DSpace at VNU: L (1) bounds for some martingale central limit theorems

Lithuanian Mathematical Journal, Vol 54, No 1, January, 2014, pp 48–60 L1 bounds for some martingale central limit theorems Le Van Dung a,1 , Ta Cong Son b,2 , and Nguyen Duy Tien b,1 a Faculty of Mathematics, Da Nang University of Education, 459 Ton Duc Thang, Da Nang, Viet Nam b Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Viet Nam (e-mail: lvdung@ud.edu.vn; congson82@hus.edu.vn; nduytien2006@gmail.com) Received September 24, 2013; revised January 7, 2014 Abstract The aim of this paper is to extend the results in [E Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann Probab., 10(3):672–688, 1982] and [J.C Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013] to the L1 -distance between distributions of normalized partial sums for martingale-difference sequences and the standard normal distribution MSC: 60F05, 60G42 Keywords: mean central limit theorems, rates of convergence, martingale Introduction and statements of results Let X1 , X2 , , Xn be a sequence of real-valued random variables with mean √ zero and finite variance σ Put S := X1 +X2 +· · ·+Xn Denote by Fn the distribution functions of S/σ n, and let Φ be the standard normal distribution function The classical central limit theorem confirms that if X1 , X2 , , Xn are independent and identically distributed, then Fn (x) converges to Φ(x) as n → ∞ for all x ∈ R In 1954, Agnew [1] showed that the convergence also holds in Lp for p > 1/2 The convergence in the case of p = is called the mean central limit theorem The rate of convergence in the mean central limit theorem was also studied by Esseen [7], who showed that Fn − Φ = O n−1/2 as n → ∞ Recently, Sunklodas [12, 13] has extended this result to independent nonidentically distributed random variables and ϕ-mixing random variables by using the Bentkus approach [2] Let X = (X1 , , Xn ) be a square-integrable martingale-difference sequence of real-valued random variables with respect to the σ -fields Fj = σ(X1 , , Xj−1 ), j = 2, 3, , n + 1; F1 = {∅, Ω} Let Mn denote the class of all such sequences of length n If X ∈ Mn , we write σj2 = E(Xj2 | Fj−1 ), σ 2j = E(Xj2 ), S = S(X) = nj=1 Xj , s2 = s2 (X) = nj=1 σ 2j , V = V (X) = nj=1 σj2 /s2 (X), and The research of the author has been partially supported by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED), grant No 101.03-2012.17 The research of the author has been partially supported by project TN-13-01 0363-1672/14/5401-0048 c 2014 Springer Science+Business Media New York 48 L1 bounds for some martingale CLT 49 X p = max1 j n Xj p for p ∞ We denote by N a standard normal random variable; the distribution function and the density function of N are denoted by Φ(x) and ϕ(x), respectively If X ∈ Mn , V (X) → in probability, and some Lindeberg-type condition is satisfied, then lim P n→∞ S(X) s(X) x = Φ(x) for all x ∈ R For bounds of the convergence rate in this central limit theorem, the following results were shown by Bolthausen [4] Theorem (See [4].) Let < α β < ∞, < γ < ∞ There exists a constant < Cα,β,γ < ∞ such that, for any n and any X ∈ Mn satisfying σj2 = σ 2j a.s., α σ 2j β for j n, and X γ , sup P x∈R S s x − Φ(x) Cα,β,γ n−1/4 Theorem (See [4].) Let γ ∈ (0; +∞) There exists a constant < Cγ < ∞, depending only on γ , such that, for any n and any X ∈ Mn satisfying X ∞ γ and V (X) = a.s., sup P x∈R S s x − Φ(x) Cγ n log n s3 Relaxing the condition that V = a.s., Bolthausen [4] also showed the following result Corollary (See [4].) Let γ ∈ (0; +∞) There exists a constant Cγ > such that, for any n X ∈ Mn satisfying X ∞ γ , sup P x∈R S s x − Φ(x) Cγ n log n + s3 1/3 , V2−1 V2−1 1/2 ∞ and any Mourrat [11] generalized Corollary and obtained the optimality of the result to any p ∈ [1; +∞) Theorem (See [11].) Let p ∈ [1; +∞) and γ ∈ (0; +∞) There exists a constant Cp,γ > such that, for any n and any X ∈ Mn satisfying X ∞ γ , sup P x∈R S s x − Φ(x) Cp,γ n log n + s3 V2−1 p p + s−2p 1/(2p+1) The aim of this article is to extend these results to L1 -bounds in the mean central limit theorem for martingale-difference sequences Theorem Let < α β < ∞, < γ < ∞ If X γ , σj2 = σ 2j a.s., and α then there exists a constant C = C(α, β, γ) ∈ (0; ∞) such that FS/s − Φ Theorem Let < γ < ∞ If X such that ∞ β for j n, Cn−1/4 γ and V (X) = a.s., then there exists a constant < C < ∞ FS/s − Φ C We have the following corollary, similar to Corollary Lith Math J., 54(1):48–60, 2014 σ 2j γ n log n s3 50 L.V Dung, T.C Son, and N.D Tien Corollary Let < γ < ∞ and p > 1/2 If X depending only on p, such that FS/s − Φ γ n log n + s3 C γ , then there exists a positive constant C = C(p), ∞ V2−1 1/2 , ∞ E V2−1 p 1/2p The following corollary is an L1 -version of Theorem Corollary Let < γ < ∞ and p > 1/2 If X depending only on p, such that FS/s − Φ γ , then there exists a positive constant C = C(p), ∞ γ n log n p + E V − + s−2p s3 C 1/2p 1/3 Note that the term V − 1 appearing in Corollary is replaced by the smaller term (E|V − 1|p )1/2p in Corollary 2, and the term ( V − pp + s−2p )1/(2p+1) appearing in Theorem is replaced by the smaller term (E|V − 1|p + s−2p )1/2p in Corollary Auxiliary lemmas For two random variables X and Y with distribution functions FX and GY , respectively, applying the Kantorovich–Rubinstein theorem (see, e.g., [6, Thm 11.8.2]), we have that ∞ FX − G Y FX (x) − GY (x) dx = sup E f (X) − E f (Y ) , = f ∈Λ1 −∞ where Λ1 is the set of 1-Lipschitzian functions from R to R For more details, we refer the reader to [8] and [5] For functions f, g : R → R, their convolution f ∗ g is defined by ∞ f ∗ g(x) = f (x − y)g(y) dy −∞ We have the following lemmas p, q, r Lemma (See [3, p 205].) If have ∞, 1/p + 1/q = + 1/r, f ∈ Lp (R), and g ∈ Lq (R), then we f ∗g f r p g q Lemma Let X and η be real random variables Then, for any p > 1/2, we have FX − Φ FX+η − Φ + 2(2p + 1) E η 2p X Proof The conclusion is trivial in the case of E(η 2p | X) γ < ∞ For any a > 0, we have that ∞ ∞ FX − Φ P(X = −∞ t − a) − Φ(t − a) dt 1/2p ∞ = ∞ So, we assume that E(η 2p | X) (2.1) ∞ = L1 bounds for some martingale CLT 51 ∞ ∞ P(X t − a) − P(X + η t) dt + −∞ Φ(t) − Φ(t − a) dt −∞ + FX+η − Φ (2.2) First, we consider the first term on the right-hand side of (2.2) We have that P(X + η t) = E P(η t − X | X) E I(X = P(X t − a) − E I(X t − a)P(η t − X | X) t − a)P(η > t − X | X) , where E I(X t − a)P(η > t − X | X) γE (t − X)−2p I(X t − a) γE (t − X)−2p I(X t − a) , Therefore, P(X t − a) − P(X + η t) + which implies ∞ P(X t − a) − P(X + η t) + dt −∞ ∞ ∞ γE (t − X) −2p I(X (t − X)−2p I(X t − a) dt = γE −∞ t − a) dt −∞ ∞ (t − X)−2p dt = γE = (2p − 1) X+a γ a2p−1 (2.3) On the other hand, P(X + η t) = E I(X t − a)P(η + E I(t − a < X t − X | X) t + a)P(η + E I(X > t + a)P(η t − X | X) t − X | X) P(X t − a) + P(t − a < X t + a) + E I(X > t + a)P(η t − X | X) P(X t − a) + P(t − a < X t + a) + γE (t − X)−2p I(X > t + a) , which implies that P(X + η t) − P(X t − a) P(t − a < X Hence, ∞ P(X −∞ Lith Math J., 54(1):48–60, 2014 t − a) − P(X + η t) − dt t + a) + γE (t − X)−2p I(X > t + a) 52 L.V Dung, T.C Son, and N.D Tien ∞ ∞ P(t − a < X γE (t − X)−2p I(X > t + a) dt t + a) dt + −∞ −∞ ∞ X−a (t − X) = 2a + γE −2p I(X > t + a) dt (t − X)−2p dt = 2a + γE −∞ = 2a + (2p − 1) −∞ γ a2p−1 (2.4) Combining (2.3) and (2.4) yields ∞ P(X t − a) − P(X + η t) dt −∞ ∞ ∞ P(X = t − a) − P(X + η − t) P(X dt + −∞ t − a) − P(X + η t) + dt −∞ 2a + 2(2p − 1) γ a2p−1 (2.5) Next, we consider the second term on the right-hand side of (2.2) We have that ∞ ∞ Φ(t − a) − Φ(t) dt = −∞ Φ(t − a) − Φ(t) dt + −∞ Φ(t − a) − Φ(t) dt 0 ∞ aϕ(t) dt + −∞ aϕ(t − a) dt 2a (2.6) Combining (2.2), (2.3), and (2.6) yields FX − Φ FX+η − Φ 1 + (2p − 1) γ a2p−1 + 2a Taking a = γ 1/2p gives conclusion (2.1) of Lemma Lemma Let ψ be a function R → R with ψ E ψ(X) ∞ < ∞ and ψ ψ ∞ |FX ∞ < ∞ If X is a random variable, then − Φ|1 + ψ ∞ Proof It is clear that ∞ E ψ(X) − E ψ(N ) ∞ ψ(x) dFX (x) − = −∞ ∞ ψ(x) dΦ(x) = −∞ FX (x) − Φ(x) ψ (x) dx −∞ ∞ ψ ∞ −∞ FX (x) − Φ(x) dx = ψ ∞ FX − Φ L1 bounds for some martingale CLT 53 and E ψ(N ) ψ ∞ Proof of Theorem Let Z1 , Z2 , , Zn , η be independent normally distributed random variables with mean and E(Zj2 ) = σ 2j , n E(η ) = n1/2 Let Um = m−1 j=1 Xj /s and Z = j=1 Zj According to Lemma with p = 1, we have FS/s − Φ F(S+η)/s − Φ 1/2 E η2 s2 +C F(S+η)/s − F(Z+η)/s F(S+η)/s − F(Z+η)/s ∞ 1/4 + Cn +C E η2 s 1/2 ∞ (3.1) On the other hand, by a proof is similar to that of Theorem in [4] we get that S+η P s n Z +η t −P s t |Xm |3 ϕ λm s E m=1 n E + m=1 |Zm |3 ϕ λm s where θm , θm and λm = ( nj=m+1 σ 2j + n1/2 )/s ∞ Applying the Fubini theorem and noting that −∞ |ϕ (t)| dt ∞ S+η s P −∞ ∞ n |Xm |3 ϕ λ3m s3 E m=1 −∞ n ∞ m=1 −∞ ∞ n E m=1 −∞ |Xm |3 ϕ λ3m s3 ∞ n E + m=1 n m=1 −∞ |Xm |3 2E λ3m s3 t dt + t − Um Zm − θm λm λm s 2E m=1 |Zm |3 λ3m s3 Combining (3.1) and (3.2) yields The theorem is proved Lith Math J., 54(1):48–60, 2014 dt t − Um Zm − θm λm λm s n , 2, we have t − Um Xm − θm λm λm s |Zm |3 ϕ λ3m s3 FS/s − Φ t − Um Zm − θm λm λm s t − Um Xm − θm λm λm s |Zm |3 ϕ λ3m s3 E + Z +η s t −P t − Um Xm − θm λm λm s Cn−1/4 dt dt dt Cn−1/4 (3.2) 54 L.V Dung, T.C Son, and N.D Tien