An almost sure central limit theorem for self-normalized products of sums of i.i.d. random variables
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(2) 30. Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. Recently, Qi [19] and Lu and Qi [16] extended (1.1) by assuming that the underlying distribution F is in the domain of attraction of a stable law with exponent α ∈ (1, 2] and α = 1, respectively. We next will recall the definition of the domain of a stable law. A sequence of i.i.d. random variables { X , X n ; n 1} is said to be in the domain of attraction of a stable law Lα if there exist constants A n > 0 and B n ∈ R such that. Sn − Bn An. d − → Lα ,. (1.2). where Lα is one of the stable distributions with index α ∈ (0, 2]. When α = 2, { X , X n ; n 1} is said to be in the domain of attraction of the normal law. In recent years, the limit theorems for self-normalized sums have received more and more attention. We refer to Griffin and Kuelbs [12] for laws of iterated logarithm, Bentkus and Götze [2] for Berry–Esseen inequalities, Lin [14] for Chungtype laws of iterated logarithm, Giné et al. [8] for the necessary and sufficient condition for the asymptotic normality, Shao [22–24] for large deviations, Csörgo˝ et al. [6,7] for Darling–Erdös theorem and Donsker’s theorem, Liu and Lin [15] for asymptotics for self-normalized random products of sums for mixing sequences. Pang et al. [17] obtained the following self-normalized products of sums for i.i.d. sequences. Theorem B. Let { X , X n ; n 1} be a sequence of i.i.d. positive random variables with EX = μ > 0, and assume that X is in the domain of attraction of the normal law. Then. n. j =1 S j n! n. μ/ Vn. d − → e. μ. √. 2N. (1.3). .. The almost sure central limit theorem (ASCLT) has been first introduced independently by Brosamler [4] and Schatte [21]. Since then many interesting results have been discovered in this field. The classical ASCLT states that when EX = 0, Var( X ) = σ 2 ,. lim. n→∞. 1 log n.
(3) n 1 Sk x = Φ(x), I √ k =1. k. kσ. a.s.. (1.4). for any x ∈ R. Here and in the sequel, I {·} denotes an indicator function and Φ(·) is the distribution function of the standard n normal random variable. Very recently, Gonchigdanzan and Rempala [11] proved the following ASCLT of products j =1 S j . Theorem C. Let { X , X n ; n 1} be a sequence of i.i.d. positive random variables with EX = μ > 0, and Var X = σ 2 < ∞ and the coefficient of variation γ = σ /μ. Then. lim. n→∞. 1 log n. k. 1/(γ √k ) n S 1 j j =1 k =1. k. I. k!μk.
(4) x = F 1 (x),. a.s.. for any x ∈ R. Here and in the sequel, F 1 (·) is the distribution function of the random variable e. (1.5) √. 2N. .. We also refer to Gonchigdanzan [9] for the ASCLT for the products of partial sums with stable distribution, Gonchigdanzan [10] for the almost sure functional limit theorem for the product of partial sums, Li and Wang [13] for the ASCLT for products of sums under association, Zhang et al. [25] for ASCLT for products of sums of partial sums under association. √ The result in (1.3) shows that when n in the classical central limit theorem is replaced by an appropriate sequence of random variables then the central limit theorem holds under a weaker moment condition than in classical case. Thus, it is natural to ask whether a self-normalized version of the ASCLT analog to Theorem C could also be valid under the same weaker assumption. As the following theorem shows, the answer to this question is affirmative. Theorem 1.1. Let { X , X n ; n 1} be a sequence of i.i.d. positive random variables with EX = μ > 0, and assume that X is in the domain of attraction of the normal law. Then. lim. n→∞. 1 log n. for any x ∈ R.. k. μ/ V n n 1 j =1 S j k =1. k. I. k!μk.
(5) x = F 1 (x),. a.s.. (1.6).
(6) Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. 31. Theorem 1.2. Let { X , X n ; n 1} be a sequence of i.i.d. positive random variables with EX = μ > 0, and assume that X is in the domain of attraction of the normal law. Then. lim. n→∞. k. μ/ n Vn 1 j =1 S j. 1 log n. k =1. I. k. k!μk.
