Berry-Esseen bounds for self-normalized martingales

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1 Berry-Essee bouds for self-ormalized martigales Xiequa Fa a, Qi-Ma Shao b a Ceter for Applied Mathematics, Tiaji Uiversity, Tiaji 30007, Chia b Departmet of Statistics, The Chiese Uiversity of Hog Kog, Shati, NT, Hog Kog Abstract A Berry-Essee boud is obtaied for self-ormalized martigales uder the assumptio of fiite momets. The boud coicides with the classical Berry-Essee boud for stadardized martigales. A example is give to show the optimality of the boud. Applicatios to Studet s statistic ad autoregressive process are also discussed. Keywords: Self-ormalized process, Berry-Essee bouds, martigales, Studet s statistic, autoregressive process 000 MSC: Primary 60G4; 60F05; Secodary 60E15 1. Itroductio Let X i i 1 be a sequece of idepedet o-degeerate real-valued radom variables with zero meas, ad let S = X i ad V = Xi be the partial sum ad the partial quadratic sum, respectively. The self-ormalized sum is defied as S /V. The study of the asymptotic behavior of self-ormalized sums has a log history. Whe X i i 1 are i.i.d. i the domai of ormal ad stable law, Loga et al. [11] obtaied the weak covergece for the self-ormalized sum, while Gié et al. [5] proved that S /V is asymptotically ormal if ad oly if X 1 belogs to the domai of attractio of a ormal law. Uder the same ecessary ad sufficiet coditio, Csörgő et al. [3] proved a self-ormalized type Dosker s theorem. For geeral idepedet radom variables with fiite + δ th momets, where 0 < δ 1, Betkus, Blozelis ad Götze [] see also Betkus ad Götze [1] for i.i.d. case have obtaied the followig Berry-Essee boud : If E X i +δ < for δ 0, 1], the these exists a absolute costat C such that sup x PS /V x Φx C B δ E X i +δ, Preprit submitted to Elsevier November 1, 017

2 where B = EX i, ad Φ x is the stadard ormal distributio fuctio. It is worth otig that the last boud coicides with the classical Berry-Essee boud for stadardized partial sums S /B ad it is the best possible. For the related error of PS /V x to 1 Φx, we refer to Shao [1], Jig, Shao ad Wag [9]. I these papers, self-ormalized Cramér type moderate deviatio theorems have bee established. We also refer to de la Peña, Lai ad Shao [4], Shao ad Wag [13] ad Shao ad Zhou [14] for surveys o recet developmets o self-ormalized limit theory. Despite the fact that the case for self-ormalized sums of idepedet radom variables is well studied, we are ot aware of Berry-Essee bouds for self-ormalized martigales i the literature. The mai purpose of this paper is to fill this gap. We first recall some Berry-Essee bouds for stadardized martigale differece sequece. Let X i, F i i=0,..., be a fiite sequece of martigale differeces defied o a probability space Ω, F, P, where X 0 = 0 ad {, Ω} = F 0... F F are icreasig σ-fields. Set S 0 = 0, S k = k X i, k = 1,...,. 1 The S = S k, F k k=0,..., is a martigale. Let [S] ad S be, respectively, the squared variace ad the coditioal variace of the martigale S, that is ad S 0 = 0, S k = [S] 0 = 0, [S] k = k X i k E[Xi F i 1 ], k = 1,...,. Suppose that E X i p < for some p > 1 ad all i = 1,...,. Defie N = E X i p + E S 1 p. Whe p 1, ], Heyde ad Brow [8] see also Theorem 3.10 of Hall ad Heyde [7] proved that there exits a costat C p depedig oly o p such that sup PS x Φ x C p N 1/p+1. 3 Later, Haeusler [6] gave a extesio of 3 to all p 1,. Moreover, Haeusler also gave a example to justify that his boud is asymptotically the best possible. It is remarked that the X i 1 i is stadardized, that is, EX i is close to 1.

