Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 7, Issue 1, Article 2, 2006 AN INEQUALITY FOR THE ASYMMETRY OF DISTRIBUTIONS AND A BERRY-ESSEEN THEOREM FOR RANDOM SUMMATION HENDRIK KLÄVER AND NORBERT SCHMITZ I NSTITUTE OF M ATHEMATICAL S TATISTICS U NIVERSITY OF M ÜNSTER E INSTEINSTR . 62 D-48149 M ÜNSTER , G ERMANY schmnor@math.uni-muenster.de Received 16 March, 2004; accepted 15 December, 2005 Communicated by S.S. Dragomir A BSTRACT. We consider random numbers Nn of independent, identically distributed (i.i.d.) PNn random variables Xi and their sums i=1 Xi . Whereas Blum, Hanson and Rosenblatt [3] proved a central limit theorem for such sums and Landers and Rogge [8] derived the corresponding approximation order, a Berry-Esseen type result seems to be missing. Using an inequality for the asymmetry of distributions, which seems to be of its own interest, we prove, under the assumption E|Xi |2+δ < ∞ for some δ ∈ (0, 1] and Nn /n → τ (in an appropriate sense), a Berry-Esseen theorem for random summation. Key words and phrases: Random number of i.i.d. random variables, Central limit theorem for random sums, Asymmetry of distributions, Berry-Esseen theorem for random sums. 2000 Mathematics Subject Classification. 60E15, 60F05, 60G40. 1. I NTRODUCTION One of the milestones of probability theory is the famous theorem of Berry-Esseen which gives uniform upper bounds for the deviation from the normal distribution in the Central Limit Theorem: Let {Xn , n ≥ 1} be independent random variables such that EXn = 0, EXn2 =: σn2 , n X 2 sn := σi2 > 0, Γ2+δ := n i=1 n X i=1 ISSN (electronic): 1443-5756 c 2006 Victoria University. All rights reserved. 056-04 E|Xi |2+δ < ∞ 2 H ENDRIK K LÄVER AND N ORBERT S CHMITZ P for some δ ∈ (0, 1] and Sn = ni=1 Xi , n ≥ 1. Then there exists a universal constant Cδ such that 2+δ Γn S n ≤ x − Φ(x) ≤ Cδ , sup P sn sn x∈R where Φ denotes the cumulative distribution function of a N (0, 1)-(normal) distribution (see e.g. Chow and Teicher [5, p. 299]). For the special case of identical distributions this leads to: Let {Xn , n ≥ 1} be i.i.d. random variables with EXn = 0, EXn2 =: σ 2 > 0, E|Xn |2+δ =: γ 2+δ < ∞ for some δ ∈ (0, 1]. Then there exists a universal constant cδ such that S cδ γ 2+δ n √ ≤ x − Φ(x) ≤ δ/2 sup P . n σ σ n x∈R Van Beek [2] showed that C1 ≤ 0.7975; bounds for other values Cδ are given by Tysiak [12], e.g. C0.8 ≤ 0.812; C0.6 ≤ 0.863, C0.4 ≤ 0.950, C0.2 ≤ 1.076. On the other hand, there exist also central limit theorems for random summation, e.g. the theorem of Blum, Hanson and Rosenblatt [3] which generalizes previous results by Anscombe [1] and Renyi [11]: P Let {Xn , n ≥ 1} be i.i.d. random variables with EXn = 0, Var Xn = 1, Sn := ni=1 Xi and P let {Nn , n ≥ 1} be N-valued random variables such that