(7) x = F 1 (x),. a.s.. (1.7). for any x ∈ R. Throughout the paper, C denotes a positive constant, which may take different values whenever it appears in different expressions. an ∼ bn means that an /bn → 1 as n → ∞. 2. Auxiliary lemmas In this section, we introduce some important lemmas which are used to prove our theorems. At first, we introduce some notations that will be used throughout this paper. Let l(x) = E ( X − μ)2 I {| X − μ| x}, b = inf{x 1: l(x) > 0} and. η j = inf s: s b + 1,. l (s) s2. 1. .
(8). j = 1, 2, . . . .. ,. j. It is easy to see that nl(ηn ) ∼ ηn2 as n → ∞. In fact, by the definition of ηn , it is not difficult to obtain that nl(ηn ) ηn2 for every n 1. Next we want to show that (n + 1)l(ηn ) ηn2 for every n 1. By the monotonicity of l(x), the definition of ηn and ηn 2, we have. nl(ηn ) nl. ηn −. 1 n+1. > ηn −. 2. 1. 2 n. η −. n+1. 1 n+1. 2 n. η = 1−. 1 n+1. ηn2 =. n n+1. ηn2 .. Thus we have. nl(ηn ) ηn2 (n + 1)l(ηn ). for every n 1.. Thus nl(ηn ) ∼ η as n → ∞. For every 1 i k n, let 2 n. . X ki = ( X i − μ) I | X i − μ| ηk ,. X ki = ( X i − μ) I | X i − μ| > ηk ,. ∗ X ki = X ki − E X ki ,. ∗ X ki = X ki − E X ki ,. S k∗ =. k . ∗ X ki ,. i =1. Yk =. k . ∗ b i ,k X ki ,. Yk =. i =1. k . ∗ b i ,k X ki ,. V k2 =. i =1. k . b i ,k =. k 1 l =i. l. ,. 2 X ki .. i =1. The first of the following lemmas can be found in Csörgo˝ et al. [7]. Lemma 2.1. If EX = 0, then the following statements are equivalent: (a) (b) (c) (d). l(x) = EX 2 I {| X | x} is a slowly varying function at ∞; x2 P (| X | > x) = o(l(x)); xE | X | I {| X | > x} = o(l(x)); E | X |α I {| X | x} = o(xα −2 l(x)) for α > 2.. Remark 2.1. The second condition (b) is well known to be equivalent to saying that X belongs to the domain of attraction of the normal law. Lemma 2.2. Let f be a real-valued function with supx | f (x)| C and supx | f (x)| C , then under the assumptions of Theorem 1.1, we have. lim. n→∞. 1 log n. n 1 k =1. k. f. . Yk. kl(ηk ). − Ef. . Yk. kl(ηk ). . = 0,. a.s.,. 2. 2 k Vk Vk 1 − Ef = 0, a.s., f n→∞ log n k kl(ηk ) kl(ηk ) k =1. . n S k∗ S k∗ 1 1 − Ef = 0, a.s. lim f n→∞ log n k k l(ηk ) k l(ηk ) k =1 lim. 1. (2.1). (2.2). (2.3).
