by a martingale. The martingale increments are stationary and almost bounded for d 3, in the sense that they have bounded L norm for every < +. Roughly seaking, for d 3, the variance of the rescaled quadratic variation u to time t decays as t 1. This bound is otimal and leads to a Berry Esseen bound of order t 1/5. Thus Theorem 1.6 demonstrates that a better exonent of decay than 1/5 cannot be obtained when relying solely on information about the variance of the quadratic variation. Theorem 1.5 is roved in Section 2, and Theorem 1.6 is roved in Section Proof of Theorem 1.5 The roof of Theorem 1.5 is essentially similar to the roof of Corollary 1.4 given by [1], with the additional ingredient of a Burkholder inequality. Let X =X 1,...,X n ) M n be such that X γ. The idea robably first suggested by [3]) is to augment the

5 Rate of convergence in the martingale central limit theorem 5 sequence to some ˆX M 2n such that V 2 ˆX)=1 a.s., while reserving the roerty that ˆX γ, and aly Theorem 1.3 to this enlarged sequence. Let τ =su { k n: } k E[Xi 2 F i 1] s 2 X). For i τ, we define ˆX i =X i. Let r be the largest integer not exceeding s 2 X) τ E[X2 i F i 1] γ 2. As X γ, clearly r n. Conditional on F τ and for 1 i r, we let ˆX i be indeendent random variables such that P[ ˆX τ+i =±γ]=1/2. If τ +r<2n, then we let ˆX τ+r+1 be such that P [ ˆX τ+r+1 =± s 2 X) ) τ E[Xi 2 F i 1] rγ 1/2]= 2 1 2, with the sign determined indeendent of everything else. Finally, if τ +r+1<2n, then we let ˆX τ+r+i =0 for i 2. Possibly enlarging the σ-fields, we can assume that ˆX i is F i -measurable for i n, and define F i to be the σ-field generated by F n and ˆX n+1,..., ˆX n+i if i>n. By construction, we have 2 E[ ˆX τ i 2 F i 1]=s 2 X) E[Xi 2 F i 1], i=τ+1 which can be rewritten as 2 E[ ˆX 2 i F i 1]=s 2 X). Consequently, s 2 ˆX) = s 2 X) and V 2 ˆX) = 1 a.s. The sequence ˆX thus satisfies the assumtions of Theorem 1.3, so DˆX) 4C γ nlogn) s 3 X). 2.1) For any x>0, we have [ ] [ SX) SX) P sx) t P sx) [ SˆX) P sx) t+x ] [ SX) SˆX) x +P sx) SX) SˆX) t, sx) ] + 1 [ SX) SˆX) x 2E sx) 2]. ] x 2.2)

6 6 J.-C. Mourrat Due to 2.1), the first term in the right-hand side of 2.2) is smaller than Φt+x)+4C γ nlogn) s 3 X) Φt)+ x 2π +4C γ nlogn) s 3 X). 2.3) To control the second term, first note that SX) SˆX)= 2 i=τ+1 X i ˆX i ), 2.4) where we ut X i =0 for i>n. Given that τ +1 is a stoing time, conditional on τ, the X i ˆX i ) i τ+2 stillformsamartingaledifferencesequence.thuswecanuseburkholder s inequality see, e.g., [5], Theorem 2.11), which states that [ 1 C E E 2 i=τ+2 [ 2 X i ˆX i ) i=τ+2 2] E[X i ˆX i ) 2 F i 1 ] ) ] [ +E max X i ˆX i 2], τ+2 i 2n 2.5) and we can safely discard the summand indexed by τ +1 aearing in 2.4), which is uniformly bounded. The maximum on the right-hand side of 2.5) is also bounded by 2γ 2. As for the other term, with X i and ˆX i as orthogonal random variables, we have 2 E[X i ˆX i ) 2 F i 1 ]= i=τ+1 i=τ+1 i=τ+1 2 E[X 2 i F i 1 ]+ 2 E[ ˆX 2 i F i 1 ] τ =s 2 X)V 2 X)+s 2 X) 2 E[Xi 2 F i 1]. }{{} 2.6) Now, if τ =n, then by definition the sum underbraced above is s 2 X)V 2 X). Otherwise, τ+1 E[X2 i F i 1] exceeds s 2 X), but as the increments are bounded, the sum underbraced is necessarily larger than s 2 X) γ 2. In any case, we thus have τ E[Xi F 2 i 1 ] mins 2 X)V 2 X),s 2 X) γ 2 ). Consequently, from 2.6), we obtain that 2 i=τ+1 E[X i ˆX i ) 2 F i 1 ] s 2 X)V 2 X) s 2 X) +2γ 2.