Nn /n −→ U where U is a positive random variable. Then √ d P SNn / Nn −→ N (0, 1). Therefore, the obvious question arises whether one can prove also Berry-Esseen type inequalities for random sums. A first result concerning the approximation order is due to Landers and Rogge [8] for random variables Xn with E|Xn |3 < ∞; this was generalized by Callaert and Janssen [4] to the case that E|Xn |2+δ < ∞ for some δ ∈ (0, 1]: Let {Xn , n ≥ 1} be i.i.d. random variables with E Xn = µ, Var Xn = σ 2 > 0, and E|Xn |2+δ < ∞ for some δ > 0. Let {Nn , n ≥ 1} be N-valued random variables, {εn , n ≥ 1} positive real numbers with εn −→ 0 where, for n large, n−δ ≤ εn if δ ∈ (0, 1] and n−1 ≤ εn if n→∞ δ ≥ 1. If there exists a τ > 0 such that Nn √ P − 1 > εn = O( εn ), nτ then sup P x∈R PNn ! √ − Φ(x) = O( εn ) sup P x∈R PNn ! √ − Φ(x) = O( εn ). i=1 (Xi − µ) √ ≤x σ nτ and i=1 (Xi − µ) √ ≤x σ Nn Moreover, there exist several further results on convergence rates for random sums (see e.g. [7] and the papers cited there); as applications, sequential analysis, random walk problems, Monte Carlo methods and Markov chains are mentioned. However, rates of convergence without any knowledge about the factors are of very limited importance for applications. Hence the aim of this paper is to prove a Berry-Esseen type result for random sums i.e. a uniform approximation with explicit constants. Obviously, due to the dependencies on the moments of the Xn as well as on the asymptotic behaviour of the sequence Nn such a result will necessarily be more complex than the original Berry-Esseen theorem. J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/ A N I NEQUALITY FOR THE A SYMMETRY OF D ISTRIBUTIONS AND A B ERRY-E SSEEN T HEOREM FOR R ANDOM S UMMATION 3 For the underlying random variables Xn we make the same assumption as in the special version of the Berry-Esseen theorem: ( Xn , n ≥ 1, are i.i.d. random variables with EXn = 0, V ar Xn = 1 and (M) : γ 2+δ := E|Xn |2+δ < ∞ for some δ ∈ (0, 1]. Similarly as Landers and Rogge [8] or Callaert and Janssen [4] resp. we assume on the random indices Nn , n ≥ 1, are integer-valued random variables and ζn , n ≥ 1, (R) : real numbers with limn→∞ ζn = 0 such that there exist √ d, τ > 0 with P (| Nnτn − 1| > ζn ) ≤ d ζn . As they are essential for applications, e.g. in sequential analysis, arbitrary dependencies between the indices and the summands are allowed. 