(9) 32. Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. Proof. Let. Tn =. n 1. 1 log n. k =1. k. . f. Yk. . − Ef. kl(ηk ). . Yk. =:. kl(ηk ). n 1. 1 log n. k =1. k. Zk .. (2.4). It is obvious that. E T n2. =. . 1 log2 n. n 1 k =1. E Z k2 k2. +2. 11 k j. k< j. =:. E Zk Z j. 1 log2 n. [ I 1 + I 2 ].. (2.5). By the fact that f is bounded, we have. 1 2. log n For any. |I1| . C. n 1. 2. log n k=1. k2. C. . log n. (2.6). .. α > 0 and j > k, it is easy to check that. log. j k. α j. C. k. (2.7). .. Then under the assumptions of Lemma 2.2, for 1 k < j n, we have. . . Yj Yk ,f | E Y k Y j | = Cov f jl(η j ) kl(ηk ) k . . Y j − i =1 b i , j X ∗ji Yj Yk ,f −f = Cov f jl(η j ) jl(η j ) kl(ηk ) k . Y j − i =1 b i , j X ∗ji Yj C E f −f jl(η j ) jl(η j ) k k E | i =1 b i , j X ∗ji | E | X − μ| I {| X − μ| > η j } C bi, j C jl(η j ) jl(η j ) i =1 k k E | X − μ| I {| X − μ| > η j } 1 C + kbk+1, j l jl(η j ) i =1 l = i k l j E | X − μ| I {| X − μ| > η j } 1 1 +k C l l jl(η j ) l =1 i =1 l=k+1 C C C. E | X − μ| I {| X − μ| > η j }. . . jl(η j ). l =1. l. j +k. 1 x. dx. k. k + k log. jl(η j ). E | X − μ| I {| X − μ| > η j }. k 1. l. jl(η j ). E | X − μ| I {| X − μ| > η j }. . . j. k. α k. j. k. ,. (2.8). where for our purpose, we fix α ∈ (0, 1). By Lemma 2.1 and jl(η j ) ∼ η2j , there exists j 0 such that. . E | X − μ| I | X − μ| > η j . l(η j ). ηj. ,. jl(η j ) 2η2j ,. for every j > j 0 . Then by (2.8) and (2.9), we have. (2.9).
(10) Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. 1 log2 n. 11. C. |I2| . log2 n k< j k j. j −1 n 1. C. . |E Yk Y j | 1. log n j =2 k=1. j 0 j −1 1. C. 1 1 E | X − μ| I {| X − μ| > η j } j α k k jl(η j ) log2 n k< j k j C. E | X − μ| I {| X − μ| > η j }. . kα j 3/2−α. 2. l(η j ). μ. 1. C. + log2 n j =2 k=1 kα j 3/2−α l(η j ) log2 n. . C. . 2. log n C. . log2 n C. . 2. log n C. . 2. log n. +. +. +. +. j −1 n 1. C. 33. 2. log n j = j +1 k=1 0. 1. kα j 3/2−α. C. j −1 n 1. C. j −1 n 1. j −1 n 1 j = j 0 +1 k =1. 1. kα j 3/2−α. . 1. l(η j ). l(η j ). ηj. . l(η j ). ηj. 1 1 √ log2 n j = j +1 k=1 kα j 3/2−α j 0. 2. log n j = j +1 k=1 0 n 1. C 2. log n j = j +1 j 0. 1. k α j 2 −α. . C log n. (2.10). .. By (2.5), (2.6) and (2.10), we have. E T n2 . C log n. τ. .. Let nk = ek , where ∞ . τ > 1. We get. E T n2k < ∞.. k =1. By Borel–Cantelli lemma, we have. T nk → 0,. a.s. as k → ∞.. Note that. log nk+1 log nk. =. (k + 1)τ kτ. → 1,. as k → ∞.. Since f is bounded, then for nk < n nk+1 , we obtain. n . . nk+1 k 1 Yk Yk 1 1 Yk Yk |T n | f f − Ef − Ef + log nk i log n i kl(ηk ) kl(ηk ) kl(ηk ) kl(ηk ) k 1. i =1. | T nk | + C. log nk+1 log nk. i =nk. − 1 → 0,. a.s. as n → ∞.. Thus (2.1) is proved. To prove (2.2), let. 2. 2 n Vk Vk 1 − Ef . Bn = f log n k kl(ηk ) kl(ηk ) 1. k =1. Then under the assumptions of Lemma 2.2, we see that. (2.11).
