Abstract
Let be a sequence of independent and identically distributed (i.i.d.) random variables and
is in the domain of attraction of the normal law and
. For
, we prove the precise asymptotics in Davis law of large numbers for
Journal of Inequalities and Applications volume 2009, Article number: 957056 (2009)
Let be a sequence of independent and identically distributed (i.i.d.) random variables and
is in the domain of attraction of the normal law and
. For
, we prove the precise asymptotics in Davis law of large numbers for
Throughout this paper, we let be a sequence of
random variables and
is in the domain of attraction of the normal law and
. Put
Also let Then by the well-known Davis laws of large numbers [1],
if and only if and
.
Gut and Spătaru [2] proved its precise asymptotics as follows.
Theorem 1 A.
Suppose that and
Then for
,
where stands for the
absolute moment of the standard normal distribution.
It is well known that, for random variables, Chow [3] discussed the complete moment convergence, and got the following result.
Theorem 1 B.
Let be a sequence of
random variables with
. Assume
,
, and
Then for any
,
On the other hand, the past decade has witnessed a significant development on the limit theorems for the so-called self-normalized sum . Bentkus and Götze [4] obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing [5] derived exponential nonuniform Berry-Esseen bound. Giné et al. [6], established asymptotic normality of self-normalized sums.
Theorem 1 C.
Let be a sequence of
random variables with
. Then for any
,
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.
Shao [7] showed a self-normalization large deviation result for without any moment conditions.
Theorem 1 D.
Let be a sequence of positive numbers with
and
as
. If
and
is slowly varying as
then
Since then, many subsequent developments of self-normalized sums have been obtained. For example, Csörgő et al. [8] have established Darling-Erdös theorem for self-normalized sums, and they [9] have also obtained Donsker's theorem for self-normalized partial sums processes.
Inspired by the above results, in this note we study the precise asymptotics in Davis law of large numbers for the moment of self-normalized sums. Our main result is as follows.
Theorem 1.1.
Suppose is in the domain of attraction of the normal law and
. Then, for
and
, one has
here and in the sequel, is the standard normal random variable.
Remark 1.2.
If and
, by the strong law of large numbers, we have
Then, we can easily obtain the following result:
Remark 1.3.
As is well known, the strong approximation method is taken in order to obtain such an analogous result, however, this method is not applicable here.
In this section, we set for
and
. Here and in the sequel,
will denote positive constants, possibly varying from place to place, and
means the largest integer
. The proof of Theorem 1.1 is based on the following propositions.
Proposition 2.1.
For , one has
Proof.
Via the change of variable , we have
Proposition 2.2.
For , one has
Proof.
Set Then, by (1.5), it is easy to see
as
Observe that
where
Thus for , it is easy to see
Now we are in a position to estimate . From (1.6), and by applying
to it, we can obtain that for large enough
and any
, there exist C and b such that
for
. In particular, for
, there exists
such that
Hence, by Markov's inequality and (2.7), we have
For , by Markov's inequality and (2.7), we have
From Cauchy inequality, it follows that
Therefore
Denote , then, since the weighted average of a sequence that converges to 0 also converges to 0, it follows that, for any
,
The proof is completed.
Proposition 2.3.
For , one has
Proof.
Note that
So this proposition is proved now.
Proposition 2.4.
For , one has
Proof.
Note that
where
For , by (2.7), we have
For , using (2.7) again, we have
By noting that (2.10), it is easily seen that
Combining (2.18), (2.19), and (2.20), the proposition is proved.
Our main result follows from the propositions using the triangle inequality.
Davis JA: Convergence rates for probabilities of moderate deviations. Annals of Mathematical Statistics 1968, 39: 2016–2028. 10.1214/aoms/1177698029
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
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.
Bentkus V, Götze F: The Berry-Esseen bound for Student's statistic. The Annals of Probability 1996,24(1):491–503.
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
Giné E, Götze F, Mason DM: When is the student -statistic asymptotically standard normal? The Annals of Probability 1997,25(3):1514–1531.
Shao Q-M: Self-normalized large deviations. The Annals of Probability 1997,25(1):285–328.
Csörgő M, Szyszkowicz B, Wang Q: Darling-Erdős theorem for self-normalized sums. The Annals of Probability 2003,31(2):676–692.
Csörgő M, Szyszkowicz B, Wang Q: Donsker's theorem for self-normalized partial sums processes. The Annals of Probability 2003,31(3):1228–1240. 10.1214/aop/1055425777
The author thanks the referees for pointing out some errors in a previous version, as well as for several comments that have led to improvements in this work. Thanks are also due to Doctor Ke-ang Fu of Zhejiang University in china for his valuable suggestion in the preparation of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Zang, Qp. A Limit Theorem for the Moment of Self-Normalized Sums. J Inequal Appl 2009, 957056 (2009). https://doi.org/10.1155/2009/957056
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DOI: https://doi.org/10.1155/2009/957056