Precise asymptotics in the law of the iterated logarithm for statistic
© Pang et al.; licensee Springer. 2014
Received: 9 May 2013
Accepted: 7 March 2014
Published: 31 March 2014
Let be a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with zero mean and possibly infinite variance, be the rescaled range statistic, where , and . Then two precise asymptotics related to probability convergence for statistic are established under some mild conditions in this paper. Moreover, the precise asymptotics related to almost surely convergence for statistic is also considered under some mild conditions.
1 Introduction and main results
Let be a sequence of i.i.d. random variables and set for , and . Hsu and Robbins  and Erdős  established the well known complete convergence result: for any , if and only if and . Baum and Katz  extended this result and proved that, for , and , holds if and only if and . Since then, many authors considered various extensions of the results of Hsu-Robbins-Erdős and Baum-Katz. Some of them studied the precise asymptotics of the infinite sums as (cf. Heyde , Chen  and Spătaru ). We note that the above results do not hold for , this is due to the fact that by the central limit theorem when , where denotes a standard normal random variable. It should be noted that is irrespective of n. However, if is replaced by some other functions of n, the results of precise asymptotics may still hold. For example, by replacing by , Gut and Spătaru  established the following results called the precise asymptotics in the law of the iterated logarithm.
Of lately, by applying strong approximation method which is different from Gut and Spătaru’s, Zhang  gave the sufficient and necessary conditions for this kind of results to be held. One of his results is stated as follows.
It is worth mentioning that the precise asymptotics in a Chung-type law of the iterated logarithm, law of logarithm and Chung-type law of logarithm were also considered by Zhang , Zhang and Lin  and Zhang , respectively.
holds only if is a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with zero mean.
Recently, based on applying a similar method to the one employed by Gut and Spătaru , a result related to the precise asymptotics in the law of the iterated logarithm for statistic was established by Wu and Wen , that is, we have the following.
Here and in what follows, we denote and be a standard Brownian bridge.
It is natural to ask whether there is a similar result for statistic when ε tends to a constant which is not equal to zero. In the present paper, the positive answer will be partially given under some mild conditions with the help of strong approximation method, and, since statistic is defined in a self-normalized form, we will not restrict the finiteness of the second moment for . Moreover, a more strong result than Wu and Wen’s is established in this paper, based on which, a precise asymptotics related to a.s. convergence for statistic is considered under some mild conditions. Throughout the paper, we denote C a positive constant whose value can be different in different places. The following are our main results.
Theorem 1.2 Suppose is a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with , and the truncated second moment satisfies for some , and . Then for , (1.7) is true.
Remark 1.1 Note that X belonging to the domain of attraction of the normal law is equivalent to being a slowly varying function at ∞. We note also that is a weak enough assumption, which is satisfied by a large class of slowly varying functions such as and , for some .
Remark 1.2 When , the truncated second moment automatically satisfies the condition for some , and . Hence, Theorems 1.1-1.3 not only hold for the random variables with finite second moments, but they also hold for a class of random variables with infinite second moments. Especially, Theorem 1.2 includes Theorem D as a special case.
Remark 1.3 From Theorem C, one can see that the finiteness of the second moment does not guarantee the results about precise asymptotics in LIL for partial sums when . Moreover, it is clear that statistic is more complicated than partial sums. Hence, it seems that it is not possible, at least not easy, to prove (1.9) for under the conditions stated in Theorem 1.1 only. However, if we impose more strong moment conditions which are similar to (1.1) and (1.2) on , it would be possible to prove (1.9) for , by following the ideas in Zhang .
holds if as and , which seems maybe more natural due to (1.6).
The remaining of this paper is organized as follows. In Section 2, Theorem 1.1 will be proved when is a sequence of normal variables with zero mean. In Section 3, truncation method and strong approximation method will be employed to approximate the probability related to statistic. In Section 4, Theorem 1.1 and Theorem 1.2 will be proved, while in Section 5 the proof of Theorem 1.3 will be given, based on some preliminaries.
2 Normal case
Now, the main results in this section are stated as follows.
The proposition is proved now. □
Proof The proof can be found in Wu and Wen . □
3 Truncation and approximation
despite a little difference for the definitions of , which are from Pang, Zhang and Wang  and this paper, respectively.
Next, we will give the main result in this section as follows.
To show this proposition, the following lemmas are useful for the proof.
whenever , . Here, A is an universal constant.
for every positive x and .
Proof It is Lemma 1.1.1 of Csörgő and Révész . □
Equations (3.7)-(3.11) yield (3.6). The proposition is proved now. □
X is in the domain of attraction of the normal law,
Proof It is Lemma 1 in Csörgő, Szyszkowicz and Wang . □
The proof is completed now. □
Now, we turn to the proof of Proposition 3.1.
Letting completes the proof by Lemmas 3.3 and 3.7. □
4 Proofs of Theorems 1.1 and 1.2
since and satisfies (1.8). Now, it follows from Proposition 2.1, (3.4) and Lemma 3.6 that Theorem 1.1 is true. □
Letting completes the proof. □
5 Proof of Theorem 1.3
Then two lemmas which play key roles in the proof of Theorem 1.3 will be given, after which, we will finish the proof of Theorem 1.3.
Proof The essential difference between this lemma and Lemma 3.3 is the different truncation levels are imposed on the random variables in two lemmas. However, by checking the proof of Lemma 3.3 carefully, one can find that the proof of Lemma 3.3 is not sensitive to the powers of . Hence, one can easily finish the proof by similar arguments to those used in Lemma 3.3. We omit the details here. □
The proof is completed. □