(11) 34. Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. E B n2 . n 1. C. log2 n k=1 k2. +. 2 2. 1 1 Vj Vk Cov f ,f 2 k j kl(ηk ) jl(η j ) log n 2. k< j. . 2 2. 2 k Vj V j − i =1 X 2ji Vk C 2 1 1 , − + Cov f f f 2 log n k j kl(ηk ) jl(η j ) jl(η j ) log n k< j. . . C log n C log n. +. +. 1 1 E|. C. k. i =1. log2 n k< j k j n 1. C. log2 n j =2 j. X 2ji |. . jl(η j ). . C log n. C log n. +. C. j −1 n 1 1 kl(η j ). log2 n j =2 k=1 k j jl(η j ) (2.12). .. The remaining part of the proof is similar to that of (2.1), then we get (2.2). To prove (2.3), let. Fn =. n 1. 1 log n. k =1. f. k. S k∗. . k l(ηk ). S k∗. . − Ef. k l(ηk ). .. (2.13). Then under the assumptions of Lemma 2.2, we obtain. E F n2 . n 1. C. log2 n k=1 k2. +. . . 1 1 S ∗j S∗ Cov f k , f 2 j l(η j ) log n k< j k j k l(ηk ) 2. . ∗ k. 1 1 S ∗j S j − i =1 X ∗ji S k∗ ,f −f + Cov f 2 log n k j j l ( η ) j l ( η ) log n k< j k l(ηk ) j j . . . . . C. C. C. C. log n C log n C log n C log n C log n. +. +. +. +. +. 1 1 E|. log2 n k< j k j. k. i =1. X ∗ji |. j l(η j ). 1 1 (E|. C. . log2 n k< j k j. k. X ∗ji |2 )1/2. i =1. . j l(η j ). j −1 n 1 1 (kl(η j ))1/2 l(η j ) log2 n j =2 k=1 k j 2. C. j −1 n 1. C. 1. log2 n j =2 k=1 k1/2 j 2 n 1. C. log2 n j =2 j 3/2. . C log n. (2.14). . 2. Then by the same argument as in (2.1), we get (2.3).. Lemma 2.3. Under the assumptions of Theorem 1.1, we have. lim. n→∞. 1 log n. n 1 k =1. . k. I. k . | X j − μ| > ηk. . . − EI. j =1. k . | X j − μ| > ηk. . = 0,. a.s.. (2.15). j =1. Proof. Let. Dn =. 1 log n. n 1 k =1. k. I. k j =1. | X j − μ| > ηk. . . − EI. k j =1. | X j − μ| > ηk. . =:. 1 log n. n 1 k =1. k. ζk ..
(12) Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. It is easy to prove that. E D n2. =. . 1 2. log n. n 1 k =1. Eζ 2 k2 k. +2. 11 i k. i <k. E ζi ζk =:. 1 log2 n. 35. [ J 1 + J 2 ].. (2.16). By the fact that {ζi } is bounded, we have. 1 log2 n. | J 1| . C. n 1. log2 n i =1 i 2. . C log n. (2.17). .. It is known that | I { A ∪ B } − I { B }| I { A } for any sets A and B, then we have. 1 2. log n. | J 2| . . . . C. 11. 2. log n i <k i k. | E ζi ζk |. i
(13) k 1 1 | X j − μ| > ηi , I | X j − μ| > ηk Cov I log2 n i <k i k j =1 j =1 i k k C 1 1 | X j − μ| > ηi , I | X j − μ| > ηk − I | X j − μ| > ηk Cov I log2 n i <k i k j =1 j =1 j = i +1 k k C 1 1 | X j − μ| > ηk − I | X j − μ| > ηk E I 2 log n i <k i k j =1 j = i +1 i C 1 1 | X j − μ| > ηk E I 2 i k log n C. j =1. i <k. . C. 11. log2 n i <k i k C. k −1 n 1 . 2. log n k=2 i =1 C. i P | X − μ| > ηk. k. P | X − μ| > ηk. n . 2. log n k=2. . . P | X − μ| > ηk .. (2.18). By Lemma 2.1 and kl(ηk ) ∼ ηk2 , there exists k0 such that. . . P | X − μ| > ηk . l(ηk ). ,. ηk2. kl(ηk ) 2ηk2 ,. (2.19). for every k > k0 . Then by (2.18) and (2.19), we have. 1 2. log n. | J 2| . C. k0 . 2. log n k=2 C log2 n. +. . P | X − μ| > ηk + C. n . log2 n k=k +1 0. l(ηk ). ηk2. C. n . 2. . log n k=k +1 0. . C log2 n. +. P | X − μ| > ηk. C. n 1. log2 n k=k +1 k 0. . . C log n. .. (2.20). Combining (2.16), (2.17) with (2.20), we have. E D n2 . C log n. .. The remaining part of the proof is similar to that of (2.1). So the proof is complete.. 2. Lemma 2.4. Under the assumptions of Theorem 1.1, we have. lim. n→∞. 1 log n. for any x ∈ R.. n 1 k =1. k. I . Yk 2kl(ηk ).
