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A General Law of Complete Moment Convergence for Self-Normalized Sums

Abstract

Let be a sequence of independent and identically distributed (i.i.d.) random variables, and is in the domain of the normal law and . In this paper, we obtain a general law of complete moment convergence for self-normalized sums.

1. Introduction and Main Results

Let be a sequence of independent and identically distributed (i.i.d.) random variables and put

(1.1)

for . We have the famous result following, that is, the complete convergence, for and ,

(1.2)

if and only if and when For , the sufficiency was proved by Hsu and Robbins [1], and the necessity by Erdös [2, 3]. For the case , we refer to Spitzer [4], and one can refer to Baum and Katz [5] for the general result. Note that the sums obviously tend to infinity as Thus it is interesting to discuss the precise rate and limit the value of as , where and are the positive functions defined on . We call and weighted function and boundary function, respectively. The first result in this direction was due to Heyde [6], who proved that

(1.3)

if and only if and Later, Chen [7] and Gut and Spătaru [8] both studied the precise asymptotics of the infinite sums as Moreover, Gut and Spătaru [9, 10] studied the precise asymptotics of the law of the iterated logarithm and the precise asymptotics for multidimensionally indexed random variables. Lanzinger and Stadtmüller [11], Spătaru [12, 13], and Huang and Zhang [14] obtained the precise rates in some different cases. While, Chow [15] discussed the complete moment convergence of i.i.d. random variables. He got the following result.

Theorem A.

Let be a sequence of i.i.d. random variables with . Suppose that and Then for any , one has

(1.4)

where

An important observation is that

(1.5)

From (1.5), we obtain that the complete moment convergence implies the complete convergence, that is, under the conditions of Theorem A, result (1.4) implies that

(1.6)

Thus, the complete moment convergence rates can reflect the convergence rates more directly than exact probability convergence rates.

For the investigation of complete moment convergence, some authors have researched it in different directions. For example, Jiang and Zhang [16] derived the precise asymptotics in the law of the iterated logarithm for the moment convergence of i.i.d. random variables by using the strong approximation method.

Theorem B.

Let be a sequence of i.i.d. random variables with , , and . Set . Then for , one has

(1.7)

Liu and Lin [17] introduced a new kind of complete moment convergence, Li [18] got precise asymptotics in complete moment convergence of moving-average processes, Zang and Fu [19] obtained precise asymptotics in complete moment convergence of the associated counting process, and Fu [20] also investigated asymptotics for the moment convergence of U-Statistics in LIL.

On the other hand, the so-called self-normalized sum is of the form . Using this notation we can write the classical Student -statistics as

(1.8)

In the recent years, the limit theorems for self-normalized sum or, equivalently, Student -statistics , have attracted more and more attention. Bentkus and Götze [21] obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing [22] derived exponential nonuniform Berry-Esseen bound. Hu et al. [23] achieved cramér type moderate deviations for the maximum of self-normalized sums. Giné et al. [24] established asymptotic normality of self-normalized sums as follows.

Theorem C.

Let be a sequence of i.i.d. random variables with . Then for any ,

(1.9)

holds if and only if is in the domain of attraction of the normal law, where is the distribution function of the standard normal random variable.

Meanwhile, Shao [25] showed a self-normalized large deviation result for without any moment conditions.

Theorem D.

Let be a sequence of positive numbers with and as . If and is slowly varying as then

(1.10)

In view of this theorem, and by applying s to it, one can obtain that for large enough and any , there exist and such that for . In particular, for , there exists such that

(1.11)

Inspired by the above results, the purpose of this paper is to study a general law of complete moment convergence for self-normalized sums. Our main result is as follows.

Theorem 1.1.

Suppose is in the domain of attraction of the normal law and . Assume that is differentiable on the interval , which is strictly increasing to , and differentiable function is nonnegative. Suppose that is monotone and . If is monotone nondecreasing, one assumes that Then, for , one has

(1.12)

Remark 1.2.

In Theorem 1.1, the condition is mild. For example, with some suitable conditions of , and and some others all satisfy this condition.

Remark 1.3.

If , by the strong law of large numbers, we have Then, we can easily obtain the following result:

(1.13)

Obviously, our main result is the generalization of i.i.d. random variables which have the finiteness of the second moments.