3 I this paper, we prove that the Berry-Essee boud 3 also holds for self-ormalized martigales S / [S] ad ormalized martigales S / S. Moreover, we also justify the optimality of our bouds. Applicatios to Studet s statistic ad autoregressive process are discussed. The paper is orgaized as follows. Our mai results are stated ad discussed i Sectio. The applicatios are give i Sectio 3. Proofs of theorems are deferred to Sectio 4.. Mai results The followig theorem gives a couterpart of Haeusler s result [6] for self-ormalized martigales. Theorem.1. Suppose that E X i p < for some p > 1 ad all i = 1,...,. The there exits a costat C p depedig oly o p such that S sup P x Φ x C p N 1/p+1, 4 [S] where N is defied by. Moreover, there exit a sequece of martigale differeces X i, F i i=0,..., ad a positive costat c p depedig oly o p such that S sup P x Φ x N 1/p+1 c p. 5 [S] Clearly, iequality 5 shows that the boud 4 is asymptotically the best possible. For a statioary martigale differece sequece, the term E X i p is of order 1 p. The iequality 4 implies the followig corollary. Corollary.1. Let X i, F i i 1 be a statioary martigale differece sequece. Suppose that E X 1 p < for some p > 1. The there exits a costat c p, which does ot deped o, such that S sup P x Φ x c p 1 p + E S 1 p 1/p+1. 6 [S] The ext theorem gives a Berry-Essee boud for ormalized martigales S / S. Theorem.. Uder the assumptios of Theorem.1, the iequalities 4 ad 5 hold whe S / [S] is replaced by S / S. For a statioary martigale differece sequece, the followig result is a cosequece of the last theorem. Corollary.. Assume the coditios of Corollary.1. Iequality 6 holds whe S / [S] is replaced by S / S. 3

4 3. Applicatios 3.1. Applicatio to Studet s t-statistic The study of self-ormalized partial sums origiates from Studet s t-statistic. The Studet s t-statistic T is defied by T = X / σ, where X = S It is kow that for all x 0, P T > x ad σ = X i X. 1 S = P > x [S] + x 1 1/. Whe X i i 1 is a sequece of i.i.d. radom variables, Betkus ad Götze [1] proved that if E X i +δ < for all i = 1,..., ad some δ 0, 1], the sup PT x Φx = O δ/. 7 For martigales, we have the followig aalogue. Corollary 3.1. Let X i, F i i 1 be a statioary martigale differece sequece. Suppose that E X 1 p < for some p > 1. The there exits a costat C p, which does ot deped o, such that 6 holds whe PS / [S] x is replaced by PT x. 3.. Applicatio to autoregressive process Cosider the autoregressive process give by Y +1 = θy + ε +1, 0, where Y ad ε represet the observatio ad the drive oise, respectively. The parameter θ is ukow ad eeds to be estimated at stage from the data Y i, i. For sake of simplicity, we assume that Y 0 = 0. We also assume that ε 0 is a statioary martigale differece sequece with E[ε i ε 1,..., ε i 1 ] = σ a.s. for a positive costat σ. We ca estimate the ukow parameter θ by the least-squares estimator give by θ = Y iy i+1 Y. i It is well kow that θ θ Σ Y i coverges i distributio to a ormal law, see Theorem 3 of Lai ad Wei [10]. By Theorem., we have the followig Berry-Essee boud for the least-squares estimator θ. 4

5 Theorem 3.1. Suppose that E ε 1 p < for some p > 1. If θ < 1, the sup P θ θ Σ Y i xσ Φ x = O 1 p + p E Yi EYi p 1/p+1, 8 where Y = θ i ε i. 4. Proofs of theorems 4.1. Proof of Theorem.1 We assume that N 1. Otherwise, 4 is trivial. Firstly, we give a lower boud for PS x [S] Φx, x 0. Let ε 0, 1/] be a positive umber, whose exact value will be chose later. It is easy to see that for x 0, where PS x [S] Φx PS x [S], [S] < 1 + ε Φx PS x 1 + ε, [S] < 1 + ε Φx PS x 1 + ε P[S] 1 + ε Φx = I 1 + I I 3, 9 I 1 = PS x 1 + ε Φx 1 + ε, I = Φx 1 + ε Φx, I 3 = P[S] 1 + ε. Next, we estimate I 1, I ad I 3. By Haeusler s iequality [6] see also 3 whe p 1, ], we get the followig estimatio for I 1 : I 1 C p,1 E X i p + E S 1 p 1/p By oe-term Taylor s expasio, we have the followig estimatio for I : I c 1 e x / x 1 + ε 1 c ε. 11 5