7 Rate of convergence in the martingale central limit theorem 7 Combining this with equations 2.5), 2.4), 2.3) and 2.2), we finally obtain that [ ] SX) P sx) t nlogn) Φt) 4C γ s 3 X) + x + C ) V 2 γ2 2π x 2 X) 1 + s 2. X) Otimizing this over x>0 leads to the correct estimate. The lower bound is obtained in the same way. 3. Proof of Theorem 1.6 Let 1 and α 1/2,1) be fixed. We let X ni ) 1 i n n α be indeendent random variables with P[X ni = ±1]=1/2. The subsequent X ni ) n nα <i n are defined recursively. Let λ ni = n i+κ 2 n, where κ n = n 1/4 in fact, any n β with 1 α<2β <α would be fine). Assuming that X n,1,...,x have been defined, we write F for the σ-algebra that they generate, and let i 1 S = X nj. For any i such that n n α <i n, we construct X ni such that δ 3/2 +δ if S 3/2 [λ ni,2λ ni ], P[X ni F ]= δ 1/2 +δ if S 1/2 [ 2λ ni, λ ni ], δ 1 +δ 1 otherwise, j=1 3.1) where δ x is the Dirac mass at oint x. Here S ni ) i n can be viewed as an inhomogeneous Markov chain. We write X n =X n1,...,x nn ) and X ni =X n1,...,x ni ) for any i n. Let Proosition 3.1. Uniformly over n, δi)=sudx ni ). 3.2) n i V 2 X ni ) 1 =Oi α 1)1+1/2) ) i + ) 3.3) and δi)=oi α 1)/2 ) i + ). 3.4) The roof goes as follows. First, we bound V 2 X ni ) 1 in terms of δj)) j i in Lemma 3.2. This gives an inequality on the sequence δi)) i N through Theorem 1.5, from which we deduce 3.4), and then 3.3).

8 8 J.-C. Mourrat Lemma 3.2. Let K i =max j i δj)j 1 α)/2. For any n and i, the following inequalities hold: E[Xni] if i n nα, 2δi 1) if n n α 3.5) <i n, s 2 X ni ) i 0 if i n nα, Ci 3α 1)/2 K i Ci α if n n α 3.6) <i n, V 2 X ni ) 1 0 if i n nα, Ci α 1)1+1/2) 1+K i ) 1/ +Ci 3α 3)/2 3.7) K i otherwise. Proof. Inequality 3.5) is obvious for i n n α. Otherwise, from the definition 3.1), we know that where we write E[X 2 ni ]= P[S I + ni ] 1 2 P[S I ni ], I + ni =[λ ni,2λ ni ] and I ni =[ 2λ ni, λ ni ]. 3.8) The random variable S /sx ) is aroximately Gaussian, u to an error controlled by δi 1). More recisely, P[S I + ni I ] dφ 2δi 1). + ni /sx) We obtain 3.5) using the fact that dφ= dφ. I + ni /sx) I ni /sx) As a by-roduct, we also learn that 0 s 2 if i n n α, X ni ) i 2 δj 1) if n n α <i n. n n α <j i Recalling that α<1, we obtain 3.6), noting that for n n α <i n, δj 1) n α n n α ) α 1)/2 K i. n n α <j i In articular, it follows that s 2 X ni )=i1+o1)). 3.9)