As examples, in Theorem 1.1, we can obtain some corollaries by choosing different and as follows.

Corollary 1.4.

Let , where , one has

(1.14)

Corollary 1.5.

Let , where , one has

(1.15)

Corollary 1.6.

Let , where , one has

(1.16)

2. Proof of Theorem 1.1

In this section, let for and is the inverse function of . Here and in the sequel, will denote positive constants, possibly varying from place to place. Theorem 1.1 will be proved via the following propositions.

Proposition 2.1.

One has

(2.1)

Here and in the sequel, denotes the standard normal random variable.

Proof.

Via the change of variable, for arbitrary , we have

(2.2)

Thus, if is monotone nonincreasing, then is nonincreasing. Hence

(2.3)

then, by (2.2), the proposition holds. If is nondecreasing, then by , for any there exists such that and for Thus we have

(2.4)

then, by (2.2) and letting we complete the proof of this proposition.

Proposition 2.2.

One has

(2.5)

Proof.

Set

(2.6)

It is easy to see, from (1.9), that , as Observe that

(2.7)

where

(2.8)

Thus for , it is easy to see that

(2.9)

Now we are in a position to estimate . From (1.11) and by Markov's inequality, we have

(2.10)

For , by Markov's inequality and (1.11), we have

(2.11)

From Cauchy inequality, it follows that

(2.12)

Therefore

(2.13)

Denote . Note that . Then, since the weighted average of a sequence that converges to 0 also converges to 0, it follows that, for any ,

(2.14)

The proof is completed.

Proposition 2.3.

One has

(2.15)

Proof.

By the similar argument in Proposition 2.1, it follows that

(2.16)

Then, this proposition holds.

Proposition 2.4.

One has

(2.17)

Proof.

By the similar argument in Proposition 2.1, it follows that

(2.18)

where

(2.19)

For , by (1.11), we have

(2.20)

For , using (1.11) again and noticing that , we have

(2.21)

By noting (2.12), it is easily seen that

(2.22)

Combining (2.20), (2.21), and (2.22), the proposition is proved.

Theorem 1.1 now follows from the above propositions using the triangle inequality.

References

  1. Hsu PL, Robbins H: Complete convergence and the law of large numbers. Proceedings of the National Academy of Sciences of the United States of America 1947, 33: 25–31. 10.1073/pnas.33.2.25

    Article  MathSciNet  MATH  Google Scholar 

  2. Erdös P: On a theorem of Hsu and Robbins. Annals of Mathematical Statistics 1949, 20: 286–291. 10.1214/aoms/1177730037

    Article  MathSciNet  MATH  Google Scholar 

  3. Erdös P: Remark on my paper "On a theorem of Hsu and Robbins". Annals of Mathematical Statistics 1950, 21: 138. 10.1214/aoms/1177729897

    Article  MathSciNet  MATH  Google Scholar 

  4. Spitzer F: A combinatorial lemma and its application to probability theory. Transactions of the American Mathematical Society 1956, 82: 323–339. 10.1090/S0002-9947-1956-0079851-X

    Article  MathSciNet  MATH  Google Scholar 

  5. Baum LE, Katz M: Convergence rates in the law of large numbers. Transactions of the American Mathematical Society 1965, 120: 108–123. 10.1090/S0002-9947-1965-0198524-1

    Article  MathSciNet  MATH  Google Scholar 

  6. Heyde CC: A supplement to the strong law of large numbers. Journal of Applied Probability 1975, 12: 173–175. 10.2307/3212424

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen R: A remark on the tail probability of a distribution. Journal of Multivariate Analysis 1978, 8(2):328–333. 10.1016/0047-259X(78)90084-2

    Article  MathSciNet  MATH  Google Scholar 

  8. Gut A, Spătaru A: Precise asymptotics in the Baum-Katz and Davis laws of large numbers. Journal of Mathematical Analysis and Applications 2000, 248(1):233–246. 10.1006/jmaa.2000.6892

    Article  MathSciNet  MATH  Google Scholar 

  9. Gut A, Spătaru A: Precise asymptotics in the law of the iterated logarithm. The Annals of Probability 2000, 28(4):1870–1883. 10.1214/aop/1019160511