6 For I 3, by Markov s iequality, it follows that I 3 = P[S] S + S 1 ε P [S] S ε + P S 1 ε c 3 ε p E [S] S p + E S 1 p. 1 We distiguish two cases to estimate I 3. Notice that [S] i S i, F i i=0,..., is also a martigale. Case 1 : If p 1, ], by the iequality of vo Bahr-Essee [15], it follows that Returig to 1, we have E [S] S p c 4 E Xi E[Xi F i 1 ] p I 3 c 6 ε p c 4 E[ X i p + E[Xi F i 1 ] p ] c 5 E X i p. 13 E X i p + E S 1. p 14 Case : If p >, by Rosethal s iequality cf. Theorem.1 of Hall ad Heyde [7], we have p/ E [S] S p C p, E E[Xi 4 F i 1 ] + E X i. p 15 Notig that X 4 i = X i p /p 1 X i p 1/p 1 for p >, we have by Hölder s iequality E[X 4 i F i 1 ] 1/p 1 p /p 1, E[ X i p F i 1 ] E[Xi F i 1 ] ad hece E[Xi 4 F i 1 ] 1/p 1 p /p 1 E[ X i p F i 1 ] E[Xi F i 1 ] 1/p 1 p /p 1. E[ X i p F i 1 ] S 6

7 By the iequality a + b q q a q + b q, a, b 0 ad q > 0, ad the fact that p p/p p, it follows that for p >, p/ E[Xi 4 F i 1 ] p p p/p p E[ X i p F i 1 ] S p/p p/p E[ X i p F i 1 ] 1 + S 1 p p/p p/p E[ X i p F i 1 ] p/p S + p E[ X i p F i 1 ] 1 pp /p. As to the secod term o the r.h.s. of the last iequality, we use the iequality ad hece Thus, x a y 1 a x + y, x, y 0 ad a [0, 1], p/ p/p E[Xi 4 F i 1 ] p E[ X i p F i 1 ] p/ E E[Xi 4 F i 1 ] + p E[ X i p F i 1 ] + S 1. p p/p ] [E p E[ X i p F i 1 ] + E X i p + E S 1 p 16 [ p/p p E X i p + E X i p + E S 1 ]. p 17 Returig to 15, we get for p >, p/p E [S] S p C p,3 E X i p + E S 1. p 7

8 From 1 ad the last iequality, we obtai for p >, p/p I 3 C p,4 ε p E X i p + E S 1. p 18 By the iequalities 14 ad 18, we always have for p > 1, I 3 C p,5 ε p E X i p + p/p E X i p + E S 1. p 19 Combiig 9, 10, 11 ad 19 together, we deduce that for p > 1, Takig PS x [S] Φx C p,1 E X i p + E S 1 p 1/p+1 c ε C p,5 ε p p/p E X i p + E X i p + E S 1. p p/p 1/p+1 ε = E X i p + E X i p + E S 1 p, 0 we obtai for x 0 ad p > 1, PS x [S] Φx C p,1 E X i p + E S 1 p 1/p+1 p/p 1/p+1 C p,6 E X i p + E X i p + E S 1 p C p,7 E X i p + E S 1 p 1/p+1, 1 where the last lie follows from the fact that p/p p + 1 1/p + 1 ad N 1. Secodly, we give a upper boud for PS x [S] Φx, x 0. It is obvious that for x 0, PS x [S] Φx 8

9 PS x [S], [S] > 1 ε Φx + PS x [S], [S] 1 ε PS x 1 ε, [S] > 1 ε Φx + P[S] 1 ε PS x 1 ε Φx 1 ε + Φx 1 ε Φx + P[S] 1 ε = I 4 + I 5 + I 6. Followig the same lies as i the proof of 1, we get for x 0 ad p > 1, PS x [S] Φx C p,8 E X i p + E S 1 p 1/p+1. Combiig 1 ad together, we get for p > 1, sup x 0 PS x [S] Φx C p,8 E X i p + E S 1 p 1/p+1. 3 Notice that S k, F k k=0,..., is also a martigale. Applyig the last iequality to S k, F k k=0,...,, we get sup PS x [S] Φx x>0 = sup PS x [S] Φx x>0 = sup Φ x P S < x [S] x>0 C p,9 E X i p + E S 1 p 1/p+1. 4 Combiig the iequalities 3 ad 4 together, we obtai sup PS x [S] Φx C p,10 E X i p + E S 1 p 1/p+1, 5 which gives the desired iequality 4. Next we give a proof of 5. We follow the example of Haeusler [6]. Let α 1 be a sequece of positive umbers such that α 0 as. Defie fuctio f : R [0, as follows { x f x = 1, if 1 α x <, 0, otherwise. 9