2. A N I NEQUALITY FOR THE A SYMMETRY OF D ISTRIBUTIONS A main tool for deriving explicit constants for the rate of convergence is an inequality which seems to be of its own interest. For a smooth formulation we use (for the different values of the moment parameter δ ∈ (0, 1]) some (technical) notation: For ϑ := 2/(1 + δ) and y ≥ 1 let gδ (y) be defined by p 2 2y − 1 + 2y y2 − 1 if δ = 1 n p min 2ϑ y 4ϑ − 1 + 2 ϑ+1 2ϑ 2 y 2ϑ−1 y 4ϑ − 1 , o p 8ϑ 4ϑ 8ϑ − 1 2y − 1 + 2y y if δ ∈ 13 , 1 gδ (y) := n p ϑ+1 ϑ (2+2k )ϑ (1+2k−1 )ϑ 2 y min 2 y − 1 + 2 2ϑ−1 y (2+2k )ϑ − 1, o i p k+1 )ϑ 1 1 (4+2k+1 )ϑ (2+2k )ϑ (4+2 2y − 1 + 2y y −1 if δ ∈ 2k +1 , 2k−1 +1 , k ≥ 2 Theorem 2.1. Let X be a random variable with EX = 0, Var X = 1 and γ 2+δ := E|X|2+δ < ∞ for some δ ∈ (0, 1]. Then P (X < 0) ≤ gδ (γ 2+δ )P (X > 0) and P (X > 0) ≤ gδ (γ 2+δ )P (X < 0). Proof. Let X be a random variable as described above. Let X+ = max{X, 0}, X− = min{X, 0}, E(X+ ) = E(X− ) = α and P (X > 0) = p, P (X = 0) = r, P (X < 0) = 1 − r − p. Since E(X) = 0, Var(X) = 1 it is obvious that p, 1 − r − p > 0. As α = pE(|X| |X > 0), we have α E(|X| |X > 0) = αp ; analogously E(|X| |X < 0) = 1−r−p . Applying Jensen’s inequality to the convex function f : [0, ∞) → [0, ∞), f (x) = x yields 3+δ 3+δ 2 3+δ 3+δ α 2 α E(|X| 2 |X > 0) ≥ and E(|X| 2 |X < 0) ≥ . p 1−r−p 3+δ 2 Defining βz := E|X|z for z > 0 we get 3+δ 3+δ β 3+δ = pE |X| 2 |X > 0 + (1 − r − p)E |X| 2 |X < 0 2 ≥α 3+δ 2 (1 − r − p) 1+δ 2 +p (p(1 − r − p)) J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 1+δ 2 1+δ 2 http://jipam.vu.edu.au/ 4 H ENDRIK K LÄVER AND N ORBERT S CHMITZ and, therefore, α (i) 3+δ 2 ≤ β 3+δ 2 (p(1 − r − p)) (1 − r − p) 1+δ 2 1+δ 2 +p 1+δ 2 Since γ 2+δ < ∞, we can apply the Cauchy-Schwarz-inequality to |X|1/2 and |X|1+δ/2 and obtain 2 3+δ E|X| 2 ≤ E|X| E|X|2+δ = 2αγ 2+δ ; hence 2 2 α≥ (ii) β 3+δ 2γ 2+δ . Combining (i) and (ii) we obtain 2 3+δ 2 1+δ β 3+δ (p(1 − r − p)) 2 2 ≤ β 3+δ 1+δ 1+δ ; 2 2γ 2+δ p 2 + (1 − r − p) 2 hence 1 1 x2 ≤ − 4 1−r ϑ+1 1 a1 2 a2 with p 1 2 − , ϑ= , 1 − r 2 1+δ Obviously x ∈ − 21 , 12 , ϑ ∈ [1, 2). Since 1 +x 2 ϑ1 + ϑ+2 a1 = β 3+δ , x= (iii) 2 1 −x 2 ϑ1 !ϑ a2 = (γ 2+δ )ϑ+1 . xµ + y µ ≥ (x + y)µ ∀µ ∈ (0, 1], x, y ≥ 0 and 0 < 1 − r ≤ 1 it follows altogether that s ϑ−1 1 1 a1 (iv) |x| ≤ 1− . 2 2 a2 For large values of |x| this estimation is rather poor, so we notice, furthermore, that ϑ1 !ϑ ϑ1 1 1 ≥ 2ϑ−1 (1 − 4x2 )1/2 ; + −x +x 2 2 hence 1 |x| ≤ 2 (v) s a1 a2 2 s a2 a1 1− . From (iv) and (v) it follows that (vi) ϑ+1 p a2 ≤ 2ϑ − 1 + 2 2 1−r−p a1 a2 2ϑ−1 a1 −1 and (vii) p ≤2 1−r−p J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 a2 a1 2 a2 −1+2 a1 s a2 a1 2 − 1. http://jipam.vu.edu.au/ A N I NEQUALITY FOR THE A SYMMETRY OF D ISTRIBUTIONS AND A B ERRY-E SSEEN T HEOREM FOR R ANDOM S UMMATION 5 4 3+δ Now we estimate β 3+δ . Due to Jensen’s inequality we have β 3+δ ≤ σ 2 = 1. Let δ = 1. 2 2 ≤ 2 + δ, so β 5−δ exists. Due Then β 3+δ = σ 2 = 1 and so aa21 = (γ 3 )2 . Let δ ∈ 13 , 1 . Then 5−δ 2 2 2 to the Cauchy-Schwarz-inequality we have 1 = (σ 2 )2 ≤ β 3+δ β 5−δ ; 2 hence β 3+δ ≥ 2 1 β 5−δ 2 1 ≥ . 