- Research
- Open Access
- Published:
A characterization of Chover-type law of iterated logarithm
SpringerPlus volume 3, Article number: 386 (2014)
Abstract
Let 0 < α ≤ 2 and − ∞ <β <∞. Let {X n ;n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set S n = X1+⋯+X n , n ≥ 1. We say X satisfies the (α,β)-Chover-type law of the iterated logarithm (and write X∈C T L I L(α,β)) if almost surely. This paper is devoted to a characterization of X ∈C T L I L(α,β). We obtain sets of necessary and sufficient conditions for X∈C T L I L(α,β) for the five cases: α = 2 and 0 < β <∞, α = 2 and β = 0, 1<α<2 and −∞<β<∞, α = 1 and −∞ <β <∞, and 0 < α <1 and −∞ <β <∞. As for the case where α = 2 and −∞ <β <0, it is shown that X∉C T L I L(2,β) for any real-valued random variable X. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm (i.e., X∈C T L I L(α,1/α)) is given; that is, X∈C T L I L(α,1/α) if and only if where whenever 1< α ≤ 2.
Mathematics Subject Classification (2000)
Primary: 60F15; Secondary: 60G50
1 Introduction
Throughout, {X n ; n ≥ 1} is a sequence of independent copies of a real-valued random variable X. As usual, the partial sums of independent identically distributed (i.i.d.) random variables X n , n ≥ 1 will be denoted by . Write L x= log(e∨x), x ≥ 0.
When X has a symmetric stable distribution with exponent α∈(0,2), i.e., for t∈(−∞,∞), Chover 1966 proved that
This is what we call the classical Chover law of iterated logarithm (LIL). Since then, several papers have been devoted to develop the classical Chover LIL. See, for example, Hedye 1969 showed that (1.1) holds when X is in the domain of normal attraction of a nonnormal stable law, Pakshirajan and Vasudeva 1977 discussed the limit points of the sequence , Kuelbs and Kurtz 1974 obtained the classical Chover LIL in a Hilbert space setting, Chen 2002 obtained the classical Chover LIL for the weighed sums, Vasudeva 1984, Qi and Cheng 1996, Peng and Qi 2003 established the Chover LIL when X is in the domain of attraction of a nonnormal stable law, Scheffler 2000 studied the classical Chover LIL when X is in the generalized domain of operator semistable attraction of some nonnormal law, Chen and Hu 2012extended the results of Kuelbs and Kurtz 1974 to an arbitrary real separable Banach space, and so on. It should be pointed out that the previous papers only gave sufficient conditions for the classical Chover LIL.
Motivated by the previous study of the classical Chover LIL, we introduce a general Chover-type LIL as follows.
Definition 1.1
Let 0 < α ≤ 2 and −∞ <β <∞. Let {X,X n ; n ≥ 1} be a sequence of real-valued i.i.d. random variables. We say X satisfies the (α,β)-Chover-type law of the iterated logarithm (and write X∈C T L I L(α,β)) if
From the classical Chover LIL and Definition 1.1, we see that X∈C T L I L(α,1/α) (i.e., (1.2) holds with β=1/α) when X has a symmetric stable distribution with exponent α∈(0,2).
This paper is devoted to a characterization of X∈C T L I L(α,β). The main results are stated in Section 2. We obtain sets of necessary and sufficient conditions for X∈C T L I L(α,β) for the five cases: α = 2 and 0 < β <∞ (see Theorem 2.1), α=2 and β = 0 (see Theorem 2.2), 1 < α < 2 and −∞<β <∞ (see Theorem 2.3), α = 1 and −∞ < β < ∞ (see Theorem 2.4), and 0 < α < 1 and −∞ < β < ∞ (see Theorem 2.5). The proofs of Theorems 2.1-2.5 are given in Section 4. For proving Theorems 2.1-2.5, three preliminary lemmas are stated in Section 3. Some llustrative examples are provided in Section 5
2 Statement of the main results
The main results of this paper are the following five theorems. We begin with the case where α = 2 and 0 < β <∞.
Theorem 2.1.
Let 0 < β <∞. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then
if and only if
For the case where α = 2 and β = 0, we have the following result.
Theorem 2.2.
Let {X,X n ;n ≥ 1} be a sequence of i.i.d. non-degenerate real-valued random variables. Then
if and only if
In either case, we have
Remark 2.1
Let c be a constant. Note that
Thus, from Theorem 2.2, we conclude that, for any −∞ <β <0, X ∉C T L I L(2,β) for any real-valued random variable X.