    Article  MathSciNet  MATH  Google Scholar 

  10. Gut A, Spătaru A: Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables. Journal of Multivariate Analysis 2003, 86(2):398–422. 10.1016/S0047-259X(03)00050-2

    Article  MathSciNet  MATH  Google Scholar 

  11. Lanzinger H, Stadtmüller U: Refined Baum-Katz laws for weighted sums of iid random variables. Statistics & Probability Letters 2004, 69(3):357–368. 10.1016/j.spl.2004.06.033

    Article  MathSciNet  MATH  Google Scholar 

  12. Spătaru A: Exact asymptotics in log log laws for random fields. Journal of Theoretical Probability 2004, 17(4):943–965. 10.1007/s10959-004-0584-z

    Article  MathSciNet  MATH  Google Scholar 

  13. Spătaru A: Precise asymptotics for a series of T. L. Lai. Proceedings of the American Mathematical Society 2004, 132(11):3387–3395. 10.1090/S0002-9939-04-07524-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang W, Zhang L: Precise rates in the law of the logarithm in the Hilbert space. Journal of Mathematical Analysis and Applications 2005, 304(2):734–758. 10.1016/j.jmaa.2004.09.052

    Article  MathSciNet  MATH  Google Scholar 

  15. Chow YS: On the rate of moment convergence of sample sums and extremes. Bulletin of the Institute of Mathematics. Academia Sinica 1988, 16(3):177–201.

    MathSciNet  MATH  Google Scholar 

  16. Jiang Y, Zhang LX: Precise rates in the law of iterated logarithm for the moment of i.i.d. random variables. Acta Mathematica Sinica 2006, 22(3):781–792. 10.1007/s10114-005-0615-4

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu W, Lin Z: Precise asymptotics for a new kind of complete moment convergence. Statistics & Probability Letters 2006, 76(16):1787–1799. 10.1016/j.spl.2006.04.027

    Article  MathSciNet  MATH  Google Scholar 

  18. Li Y-X: Precise asymptotics in complete moment convergence of moving-average processes. Statistics & Probability Letters 2006, 76(13):1305–1315. 10.1016/j.spl.2006.04.001

    Article  MathSciNet  MATH  Google Scholar 

  19. Zang Q-P, Fu K-A: Precise asymptotics in complete moment convergence of the associated counting process. Journal of Mathematical Analysis and Applications 2009, 359(1):76–80. 10.1016/j.jmaa.2009.05.032

    Article  MathSciNet  MATH  Google Scholar 

  20. Fu K-A: Asymptotics for the moment convergence of U -statistics in LIL. Journal of Inequalities and Applications 2010, 2010:-8.

    Google Scholar 

  21. Bentkus V, Götze F: The Berry-Esseen bound for Student's statistic. The Annals of Probability 1996, 24(1):491–503.

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang Q, Jing B-Y: An exponential nonuniform Berry-Esseen bound for self-normalized sums. The Annals of Probability 1999, 27(4):2068–2088. 10.1214/aop/1022677562

    Article  MathSciNet  MATH  Google Scholar 

  23. Hu Z, Shao Q-M, Wang Q: Cramér type moderate deviations for the maximum of self-normalized sums. Electronic Journal of Probability 2009, 14(41):1181–1197.

    MathSciNet  MATH  Google Scholar 

  24. Giné E, Götze F, Mason DM: When is the Student -statistic asymptotically standard normal? The Annals of Probability 1997, 25(3):1514–1531.

    Article  MathSciNet  MATH  Google Scholar 

  25. Shao Q-M: Self-normalized large deviations. The Annals of Probability 1997, 25(1):285–328.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank an associate editor and the reviewer for their pointing out some serious problems in previous version. Thanks are also paid to the author's supervisor professor Zhengyan Lin of Zhejiang University in China and Dr. Keang Fu of Zhejiang Gongshang University in China for their help.

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Zang, Qp. A General Law of Complete Moment Convergence for Self-Normalized Sums. J Inequal Appl 2010, 760735 (2010). https://doi.org/10.1155/2010/760735

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