5−δ (γ 2+δ ) 2(2+δ) 2 altogether a2 ≤ (γ 2+δ )4ϑ . a1 1 ≤ 2 + δ; hence β2+ 1−δ exists. Due to Let k ∈ N, k ≥ 2. For δ ≥ 2k +1 it follows that 2 + 1−δ 2k 2k the Cauchy-Schwarz-inequality we have 1 = (σ 2 )2 ≤ β2+ 1−δ β2− 1−δ 2k and β2− 1−δ j+1 2 j = 2 β2− 1−δ j+1 2 2j−1 2 for j ∈ {1, . . . , k − 1}. This yields 2k−1 β 3+δ ≥ β2− 1−δ ≥ 2 2k 2k 2j−1 ≤ β2− 1−δ β 2 j 2 1 2k−1 ≥ 1 (γ 2+δ ) β2+ 1−δ 2k h a2 1 1 2+δ (2k +2)ϑ Altogether we get a1 ≤ (γ ) for δ ∈ 2k +1 , 2k−1 +1 . Combining this with (vi) and (vii) we obtain the assertion. 2k+1 +1−δ 2(2+δ) . Remark 2.2. For each δ ∈ (0, 1] equality holds in Theorem 2.1 iff P X = 12 (ε1 + ε−1 ) (where εx denotes the Dirac measure in x). Proof. (i) Let X be a real random variable with P X = 12 (ε1 +ε−1 ). Then E(X) = 0, Var(X) = 1, γ 2+δ = 1 and so gδ (γ 2+δ ) = 1 for all δ ∈ (0, 1]. Since P (X < 0) = P (X > 0) = 12 we get equality. (ii) In Theorem 2.1 we have β 23+δ ! 3+δ 2 2 2γ 2+δ ≤α 3+δ 2 ≤ (p(1 − r − p)) (1 − r − p) 1+δ 2 1+δ 2 +p 1+δ 2 β 3+δ . 2 In the first “≤” there is equality iff |X|1/2 and |X|1+δ/2 are linearly dependent P -almost surely, i.e. P (|X| ∈ {0, c}) = 1 for some c > 0. As E(X) = 0, we obtain P X = p(εc + ε−c ) + (1 − 2p)ε0 . So the inequality is sharp iff gδ (γ 2+δ ) = 1. With 1 = E(X 2 ) = 2c2 p we obtain γ 2+δ = 2pc2+δ = 1 . (2p)δ/2 The functions hδ : (0, 12 ] → R, δ ∈ (0, 1], defined by hδ (p) = gδ (2p)1δ/2 , are strictly de creasing. Due to p ∈ 0, 21 and hδ 12 = 1 we get p = 12 , r = 0 and c = 1, therefore P X = 12 (ε1 + ε−1 ). J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/ 6 H ENDRIK K LÄVER AND N ORBERT S CHMITZ Since the Central Limit Theorem is concerned with sums of random variables (instead of single variables) we need a corresponding generalization of Theorem 2.1. Corollary 2.3. Under assumption (M), 2+δ n 1 X γ 2+δ − 1 Xi ≤ + n2−δ/2 E √ δ/2 n n i=1 holds, and therefore, P n X ! ≤ gδ Xi < 0 i=1 n X γ 2+δ − 1 2−δ/2 +n P nδ/2 ! Xi > 0 . i=1 Proof. 2+δ 3 2+δ 3 n n X X Xi = Xi i=1 i=1 ≤ n X |Xi |3 + 3 i=1 ≤ n X |Xk2 Xi | + k,i=1 k6=i 2+δ |Xi | i=1 n X +3 n X 2+δ 3 n X |Xi Xk Xl | i,k,l=1 i6=k6=l6=i 2+δ |Xk2 Xi | 3 k,i=1 k6=i n X + |Xi Xk Xl | 2+δ 3 . i,k,l=1 i6=k6=l6=i Since the Xi are independent and, due to E|Xi |2 = 1 and Jensen’s inequality, E|Xi |α ≤ 1 for α ≤ 2, we obtain 2+δ n 1 X 1 γ 2+δ − 1 2+δ 3 √ E Xi ≤ 1+δ/2 (nγ + (n − n)) = + n2−δ/2 . δ/2 n n n i=1 P P 2 From E √1n ni=1 Xi = 0, E √1n ni=1 Xi = 1 and Theorem 2.1 the assertion follows. We need a bound which does not depend on the number of summands. For this aim we use for n < κ := (4cδ γ 2+δ )2/δ the asymmetry inequality of Corollary 2.3 and for n ≥ κ the Berry-Esseen bound (the choice of κ as boundary between “small” and “large” n is somewhat arbitrary). Corollary 2.4. Under Assumption (M) ! n X P Xi < 0 ≤ (fδ (γ 2+δ ) − 1)P i=1 where n X ! Xi > 0 , i=1 4 y−1 −1 fδ (y) := max 3, gδ (y), gδ + (4cδ y) δ + 1. 