(14) x = Φ(x),. a.s.. (2.21).
(15) 36. Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. Proof. By the formula (2.10) in Pang et al. [17], we have. . Yk 2kl(ηk ). d − → N.. (2.22). Let f be a bounded Lipschitz function and have a Radon–Nikodyn derivative h bounded by C . From (2.22), we have. . Ef. Yk. → E f N (0, 1) ,. 2kl(ηk ). as k → ∞.. (2.23). On the other hand, note that (2.21) is equivalent to. lim. n→∞. n 1. 1 log n. k =1. k. f. . Yk 2kl(ηk ). = E f N (0, 1) ,. a.s.. (2.24). from Section 2 of Peligrad and Shao [18] and Theorem 7.1 of Billingsley [3]. Hence, to prove (2.21), it suffices to show that. R n =:. n 1. 1 log n. k =1. f. k. . Yk 2kl(ηk ). . − Ef. . Yk. → 0,. 2kl(ηk ). a.s. as n → ∞.. (2.25). By the formula (4) in Rempala and Wesolowski [20], we know k . b2i ,k = 2k − b1,k 2k.. (2.26). i =1. By (2.7) and (2.26), similarly to (2.8), for k < j, we get. k . . . . Y j − i =1 b i , j X ∗ji Yj Yj Yk Yk = Cov f Cov f f f f − , , 2 jl(η j ) 2 jl(η j ) 2 jl(η j ) 2kl(ηk ) 2kl(ηk ) 2 1/2 k k E | i =1 b i , j X ∗ji | 1 ∗ C C E b i , j X ji 2 jl(η j ) 2 jl(η j ) i =1 k 1/2 k 1/2 C C 2 1/2 2 bi, j l (η j ) √ (b i ,k + bk+1, j ) j 2 jl(η j ) i =1 i =1 k 1/2. . 1/2 k C C j 2 2 √ b i ,k + b k +1 , j √ k + k log2 j. i =1. C. √ k1/2 log j. where we can select. E R n2 . C log n. j k. . α. C. k. j. j. k. k. j. i =1. (2.27). ,. α ∈ (0, 1/2). Then it is easy to prove. .. The remaining part of the proof is similar to that of (2.1). So the proof is complete.. 2. 3. Proof of the main theorems Proof of Theorem 1.1. Let C i = S i /(i μ), we have k 1 ( C i − 1) = 2kl(ηk ) i =1 2kl(ηk ). μ. . k k 1 j =1 l = j. l. ∗. X kj +. k k 1 j =1 l = j. l. X∗. kj. =. 1 2kl(ηk ). [Y k + Y k ].. (3.1). Note that in order to prove (1.6), it is sufficient to show that. lim. n→∞. 1 log n. n 1 k =1. k. . μ. k . . log C i x = Φ(x), I √ 2V k i =1. a.s.. (3.2).