10 Furthermore, let X 1,..., X 1 be idepedet ad ormally distributed radom variables with mea 0 ad variace 1/ 1. Deote ν x the oe-poit mass cocetratio at x. Defie the radom variable X such that its coditioal distributio, give X 1,..., X 1, is PX S 1 = x = 1 ν α f x + 1 ν α f x, where S 1 = 1 X i. Deote F i the atural filtratio of X 1,..., X, that is F 0 beig the trivial σ field ad F i = σ{x 1,..., X i }, i = 1,...,. Clearly, X i, F i i=0,..., is a fiite sequece of martigale differeces. Moreover, it holds ad E X p = 1 E X i p = 1 1 E N 0, p 1 p C p,11 1 p = 1 π 1 y p PX dy N 0, 1 = xpn 0, 1 dx α α x p e 1 x dx C p,1 α p+ 1 6 for some costats 0 < C p,11, C p,1 <, where N 0, 1 is a stadard radom variable. Similarly, we have 1 1 E[Xi F i 1 ] = EXi = 1 ad E S 1 p = E E[X F 1 ] p = y PX dy N 0, 1 = x p PN 0, 1 dx = 1 π 1 α α x p e 1 x dx C p,1 α p Thus N C p α p+ 1, 10

11 where C p is a positive costat depedig oly o p. O the other had, we have S P 0 = P X + S 1 0 [S] Hece, we deduce that sup P P = = = 1 α + PX x N 0, 1 = xpn 0, 1 dx PX x N 0, 1 = xpn 0, 1 dx PX 1 x N 0, 1 = xpn 0, 1 dx α e 1 x dx + 1 α x dx π 1 0 π = Φ π S α 1 + o1 x Φ x N 1/p+1 [S] S 1 1 α. 1 e 0 Φ 0 C p α p /p+1 + o1 [S] = 1 4 α C p α p /p+1 + o1 π 1 4 π C p 1/p+1. This completes the proof of Theorem Proof of Theorem. First, we give a lower boud for PS x S Φx, x 0. Let ε 0, 1/] be a positive umber, whose exact value will be chose later. It is easy to see that for x 0, where PS x S Φx PS x S, S < 1 + ε Φx PS x 1 + ε, S < 1 + ε Φx PS x 1 + ε P S 1 + ε Φx = I 1 + I I 7, I 7 = P S 1 + ε. 11

12 For I 7, by Markov s iequality, it follows that I 7 ε p E S 1 p. 8 Combiig the estimatios 10, 11 ad 8 together, we have for p > 1, PS x [S] Φx C p,1 E X i p + E S 1 p 1/p+1 c ε ε p E S 1 p. Next we carry out a argumet as the proof of Theorem.1 with what we obtai is sup P S ε = E S 1 p 1/p+1, 9 x Φ x C p, E X i p + E S 1 p 1/p+1, 30 S that is iequality 4 holds whe S / [S] is replaced by S / S. The proof of optimality is similar to the proof of 5. This completes the proof of Theorem Proof of Theorem 3.1 The proof of theorem is based o Theorem.. It is easy to see that 1 σ θ θ Σ Y i = Y iε i+1 σ Y. i Notice that Y = θ i ε i. Set X i = Y i ε i+1 σ EY i ad F i = σ{ε k, 1 k i + 1}. The it is easy to see that X i, F i i=0,..., is a sequece of martigale differeces, ad that 1 σ θ θ Σ Y i = S. S Moreover, we have EY = θ i σ = 1 θ 1 θ σ 1

13 ad By Rosethal s iequality, we also have ad EY i = E Y p C p EY p + E Y i p C p 1 θ i 1 θ σ. E θ i ε i p 1 θ pσ C p p + 1 θ p 1 θ 1 θ E ε 1 p p 1 θ i pσ p + 1 θ p i 1 θ 1 θ E ε 1 p. p Thus E X i p EX i p C p If θ < 1, iequality 31 implies that E X i p It is obvious that EX i p C p 1 θ i pσ p + 1 θ p i / 1 θ 1 θ E ε 1 p p 1 C p 1 θ p + σ p 1 θ p + E ε 1 p / 1 θ p 1 E ε 1 p 1 θ p σ p i 1 p. 1 θ i 1 θ σ p. 31 σ p S = Y i EY. 3 By Theorem., we obtai sup P θ θ Σ Y i xσ Φ x C E X i p p,θ + E Y i EX i p EY 1 i = O 1 p + p E Yi EYi p 1/p+1. This completes the proof of theorem. 13 p 1/p+1

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