4cδ Proof. (i) Let n < κ. Then Corollary 2.3 yields ! 2+δ n X γ −1 2−δ/2 P Xi < 0 ≤ gδ +n P nδ/2 i=1 J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 n X ! Xi > 0 . i=1 http://jipam.vu.edu.au/ A N I NEQUALITY FOR THE A SYMMETRY OF D ISTRIBUTIONS AND A B ERRY-E SSEEN T HEOREM For the function h : [1, ∞) → [1, ∞) defined by h(y) := δ h (y) = 2 00 δ 1+ 2 (γ 2+δ − 1)y γ 2+δ −1 y δ/2 δ + 2− 2 −2−δ/2 FOR R ANDOM S UMMATION 7 + y 2−δ/2 we obtain δ 1− y −δ/2 > 0, 2 i.e., h is convex. Hence the maximum of h : {1, . . . , bκc} → [1, ∞) is attained either for n = 1 or for n = bκc. Since gδ is strictly increasing this yields 2+δ 2+δ γ −1 γ −1 2−δ/2 2+δ 2+δ 4/δ−1 gδ +n ≤ max gδ (γ ), gδ + (4cδ γ ) nδ/2 4cδ γ 2+δ for all n < κ. (ii) Let n ≥ κ, i.e. nδ/2 ≥ 4cδ γ 2+δ . Then the (special case of the) theorem of Berry-Esseen yields ! n X γ 2+δ 1 1 ≤ c P X < 0 − i δ δ/2 ≤ 2 n 4 i=1 and, therefore, n X P !, Xi < 0 n X P i=1 ! Xi > 0 ≤ i=1 3/4 = 3. 1/4 Combining (i) and (ii) we obtain the assertion. 3. S OME F UTHER I NEQUALITIES The next inequality represents a quantification of Lemma 7 by Landers and Rogge [8]. Lemma 3.1. Under Assumption (M) P min p≤n≤q n X ! Xi ≤ r −P i=1 max p≤n≤q n X ! Xi ≤ r i=1 p ≤ fδ (γ 2+δ ) P X Xi ≤ r, i=1 q X ! Xi ≥ r +P i=1 p X i=1 Xi ≥ r, q X !! Xi ≤ r i=1 for all p, q ∈N s.t. p < q and for all r ∈ R. Proof. Using the same notation (A, α, β, Ak ) as Landers and Rogge [8, p. 280], we have to prove that P (A) ≤ fδ (γ 2+δ )(α + β). P (i) First we show that P (A ∩ { pi=1 Xi ≤ r}) ≤ fδ (γ 2+δ ) · α; for this it is sufficient to prove that ) ( q )! ( p X X Xi ≤ r ≤ (fδ (γ 2+δ ) − 1) · α. Xi ≤ r ∩ P A∩ i=1 J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 i=1 http://jipam.vu.edu.au/ 8 H ENDRIK K LÄVER AND N ORBERT S CHMITZ P But due to the independence of the Ak and qi=k+1 Xi ) ( q )! ( p X X Xi ≤ r ∩ Xi ≤ r P A∩ i=1 i=1 q−1 X ≤ P (Ak )P k=p+1 q X ! Xi ≤ 0 i=k+1 q−1 ≤ (fδ (γ 2+δ ) − 1) X q X P (Ak )P k=p+1 ! Xi ≥ 0 i=k+1 according to Corollary 2.4 ≤ (fδ (γ 2+δ ) − 1) · α. (ii) Similarly, it follows that A∩ P ( k X )! Xi > r ≤ fδ (γ 2+δ ) · β. i=1 (i) and (ii) yield the assertion. A thorough examination of the proof of Lemma 8 of Landers and Rogge [8] allows a generalization and quantification of their result: Lemma 3.2. Under Assumption (M), P Pn+k n X ≥ t ≤ X ≤ t, (i) P i i i=1 i=1 (ii) P P n i=1 Xi ≥ t, Pn+k i=1 Xi ≤ t ≤ 2cδ γ 2+δ nδ/2 + √1 2π 2cδ γ 2+δ nδ/2 + √1 2π q q k . n k . n Proof. (ii) follows from (i) by replacing Xi by −Xi . Analogously to the proof of [8], Lemma 8, we obtain ! n n+k X X P Xi ≤ t, Xi ≥ t i=1 i=1 ! Z k t 1 X t −Φ √ − √ xi Q(dx1 , . . . , dxk ) Φ √ n n n i=1 r Z k k 1 X 1 xi Q(dx1 , . . . , dxk ) +√ √ 2π n k i=1 r k 1 , +√ 2π n 2cδ γ 2+δ ≤ + nδ/2 ≤ 2cδ γ 2+δ nδ/2 ≤ 2cδ γ 