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Precise asymptotics in the law of the iterated logarithm for statistic
Journal of Inequalities and Applications volume 2014, Article number: 137 (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.
Theorem A Suppose is a sequence of i.i.d. random variables with , and for some , and let for some . Then
Theorem B Suppose is a sequence of i.i.d. random variables with and . Then
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.
Theorem C Let and and let be a function of ε such that
Here , and here and in what follows is a gamma function. Conversely, if either (1.3) or (1.4) holds for , and some , then (1.1) holds and
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.
The above-mentioned results are all related to partial sums. This paper is devoted to the study of some precise asymptotics for the rescaled range statistic (or the statistic), defined by , where
This statistic, introduced by Hurst  when he studied hydrology data of the Nile river and reservoir design, plays an important role in testing statistical dependence of a sequence of random variables and has been used in many practical subjects such as hydrology, geophysics and economics, etc. Because of the importance of this statistic, some people studied some limit theorems for statistic. Among them, Feller  established the limit distribution of for i.i.d. case, Mandelbrot  studied weak convergence of for a more general case, while Lin [15–17] and Lin and Lee  established the law of the iterated logarithm for under various assumptions. Among Lin’s results, we notice that Lin  proved that
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.
Theorem D Suppose is a sequence of i.i.d. random variables with , . Then for ,
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 1.1 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 . Let , and be a function of ε such that
Then we have
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.
Theorem 1.3 Suppose is a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with , and satisfies for some , and . Then for any , we have
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 .
Remark 1.4 Checking the proof of Theorem 1.1, one can find that
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
In this section, Theorem 1.1 in the case that is a sequence of normal random variables with zero mean is proved. In order to do it, we firstly recall that is a standard Brownian bridge and . The distribution of Y plays an important role in our first result, and, fortunately, it has been given by Kennedy :
Now, the main results in this section are stated as follows.
Proposition 2.1 Let , and be a function of ε such that
Then we have
Proof Firstly, it follows easily from (2.1) that
as . Then, by condition (2.2), one has
as uniformly in for some . Hence, for above-mentioned and any , there exists an integer such that, for all and ,
Obviously, it suffices to show
for proving Proposition 2.1 by the arbitrariness of θ. To this end, by noting that the limit in (2.3) does not depend on any finite terms of the infinite series, we have
The proposition is proved now. □
Proposition 2.2 For any ,
Proof The proof can be found in Wu and Wen . □
3 Truncation and approximation
In this section, we will use the truncation method and strong approximation method to show that the probability related to with suitable normalization can be approximated by that for Y. To do this, we first give some notations. Put and
For each n and , we let
It follows easily that
Furthermore, we denote be the truncated R statistic which is defined by the first expression of (1.5) with every being replaced by , . In addition, for any , all and k large enough, following the lines of the proof of (2.4) in Pang, Zhang and Wang , we easily have
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.
Proposition 3.1 For any , and , there exists a sequence of positive numbers such that, for any ,
To show this proposition, the following lemmas are useful for the proof.
Lemma 3.1 For any sequence of independent random variables with zero mean and finite variance, there exists a sequence of independent normal variables with and such that, for all and ,
whenever , . Here, A is an universal constant.
Lemma 3.2 Let be a standard Wiener process. For any there exists a constant such that
for every positive x and .
Proof It is Lemma 1.1.1 of Csörgő and Révész . □
Lemma 3.3 For any , and , there exists a sequence of positive numbers such that, for any ,
Proof Let , then obviously, satisfies (3.5). For each n, let be a standard Wiener process, then we have and
We consider first. Clearly,
It follows from Lemma 3.1 and (3.3) that, for all ,
Next, we treat with . Clearly, one has
It follows from Lemma 3.2 that
which obviously leads to
On the other hand,
which also obviously leads to
Equations (3.7)-(3.11) yield (3.6). The proposition is proved now. □
Lemma 3.4 For any and , one has
Proof It follows from (3.3) that
Lemma 3.5 Let X be a random variable. Then the following statements are equivalent:
X is in the domain of attraction of the normal law,
Proof It is Lemma 1 in Csörgő, Szyszkowicz and Wang . □
Lemma 3.6 For any and , one has, for ,
Proof It is easy to see that, for large n,
by Lemma 3.5. Applying Lemma 3.4, we only need to show
for proving Lemma 3.6. Consider the first part of (3.13) first. By employing Lemma 3.5 and Bernstein’s inequality (cf. Lin and Bai ), we have for any fixed
The second part of (3.13) can be proved by similar arguments. Now, let us consider the third part of (3.13). It follows from Markov’s inequality that
The proof is completed now. □
Lemma 3.7 Define . Then for any and , one has
Proof Firstly, notice that statistic has an equivalent expression
and so does with being replaced by in (3.15), . That is,
Let , then
then it is easily seen that, for ,
since . Hence, it follows from Lemma 3.4 that
When , applying (3.3) yields
Now, we turn to the proof of Proposition 3.1.
Proof of Proposition 3.1 Applying Lemma 3.3, one easily has
Also, one has
Letting completes the proof by Lemmas 3.3 and 3.7. □
4 Proofs of Theorems 1.1 and 1.2
Proof of Theorem 1.1 For any and , we have
Noting that , one easily has
and for large n,
which tends to zero as and . Hence, we have
since and satisfies (1.8). Now, it follows from Proposition 2.1, (3.4) and Lemma 3.6 that Theorem 1.1 is true. □
Proof of Theorem 1.2 For any , applying similar arguments to those used in (4.1), we have for large n,
as . Hence, Proposition 2.2, (3.4) and Lemma 3.6 guarantee that
Letting completes the proof. □
5 Proof of Theorem 1.3
In this section, we first modify the definition in (3.1) as follows:
Then one easily has . Moreover, we define for each n and ,
Secondly, we give two notations related to the truncated statistic. That is,
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.
Lemma 5.1 Suppose is a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with , and satisfies for some , and . Then, for any and , there exists a sequence of positive numbers such that, for any ,
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. □
Lemma 5.2 Suppose is a sequence of i.i.d. random variables which is in the domain of attraction of the normal law with , and let be a real function such that and . Then for any , and , we have
Proof Firstly, we consider the first part of (5.2). For , since is independent of
It follows that
Hence, for any and , we have
Consider the second part of (5.2). Similar arguments used in (5.3) leads easily to
It follows that
Consider the third part of (5.2). The similar arguments used in (5.3) also lead easily to
which implies that
Finally, we turn to handling the fourth part of (5.2). By employing Lemma 3.5 one has
The proof is completed. □
Proof of Theorem 1.3 At the beginning of the proof, we first give an upper bound and a lower bound for the indicator function of statistic. For any with , and large n, one has the following fact:
since one easily has
Also, one has, for any with , and large n,
Denote for any and fixed . Let be real functions such that for and
Define , . Then it follows from Lemma 5.2 that
which together with the Borel-Cantelli lemma easily yield
as . Similar arguments also yield
as . Denote . Using the inequality (5.4), one has
We are going to treat the above terms, respectively. In view of (5.5), (5.6), Lemma 5.1 and Proposition 2.2, one has
Applying (5.5), (5.7), and Bernstein’s inequality, one has, for any ,
Similarly, one can prove
For the fourth part of (5.8), by similar arguments to those used in (5.9) and Lemma 3.4, we have
and the details are omitted here. As for the fifth part of (5.8), one can easily show that, for any fixed ,
Hence, it follows from (5.8)-(5.12) that
On the other hand,