9 Rate of convergence in the martingale central limit theorem 9 Turning now to 3.7), V 2 X ni ) 1 is clearly equal to 0 for i n n α, so let us assume the contrary. We have: V 2 X ni ) 1 =s 2 i X ni ) E[Xnj 2 F n,j 1] s 2 X ni ) 1 s 2 X ni ) 1 2s 2 X ni ) j=1 i j=1 E[X 2 nj F n,j 1] 1 + s2 X ni ) i s 2 X ni ) n n α <j i P[S n,j 1 I + nj I nj ])1/ + s2 X ni ) i s 2. X ni ) 3.10) We consider the two terms in 3.10) searately. First, by the definition of δ, we know that P[S n,j 1 I + nj I nj I ] dφ 2δj 1). + nj I nj )/sxn,j 1) Equation 3.9) imlies that, uniformly over j >n n α, I + nj I nj )/sxn,j 1) dφ=2π) 1/2 2λ nj sx n,j 1 ) 1+o1)) Cnα 1)/2, and so the first term of 3.10) is bounded by C i n n α <j i C i n n α <j i n α 1)/2 +2δj 1)) 1/ n α 1)/2 +2n n α ) α 1)/2 K i ) 1/ 3.11) Ci α 1)1+1/2) 1+K i ) 1/. The second term in 3.10) is controlled by 3.6), and we obtain inequality 3.7). Proof of Proosition 3.1. Alying Theorem 1.5 with the information given by Lemma 3.2, we obtain that, u to a multilicative constant that does not deend on n and i n, DX ni ) is bounded by logi) i +i α 1)/2 1+K i ) 1/2+1) +i 31 α)/4+2) K /2+1) i +i /2+1). 3.12) The first term can be disregarded, because it is dominated by i /2+1). Also note that as 1, we have 31 α) α, 2

10 10 J.-C. Mourrat and as α>1/2>1/2+1), we also have α. 2 Multilying 3.12) by i 1 α)/2, we thus obtain K i C1+K i ) 1/2+1) +CK /2+1) i, where we recall that the constant C does not deend on i. Observing that the set {x 0: x C1+x) 1/2+1) +Cx /2+1) } is bounded, we obtain that K i is a bounded sequence, so 3.4) is roved. The relation 3.3) then follows from 3.4) and 3.7). Proosition 3.3. We have lim sui 1 α)/2 δi)>0. i + Proof. Our aim is to contradict, by reductio ad absurdum, the claim that δi)=oi α 1)/2 ) i + ). 3.13) Let Z 1,...,Z n be indeendent standard Gaussian random variables, and let ξ n be an indeendent centered Gaussian random variable with variance κ 2 n, all indeendent of X n. Assuming 3.13), we contradict the fact that DX n )=on α 1)/2 ). 3.14) Let W ni = n j=i+1 Z j +ξ n. Noting that n 1/2 n j=1 Z j is a standard Gaussian random variable, and with the aid of [1], Lemma 1, we learn that P[W n0 0] 1 2 C κ n n and, similarly, P[S nn+ξ n 0] 1 2 C DX n )+ κ ) n. sx n ) Combining these two observations with 3.6), we thus obtain that P[S nn +ξ n 0] P[W n0 0] C DX n )+ κ n ). 3.15) n As κ n =n 1/4 and α>1/2, we know that κ n / n=on α 1)/2 ). We decomose the lefthand side of 3.15) as P[S +X ni +W ni 0] P[S +Z i +W ni 0].

11 Rate of convergence in the martingale central limit theorem 11 The random variable W ni is Gaussian with variance λ 2 ni =n i+κ2 n and is indeendent of X n ; thus the sum can be rewritten as [ E Φ S ) +X ni Φ S )] +Z i. 3.16) λ ni λ ni Let ϕx)=2π) 1/2 e x2 /2. We can relace Φ S ) +X ni λ ni 3.17) by its Taylor exansion, Φ S λ ni ) X ni ϕ S ) + X2 ni λ ni λ ni 2λ 2 ϕ S ), 3.18) ni λ ni u to an error bounded by X ni 3 6λ 3 ϕ. 3.19) ni Ste 1. We show that the error term 3.19), after integration and summation over i, is on α 1)/2 ). Because X ni is uniformly bounded, it suffices to show that The foregoing sum equals 1 λ 3 ni =on α 1)/2 ). 3.20) 1 κ 2 n +n)/n n i+κ 2 n 1/2 x 3/2 dx=oκ 1 3/2 n) κ 2 n 1)/n n ). Because we defined κ n to be n 1/4 and α>1/2, equation 3.20) is roved. Ste 2. Forthesecondartofthesummandsin3.16),thesameholdswith X ni relaced by Z i and, similarly, E[ Z i 3 ] λ 3 =on α 1)/2 ). 3.21) ni Ste 3. Combining the results of the two revious stes, we know that u to a term of order on α 1)/2 ), the sum in 3.16) can be relaced by [ Zi X ni E ϕ S ) + X2 ni Z2 i λ ni λ ni 2λ 2 ϕ S )]. ni λ ni