2+δ nδ/2 since v 2 u k k 1 X u 1 X t E √ Xi ≤ E √ Xi = 1. k k i=1 i=1 J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/ A N I NEQUALITY A SYMMETRY FOR THE OF D ISTRIBUTIONS AND A B ERRY-E SSEEN T HEOREM 4. A B ERRY-E SSEEN T HEOREM FOR FOR R ANDOM S UMMATION 9 R ANDOM S UMS As an important application of these inequalities we state our main result: Theorem 4.1 (Berry-Esseen type result for random sums). Let {Xn , n ≥ 1} be i.i.d. random variables with EXn = 0, Var Xn = 1 and γ 2+δ := E|Xn |2+δ < ∞ for some δ ∈ (0, 1]; let {Nn , n ∈ N} be integer-valued random variables and {ζn , n ∈ N} real numbers with limn→∞ ζn = 0 such that there exist d, τ > 0 with p Nn P − 1 > ζn ≤ d ζn . nτ Then1 ! Nn 1 X √ Xi ≤ t − Φ(t) nτ i=1 sup P t∈R (i) 1 1 +p δ/2 bnτ c 2πebnτ c s p 2 + 1/τ 1 2+δ + 2fδ (γ ) max , ζn + 2d ζn . π n ≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ )) (ii) sup P t∈R ! Nn 1 X √ Xi ≤ t − Φ(t) Nn i=1 1 1 p + bnτ cδ/2 2πebnτ c s p 2 + 1/τ 1 2+δ + 2fδ (γ ) max , ζn + (3d + 1) ζn π n ≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ )) for all n ∈ N s.t. Since nτ 2 nτ 2 − nτ ζn ≥ 1. − nτ ζn −→ ∞ there exists an n0 s.t. n→∞ nτ 2 − nτ ζn ≥ 1 for all n ≥ n0 . Proof. (i) Let n ∈ N fulfill nτ − nτ ζn ≥ 1 and define as Landers and Rogge [8, p. 271] 2 √ bn (t) := t nτ and In := {k ∈ N : bnτ − nτ ζn c ≤ k ≤ bnτ + nτ ζn c}. Due to the assumption on Nn we have p P (Nn ∈ / In ) ≤ d ζn . For ( An (t) := max k∈In k X ) Xi ≤ bn (t) , ( Bn (t) := i=1 min k∈In k X ) Xi ≤ bn (t) i=1 follows (see Landers and Rogge [8, p. 272]) for each t ∈ R P (An (t) ∩ {Nn ∈ In }) ≤ P Nn X ! Xi ≤ bn (t), Nn ∈ In ≤ P (Bn (t)) i=1 1 bxc := max{n ∈ N : n ≤ x}. J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/ 10 H ENDRIK K LÄVER AND N ORBERT S CHMITZ and bnτ c P (An (t)) ≤ P X Xi ≤ bn (t) ≤ P (Bn (t)). i=1 Using the Berry-Esseen theorem and the result (3.3) of Petrov [10, p. 114], we obtain bnτ c X sup P Xi ≤ bn (t) − Φ(t) t∈R i=1 r r r bnτ c X 1 nτ nτ nτ −Φ t ≤ sup P p Xi ≤ t + sup Φ t − Φ(t) bnτ c bnτ c t∈R bnτ c t∈R bnτ c i=1 ≤ 1 cδ γ 2+δ p + . bnτ cδ/2 2πebnτ c For p(n) := bnτ − nτ ζn c, q(n) := bnτ + nτ ζn c we obtain from Lemma 3.1 P (Bn (t)) − P (An (t)) ≤ fδ (γ 2+δ ) · P p(n) X Xi ≤ bn (t) ≤ i=1 q(n) X Xi + P i=1 p(n) X Xi ≥ bn (t) ≥ i=1 q(n) X Xi . i=1 According to Lemma 3.2 it follows that s p(n) q(n) 2+δ X X 1 2cδ γ q(n) − p(n) sup P Xi ≤ bn (t) ≤ Xi ≤ +√ δ/2 (p(n)) p(n) 2π t∈R i=1 i=1 s 21+δ/2 cδ γ 2+δ 2 + 1/τ 1 ≤ + max , ζn , (nτ )δ/2 π n since p(n) ≥ nτ − nτ ζn − 1 ≥ nτ /2 and, therefore, s q(n) − p(n) ≤ p(n) s 2nτ ζn + 1 = nτ /2 r 4ζn + 2 ; nτ analogously s p(n) q(n) 1+δ/2 2+δ X X 2 c γ 2 + 1/τ 1 δ sup P Xi ≥ bn (t) ≥ Xi ≤ + max , ζn . δ/2 (nτ ) π n t∈T i=1 i=1 J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/ A N I NEQUALITY FOR THE A SYMMETRY OF D ISTRIBUTIONS AND A B ERRY-E SSEEN T HEOREM FOR R ANDOM S UMMATION 11 Altogether we obtain ! Nn X Xi ≤ bn (t) − Φ(t) sup P t∈R i=1 ! Nn X ≤ sup P Xi ≤ bn (t), Nn ∈ In − Φ(t) + P (Nn ∈ / In ) t∈R i=1 ≤ P (Bn (t)) − P (An (t)) + ≤ 2fδ γ 2+δ 21+δ cδ γ 2+δ (nτ )δ/2 p cδ γ 2+δ 1 p + + 2d ζn bnτ cδ/2 2πebnτ c s ! 