Proof of Theorem For n ∈ N, s > 0, and γ > 0, let G(s, γ) = X ∈ Mn : s(X) = s, X γ, V (X) = a.s ∞ and Δ(n, s, γ) = sup sup E f f ∈Λ1 S(X) s − E f (N) : X ∈ G(s, γ) It is clear that Δ(n, s, γ) Δ(n − 1, s, 2γ) For a fixed element X ∈ G(s, γ), where we assume that γ 1, let Z1 , Z2 , , Zn be i.i.d standard normal variables, and let η be a centered normal r.v with variance κ2 such that η is independent of anything else The variance κ2 will be specified later, but in any case, κ2 > 2γ Let m−1 j=1 Xj Um = n j=m+1 σj Zj +η λ2m n j=m+1 σj s2 + κ2 s s Conditioned on σ(Fn+1 , Zm ), Wm is normally distributed with mean and variance λ2m , and Z = n j=1 σj Zj /s is a standard normal variable Hence, by Lemma we have , Wm = FS/s − Φ F(S+η)/s − F( n j=1 , = σj Zj +η)/s κ +C s (4.1) Now we consider the first term on the right-hand side of (4.1) Let ϕλm (x) be the density function of Wm For any 1-Lipschitzian f , according to an idea that goes back to Lindeberg [10], we write E f S+η s −E f n j=1 σj Zj s n = E f Wm + U m + Xm s − E f Wm + U m + E f ∗ ϕλ m U m + Xm s − E f ∗ ϕλ m U m + m=1 n = m=1 n E gm Um + = m=1 n E = m=1 + +η Xm s − E gm Um + − σm Z m s σm Z m s (where gm = f ∗ ϕλm ) Z2 − X2 σm m m gm (Um ) s2 σm Z m − X m gm (Um ) − s θm σm Z m (σm Zm )3 gm Um − s s σm Z m s Xm θ Xm gm Um − m s s Since Um and λm are F m−1 -measurable, where F m−1 is the completion of Fm−1 , from E(Xm | F m−1 ) = Z 2F 2 E(σm Zm | F m−1 ) = a.s and E(σm j m−1 ) = E(Xj | F m−1 ) = σj a.s it follows that the first two sums in the above expression must vanish Moreover, since gm (x) = f ∗ ϕλm (x), we get that S+η E f s −E f n j=1 σj Zj s +η L1 bounds for some martingale CLT n |σm Zm |3 s3 E m=1 n 6s3 + := m=1 θσm Zm s E |σm Zm |3 f ∗ ϕλm Um − m=1 +E 3| |Xm s3 gm Um + θm Xm s θm Xm s E |Xm |3 f ∗ ϕλm Um − n 6s3 gm Um + 55 θm σm Z m s (I + II) 6s3 (4.2) We define the sequence of stopping times τj (1 k τ0 = 0, τj = inf k: j n) by j n σi2 i=1 for j n − 1, τn = n θm Xm s Then n m=1 E |Xm |3 f ∗ ϕλm Um − τj n E = m=τj−1 +1 j=1 If τj−1 < m θm Xm s |Xm |3 f ∗ ϕλm Um − (4.3) τj , then λ2m n j=τj +1 σj s2 n λ2m + κ2 s2 − js2 /n − γ + κ2 := λ2j , s s2 − (j − 1)s2 /n + κ2 := λj s σj2 + κ2 s2 j=τj−1 +1 We denote Rm = be defined by m−1 i=τj−1 +1 Xi and Amt = {|Rm | ψ(x) = sup |Uτj−1 +1 − t|/2} for t ∈ R Let the function ψ : R → R ϕ (y) : |y| |x| −1 We conclude that, for every t, ϕ holds on Amt ∩ {τj−1 < m τj E m=τj−1 +1 Lith Math J., 54(1):48–60, 2014 U m − t θm Xm − λm λm s ψ τm } Then, |Xm |3 f ∗ ϕλm Um − θm Xm s Uτj−1 +1 − t λj 56 L.V Dung, T.C Son, and N.D Tien τj γE m=τj−1 +1 Xm f ∗ ϕλ m U m − ∞ τj γE m=τj−1 +1 τj γE m=τj−1 +1 Xm Xm ∞ m=τj−1 +1 γE m=τj−1 +1 ∞ m=τj−1 +1 Xm τj m=τj−1 +1 Xm ψ Uτj−1 +1 − t λj −∞ f (t) dt ∞ m=τj−1 +1 m=τj−1 +1 U m − t θm Xm − f (t) IAcmt dt λm λm s λ−3 m ϕ −∞ ∞ τj τj γλ−3 j E U m − t θm Xm − f (t) IAmt dt λm λm s λ−3 m ϕ −∞ + γE + −∞ Xm τj γλ−3 j E U m − t θm Xm − f (t)IAcmt dt λm λm s λ−3 m ϕ ∞ τj + Xm θm Xm − t f (t) dt s U m − t θm Xm − f (t)IAmt dt λm λm s λ−3 m ϕ −∞ + γE γλ−3 j E ϕλ m U m − −∞ ∞ τj γλ−3 j E θm Xm s Xm IAcmt