(16) Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. 37. for any x ∈ R. Furthermore, (3.2) is equivalent to the following two inequalities. lim sup n→∞. n 1. 1 log n. k =1. k. and. lim inf n→∞. n 1. 1 log n. k =1. k. I √. I √. k . μ. 2V k i =1 k . μ. 2V k i =1. log C i x Φ(x),. a.s.. (3.3). log C i x Φ(x),. a.s.. (3.4). for any x ∈ R. Firstly, we prove (3.3). For x 0 and 0 < δ1 < 1/2, we have. lim sup n→∞. n 1. 1 log n. lim sup n→∞. lim sup n→∞. +I. k =1. k. log n. k =1. log n. k =1. k . μ. 2V k i =1. . k. log C i x. . . log C i x 2(1 + δ1 )kl(ηk ) + I. i =1. . k . μ. I. . k . μ. I. k. n 1. 1. I √. n 1. 1. k . . . . V k2. > (1 + δ1 )kl(ηk ). . . log C i x 2(1 + δ1 )kl(ηk ) + I V k2 > (1 + δ1 )kl(ηk ). i =1. | X j − μ| > ηn. . . j =1. =: I 1 + I 2 + I 3 .. (3.5). For x < 0, we have. lim sup n→∞. n 1. 1 log n. lim sup n→∞. k =1. k. 1 log n 1. lim sup. n→∞ log n. +I. k . I √. n 1 k =1. k =1. 2V k i =1. . I. log C i x. . . log C i x 2(1 − δ1 )kl(ηk ) + I. i =1. . k. . k . μ. I. k. n 1. k . μ. k . μ. . . V k2. < (1 − δ1 )kl(ηk ). . . log C i x 2(1 + δ1 )kl(ηk ) + I V k2 < (1 − δ1 )kl(ηk ). i =1. | X j − μ| > ηn. . . j =1. =: J 1 + J 2 + J 3 .. (3.6). We need only to prove (3.3) holds for x 0. For x < 0, we have the same conclusion. Let f 1 be real-valued function such that. . . I |x| 1 + δ1 f 1 (x) I |x| 1 + δ1 /2 By Lemma 2.1 and kl(ηk ) ∼ ηk2 , for arbitrary. . E | X | I | X | ηk l(ηk )/ηk ,. and. . . sup f 1 (x) < ∞. x. > 0, there exists k1 such that. kl(ηk ) ηk2 /2,. (3.7). for every k > k1 . By Lemma 2.2 and (3.7), we have. I 2 lim sup n→∞. lim sup n→∞. 1 log n. n 1 k =1. k. f1. V k2. kl(ηk ). . . n n V k2 V k2 V k2 1 1 1 − E f1 + lim sup f1 E f1 log n k kl(ηk ) kl(ηk ) k kl(ηk ) n→∞ log n 1. k =1. k =1.
(17) 38. Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. n 1. 1. lim sup. log n. n→∞. log n. n→∞. V k2 kl(ηk ). n 1. 1. lim sup. k. k =1. E f1. lim sup n→∞. E ( V k2 )2. k (1 + δ/2)2 k2 l2 (ηk ). k =1. 1 log n. n 1 . k. k =1. lim sup n→∞. . P V k2 > (1 + δ/2)kl(ηk ). n 1 kE | X − μ|4 I {| X − μ| ηk }. C log n. k =1. (1 + δ/2)2k2l2 (ηk ). k. k1 n E | X − μ|4 I {| X − μ| ηk } C E | X − μ|4 I {| X − μ| ηk } lim sup + log n k2l2 (ηk ) k2l2 (ηk ) n→∞ log n. C. lim sup n→∞. k =1. k=k1 +1. n . ηk2l(ηk ) C C lim sup + lim sup k2l2 (ηk ) n→∞ log n n→∞ log n k=k1 +1. n . C. lim sup. log n. n→∞. = ,. k. k=k1 +1. a.s.. → 0, we obtain. Now, letting. I 2 = 0,. a.s.. (3.8). By Lemma 2.1 and Lemma 2.3, following the proof of (3.8), we have. I 3 = 0,. a.s.. (3.9). Now we estimate I 1 . Note that E | X | < ∞ for all 1 < p < 2 (since X belongs to the domain of attraction of the normal law). For our purpose, we fix 4/3 < p < 2. By Marcinkiewicz–Zygmund’s strong law of large numbers (see Chow and Teicher [5, p. 125]), for i large enough, we have p. | C i − 1| i 1 / p −1 ,. a.s.. It is