2 + 1/τ 1 + max , ζn π n p 1 cδ γ 2+δ p + + 2d ζn bnτ cδ/2 2πebnτ c 1 1 ≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ )) +p δ/2 bnτ c 2πebnτ c s p 2 + 1/τ 1 2+δ + 2fδ (γ ) max , ζn + 2d ζn . π n + (ii) Applying Lemma 1 of Michel and Pfanzagl [9] for r = ζn , Nn 1 X f=√ Xi , nτ i=1 r g= Nn nτ q √ and using the fact that Nnτn − 1 > ζn implies Nnτn − 1 > ζn , hence r ! N p p Nn n − 1 > ζn ≤ d ζn , P − 1 > ζn ≤ P nτ nτ we obtain from part (i) ! Nn X 1 sup P √ Xi ≤ t − Φ(t) Nn i=1 t∈T 1 1 +p δ/2 bnτ c 2πebnτ c s p 2 + 1/τ 1 2+δ + 2fδ (γ ) max , ζn + (3d + 1) ζn . π n ≤ cδ γ 2+δ (1 + 22+δ/2 fδ (γ 2+δ )) R EFERENCES [1] F.J. ANSCOMBE, Large sample theory of sequential estimation, Proc. Cambridge Philos. Soc., 48 (1952), 600–607. [2] P. VAN BEEK, An application of Fourier methods to the problem of sharpening the Berry-Esseen inequality, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 23 (1972), 187–196. J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/ 12 H ENDRIK K LÄVER AND N ORBERT S CHMITZ [3] J.R. BLUM, D.L. HANSON AND J.I. ROSENBLATT, On the central limit theorem for the sum of a random number of independent random variables,. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1 (1963), 389–393. [4] H. CALLAERT AND P. JANSSEN, A note on the convergence rate of random sums, Rev. Roum. Math. Pures et Appl., 28 (1983), 147–151. [5] Y.S. CHOW AND H. TEICHER, Probability Theory: Independence, Interchangeability, Martingales. Springer, New York, 1978. [6] H. KLÄVER, Ein Berry-Esseen-Satz für Zufallssummen. Diploma Thesis, Univ. Münster, 2002 [7] A. KRAJKA AND Z. RYCHLIK, The order of approximation in the central limit theorem for random summation, Acta Math. Hungar., 51 (1988), 109–115. [8] D. LANDERS AND L. ROGGE, The exact approximation order in the central limit theorem for random summation, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 36 (1976), 269–283. [9] R. MICHEL AND J. PFANZAGL, The accuracy of the normal approximations for minimum contrast estimates, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 18 (1971), 73–84. [10] V.V. PETROV, Sums of Independent Random Variables. Springer, Heidelberg, 1975. [11] A. RENYI, On the central limit theorem for the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hung., 11 (1960), 97–102. [12] W. TYSIAK, Gleichmäßige Berry-Esseen-Abschätzungen. Ph. Thesis, Univ. Wuppertal, 1983. J. Inequal. Pure and Appl. Math., 7(1) Art. 2, 2006 http://jipam.vu.edu.au/