12 12 J.-C. Mourrat Conditional on S, both Z i and X ni are centered random variables; thus the first art of the summands vanishes, and only the following remains: [ X 2 E ni Zi 2 2λ 2 ϕ S )] = ni λ ni [ E[X 2 E ni 1 S ] 2λ 2 ϕ S )]. 3.22) ni λ ni Fromthe definition of X ni, we learnthat E[Xni 2 1 S ] is 0 if i n n α but otherwise equals 1/2 if S I ni +, 1/2 if S I ni, 0 otherwise, where I + ni and I ni are as defined in 3.8). Consequently, it is clear that the contribution of each summand in the right-hand side of 3.22) is ositive. Moreover, for i>n n α and in the case where S I ni I+ ni, we have E[Xni 2 1 S ]ϕ S ) 1 λ ni 2 inf [1,2] ϕ >0. Letusassumetemorarilythat,uniformlyovernandisuchthat n n α <i n n α )/2, we have P[S I ni I+ ni ] Cλ ni n. 3.23) Then the sum in the right-hand side of 3.22) is, u to a constant, bounded from below by 1 Cn α 1 λ ni n n α/2 n =Cnα 1)/2. n n α <i n n α )/2 This contradicts 3.14) via inequality 3.15), and thus comletes the roof of the roosition. Ste 4. There remains to show 3.23), for n n α <i n n α )/2. We have P[S I + ni I ] dφ 2δi 1). + ni /sx) Using inequality 3.6), it follows that dφ C λ ni. I + ni n /sx) Because we choose i inside [n n α,n n α )/2], λ ni is larger than Cn α/2, whereas δi 1)=oi α 1)/2 ) by assumtion 3.13). This roves 3.23).

13 Rate of convergence in the martingale central limit theorem 13 Remark. To match the examle roosed by [1], α=1/3 and κ n =1 should be used in the definition of the sequences X n ). In this case, Proositions 3.1 and 3.3 still hold. Although the roof of Proosition 3.1 can be ket unchanged, Proosition 3.3 requires a moresubtleanalysis.first, ξ n ofvarianceκ 2 n 1must bechosen,whichrequireschanging the λ ni aearing in 3.16) by, say, λ ni = n i+κ 2 n. The sequence κ2 n should grow to infinity with n, while remaining on α ). In Ste 1, bounding the difference between 3.17) and 3.18) by 3.19) is too crude. Instead, it can be bounded by C Ψ S ), λ ni λ 3 ni where Ψx)=su y 1 ϕ x+y). One can then aeal to [1], Lemma 2, and get through this ste, using the fact that κ n tends to infinity. Ste 2 is similar, but with some additional care required because Z i is unbounded. The rest of the roof then alies, taking note of the discreancy between λ ni and λ ni when necessary. Acknowledgements I would like to thank an anonymous referee for his careful review, and for his mention of the reference [7], which I was not aware of. References [1] Bolthausen, E. 1982). Exact convergence rates in some martingale central limit theorems. Ann. Probab MR [2] Durrett, R. 1996). Probability: Theory and Examles, 2nd ed. Belmont, CA: Duxbury Press. MR [3] Dvoretzky, A. 1972). Asymtotic normality for sums of deendent random variables. In Proceedings of the Sixth Berkeley Symosium on Mathematical Statistics and Probability Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory Berkeley, CA: Univ. California Press. MR [4] Haeusler, E. 1988). On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab MR [5] Hall, P. and Heyde, C.C. 1980). Martingale Limit Theory and Its Alication. New York: Academic Press. MR [6] Heyde, C.C. and Brown, B.M. 1970). On the dearture from normality of a certain class of martingales. Ann. Math. Statist MR [7] Joos, K. 1989). Abschätzungen der Konvergenzgeschwindigkeit in asymtotischen Verteilungsaussagen für Martingale. Ph.D. thesis, Ludwig-Maximilians-Universität München. [8] Joos, K. 1993). Nonuniform convergence rates in the central limit theorem for martingales. Studia Sci. Math. Hungar MR [9] Mourrat, J.C. A quantitative central limit theorem for the random walk among random conductances. Prerint. Available at arxiv: v1. Received June 2011 and revised October 2011