easy to see that log(1 + x) − x = O (x2 ) as x → 0. Thus. k k k log C i − (C i − 1) C (C i − 1)2 Ck2/ p −1 , i =1. i =1. a.s.. i =1. Let {xn }, { yn } be two sequences of real numbers, then the following inequalities are satisfied. lim inf xn + lim sup yn lim sup(xn + yn ) lim sup xn + lim sup yn ,. (3.10). lim inf xn + lim inf yn lim inf(xn + yn ) lim inf xn + lim sup yn .. (3.11). n→∞. n→∞. n→∞. n→∞. n→∞. n→∞. n→∞. n→∞. By Lemma 2.1 and (3.10), for arbitrary small. lim sup n→∞. n 1. 1 log n. lim inf n→∞. k=k0 +1. log n. n→∞. lim sup n→∞. lim sup n→∞. I . k0 1. 1. + lim sup. k. . k =1. k. 1 log n 1 log n. k=k0 +1. k =1. k. k0 1 k =1. k. k ( C i − 1) x − ε. 2(1 + δ1 )kl(ηk ) i =1. I . n 1. n→∞. ε > 0, there exists k0 = k0 (ω, ε, x) such that for k > k0 ,. μ. k . μ. 2(1 + δ1 )kl(ηk ) i =1. n 1. 1 log n. . k. . I . I . . I . log C i x k . 2(1 + δ1 )kl(ηk ) i =1 k . 2(1 + δ1 )kl(ηk ) i =1. μ. . . μ. μ. n→∞. k . 2(1 + δ1 )kl(ηk ) i =1. log C i x. . log C i x = I 1. log C i x.
(18) Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. + lim sup n→∞. lim sup n→∞. log n. k=k0 +1. n 1. 1 log n. . n 1. 1. k. k=k0 +1. I . k. . k . μ. 2(1 + δ1 )kl(ηk ) i =1. 39. log C i x. k ( C i − 1) x + ε .. μ. I 2(1 + δ1 )kl(ηk ) i =1. (3.12). For any 0 < δ2 < 1/2, we have. lim sup n→∞. n 1. 1 log n. lim sup n→∞. k=k0 +1. k. . I 2(1 + δ1 )kl(ηk ) i =1. n 1. 1 log n. k ( C i − 1) x + ε. μ. k. k=k0 +1. I . Yk 2(1 + δ1 )kl(ηk ). .
(19) 1 x + ε + δ2 + lim sup n→∞. log n. n 1 k=k0 +1. I . k. |Y k | 2(1 + δ1 )kl(ηk ).
(20) δ2 . (3.13). By Lemma 2.4, we have. lim. n 1. 1. n→∞. log n. k=k0 +1. k. I .
(21). Yk 2(1 + δ1 )kl(ηk ). x + ε + δ2 = Φ. . . 1 + δ1 (x + ε + δ2 ) ,. a.s.. (3.14). By Lemma 2.1 and Lemma 2.2, following the proof of (3.8), we have. lim sup n→∞. n 1. 1 log n. k=k0 +1. k. I .
(22) δ2 = 0,. |Y k | 2(1 + δ1 )kl(ηk ). a.s.. (3.15). By (3.5)–(3.17) and letting δ1 → 0, δ2 → 0 and ε → 0, then we obtain that (3.3) holds for x 0. Thus (3.3) holds for all x ∈ R. Next we want to prove (3.4). For x 0 and 0 < δ1 < 1/2, we have. lim inf n→∞. n 1. 1 log n. lim inf n→∞. lim inf n→∞. −I. k =1. 1 log n 1 log n. k . k. I √. n 1 k =1. k =1. 2V k i =1. I. k. n 1. k . μ. μ. k. log C i x. k . . μ. . log C i x 2(1 − δ1 )kl(ηk ) − I. i =1. I. . k . . . V k2. < (1 − δ1 )kl(ηk ). . . log C i x 2(1 − δ1 )kl(ηk ) − I V k2 < (1 − δ1 )kl(ηk ). i =1. | X j − μ| > ηn. . .. j =1. For x < 0, we have. lim inf n→∞. n 1. 1 log n. lim inf n→∞. lim inf n→∞. −I. k =1. 1 log n 1 log n. k . k. I √. n 1 k =1. k. n 1 k =1. k. k . μ. 2V k i =1. I. μ. μ. | X j − μ| > ηn. log C i x. k . . . log C i x 2(1 + δ1 )kl(ηk ) + I. i =1. I. . k . . . V k2. > (1 + δ1 )kl(ηk ). . log C i x 2(1 + δ1 )kl(ηk ) − I V k2 > (1 + δ1 )kl(ηk ). i =1. . .. j =1. The remaining part of the proof is similar to the proof of (3.3), thus the proof is complete.. 2. .