dt −∞ ∞ Xm ψ Uτj−1 +1 − t λj −∞ f (t) dt ∞ τj m=τj−1 +1 Xm IAcmt dt −∞ := γλ−3 j (Mj + Nj ) (4.4) We first consider Mj Since Uτj−1 +1 is Fτj−1 -measurable, we obtain ∞ Mj = τj E m=τj−1 +1 −∞ ∞ τj E −∞ ∞ ψ −∞ Uτj−1 +1 − t λj Uτj−1 +1 − t λj Uτj−1 +1 − t m=τj−1 +1 E ψ = Xm ψ λj f (t) dt E Xm Fτj−1 τj m=τj−1 +1 E Xm Fτj−1 f (t) dt f (t) dt L1 bounds for some martingale CLT ∞ 2γ Uτj−1 +1 − t E ψ ∞ = 2γ E f (t) dt λj −∞ Uτj−1 +1 − t ψ f (t) dt = 2γ E gj (Uτj−1 +1 ) , λj −∞ 57 where ∞ gj (x) = x−t λj ψ −∞ f (t) dt Now, since n n Xi2 E Fτj−1 σi2 Fτj−1 =E i=τj−1 +1 s2 − i=τj−1 +1 j−1 n a.s., from Lemma we obtain E gj (Uτj−1 +1 ) C FS/s − Φ j−1 + λj n + 1− 1− j−1 + λj n Hence, Mj Cγ FS/s − Φ + (4.5) Next, we consider Nj Let k Bj = Xi > max τj−1 t dt + |x|3 P |Zm | > |x| =2 |x|} |x| Using this expression as above, we obtain C γ n κ2 − 2γ II −1/2 Δ(n, s, γ) + γ n log n Combining this with (4.1), (4.2), and (4.7), we obtain Cs−3 γ n κ2 − 2γ Δ(n, s, γ) −1/2 Taking κ2 = 2γ + C 22 γ n2 and putting Kn = supγ κ Δ(n − 1, s, 2γ) + γ n log n + C s 1, 0 if C is suitably chosen ˆ = For the estimate V − 1 , we again let X = (X1 , , Xn ) ∈ Mn with X γ and define X ˆ ˆ (X1 , , X2n ) as in the proof of Corollary in [4] Then, by Theorem 2, FS/s − Φ Cγ n log n/s3 , and by Burkholder’s inequality (see, e.g., ˆ Lith Math J., 54(1):48–60, 2014 60 L.V Dung, T.C Son, and N.D Tien [9, Thm 2.11]) it is easy to see that E(Sˆ − S)2p p CE s2 V − s2 (5.2) For any x > 0, we have ∞ FS/s − Φ FS/s ˆ −Φ + −∞ Sˆ P s C E(Sˆ − S)2p + 2x x2p−1 s2p E|V − 1|p FS/s − Φ + C + 2x, ˆ x2p−1 FS/s ˆ −Φ ∞ S t+x −P s + t dt + Φ(t + x) − Φ(t) dt −∞ (by (2.5) and (2.6)) (5.3) and now putting x = (E V − )1/2p , we have |FS/s − Φ|1 C γ n log n +C E V2−1 s3 Combining this with (5.1) yields the conclusion of Corollary Substituting, instead of (5.2), the inequality E(Sˆ − S)2p immediately get the conclusion of Corollary 1/2p C(E|s2 V − s2 |p + 2γ 2p ) into (5.3), we Acknowledgment We would like to express our gratitude to the referee for his/her detailed comments and valuable suggestions, which helped us to improve the manuscript References R.P Agnew, Global versions of the central limit theorem, Proc Natl Acad Sci USA, 40(9):800–804, 1954 V Bentkus, A new method for approximation in probability and operator theories, Lith Math J., 43(4):367–388, 2003 V.I Bogachev, Measure Theory, Vol 1, Springer-Verlag, Berlin, 2007 E Bolthausen, Exact convergence rates in some 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Vol 1, Springer-Verlag, Berlin, 2007 E Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann Probab., 10(3):672–688, 1982 J Dedecker and E Rio, On mean central limit. .. mean central limit theorems, Kungl Tekn Högsk Handl., Stockholm, 121(3):1–30, 1958 L Goldstein, Bounds on the constant in the mean central limit theorem, Ann Probab., 38(4):1672–1689, 2010 P Hall... respectively If X ∈ Mn , V (X) → in probability, and some Lindeberg-type condition is satisfied, then lim P n→∞ S(X) s(X) x = Φ(x) for all x ∈ R For bounds of the convergence rate in this central limit