(23) 40. Y. Zhang, X.-Y. Yang / J. Math. Anal. Appl. 376 (2011) 29–41. Proof of Theorem 1.2. The proof of Theorem 1.2 is similar to that of Theorem 1.1, and we will give a short explanation. In order to prove (1.7), it is sufficient to show that. lim. n→∞. . n 1. 1 log n. k =1. . k . μ. log C i x = Φ(x), I √ 2 V k i =1. k. a.s.. (3.16). for any x ∈ R. However, (3.16) is equivalent to the following two inequalities. lim sup n→∞. n 1. 1 log n. k =1. k. and. lim inf n→∞. 1 log n. n 1 k =1. k. . I √. . μ. k . 2 V k i =1. . log C i x Φ(x),. (3.17). . k . μ. a.s.. log C i x Φ(x), I √ 2 V k i =1. a.s.. (3.18). for any x ∈ R. V n2 = V n2 − n( X n − μ)2 V n2 , then by (3.3), we have Firstly, we want to prove (3.17). For x 0, note that . lim sup n→∞. 1 log n. n 1 k =1. k. I √. μ. k . 2 V k i =1. log C i x lim sup n→∞. n 1. 1 log n. k =1. k. I √. μ. k . 2V k i =1. log C i x Φ(x),. a.s.. (3.19). For x < 0, 0 < δ3 < 1/2 and 0 < δ4 < 1/2, note that V n2 = V n2 − n( X n − μ)2 , we have. . μ. k . . I √ log C i x 2 V k i =1. = I √. μ. 2 V k i =1. . +I √ I √ = I √ I √. k . μ. μ. 2 V k i =1 k . k . 2V k i =1. μ. k. k . 2V k i =1. μ. log C i x, V 2 (1 − δ3 ) V 2. k . 2V k i =1. k. log C i . x, V k2. <. (1 − δ3 ) V k2. . log C i (1 − δ3 )x + I V k2 < (1 − δ3 ) V k2. . . log C i (1 − δ3 )x + I k( X k − μ)2 > δ3 V k2. . . . log C i (1 − δ3 )x + I k( X k − μ)2 δ3 (1 − δ4 )kl(ηk ). + I V k2 < (1 − δ4 )kl(ηk ) k . μ I √ log C i (1 − δ3 )x + I | S k∗ | k δ3 (1 − δ4 )l(ηk ) 2V k i =1. +I. V k2. k | X j − μ| > ηk < (1 − δ4 )kl(ηk ) + 2I j =1. =: T 11 + T 12 + T 13 + 2T 14 . The remaining part of the proof is similar to that of Theorem 1.1, so we omit it here.. (3.20). 2. References [1] [2] [3] [4] [5] [6] [7]. B.C. Arnold, J.A. Villaseñor, The asymptotic distribution of sums of records, Extremes 1 (1998) 351–363. V. Bentkus, F. Götze, The Berry–Esseen bound for Student’s statistic, Ann. Probab. 24 (1996) 491–503. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. G.A. Brosamler, An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561–574. Y.S. Chow, H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, second ed., Springer, New York, 1988. ˝ B. Szyszkowicz, Q.Y. Wang, Darling–Erdös theorem for self-normalized partial sums, Ann. Probab. 31 (2003) 676–692. M. Csörgo, ˝ B. Szyszkowicz, Q.Y. Wang, Donsker’s theorem for self-normalized partial sums processes, Ann. Probab. 31 (2003) 1228–1240. M. Csörgo,.
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