In the next three theorems, we provide necessary and sufficient conditions for X ∈C T L I L(α,β) for the three cases where 1 < α < 2 and −∞ < β < ∞, α = 1 and −∞ <β <∞, and 0 < α <1 and −∞ <β <∞ respectively.
Theorem 2.3.
Let 1<α<2 and −∞ <β <∞. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then
if and only if
Theorem 2.4.
Let −∞ <β <∞. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then
if and only if
In particular, and imply that
Theorem 2.5.
Let 0 < α <1 and −∞ <β <∞. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then
if and only if
Remark 2.2.
From our Theorems 2.1, 2.3, 2.4, and 2.5, a simple and precise characterization of the classical Chover LIL (i.e., X∈C T L I L(α,1/α)) is obtained as follows. For 0 < α≤ 2, we have
if and only if
Our Theorems 2.1-2.5 also imply the following two interesting results.
Corollary 2.1
Let 0 < α≤ 2. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then
if and only if
Corollary 2.2.
Let 0 < α≤ 2. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then
if and only if
From our main results Theorems 2.1-2.5 and Corollaries 2.1-2.2 above, an almost unified characterization for 0 < α≤ 2 stated as the following result was so kindly presented to us by a referee.
Theorem 2.6.
Let 0 < α≤ 2. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Assume that and whenever . Then
where
3 Preliminary lemmas
To prove the main results, we use the following three preliminary lemmas. The first lemma is new and may be of independent interest.
Lemma 3.1
Let {a n ; n ≥ 1} be a sequence of real numbers. Let {c n ; n ≥ 1} be a sequence of positive real numbers such that
Then we have
(i) There exists a constant −∞<β<∞ such that
if and only if
(ii) There exists a constant −∞<β<∞ such that
if and only if
(iii) There exists a constant −∞<β<∞ such that
if and only if
Proof We prove the sufficiency part of Part (i) first. It follows from (3.3) that the set
Note that
We thus conclude that
which ensures (3.2).
We now prove the necessity part of Part (i). For all b≠β, let h=(b+β)/2. Then
It follows from (3.2) that the set
Note that
We thus have that
Note that (3.1) implies that
Thus it follows from (3.4) that
i.e., (3.3) holds.
We leave the proofs of Parts (ii) and (iii) to the reader since they are similar to the proof of Part (i). This completes the proof of Lemma 3.1. □
The following result is a special case of Corollary 2 of Einmahl and Li 2005.
Lemma 3.2.
Let b>0. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. random variables. Then
if and only if
where , x ≥ 0.
The following result is a generalization of Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and follows easily from Theorems 1 and 2 of Feller 1946.
Lemma 3.3.
Let 0 < α<2 and −∞<b<∞. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. random variables. Then
if and only if
where whenever either 1<α<2 or α=1 and −∞<b≤ 0.
4 Proofs of the main results
In this section, we only give the proofs of Theorems 2.1-2.2. By applying Lemmas 3.1 and 3.3, the proofs of Theorems 2.3-2.5 involves only minor modifications of the proof of Theorem 2.1 and will be omitted.
Proof of Theorem 2.1 We prove the sufficiency part first. Note that the second part of (2.2) implies that
We thus see that for all b>β,
where h=(b+β)/2. Since β<h<b, it follows from (4.1) that
We thus conclude from (2.2) that
which, by applying Lemma 3.2, ensures that
Since β>0, the second part of (2.2) implies that
which ensures that
Let and c n =L L n, n ≥ 1. It then follows from (4.2) and (4.3) that
By Lemma 3.1, we see that (4.4) is equivalent to
i.e., (2.1) holds.
We now prove the necessity part. By Lemma 3.1, (2.1) is equivalent to (4.4) which ensures that (4.2) holds. By Lemma 3.2, we conclude from (4.2) that
Since 0 < β<∞, it follows from (4.5) that
If β1<β then, using the argument in the proof of the sufficiency part, we have that
Hence, by Lemma 3.1, (4.6) implies that
which is in contradiction to (2.1). Thus (2.2) holds. The proof of Theorem 2.1 is complete. □
Proof of Theorem 2.2 Using the same argument used in the proof of the sufficiency part of Theorem 2.1, we have from (2.4) that
Since X is a non-degenerate random variable, by the classical Hartman-Wintner-Strassen LIL, we have that
which implies that
Let and c n =L L n, n ≥ 1. It then follows from (4.7) and (4.8) that
By Lemma 3.1, we see that (4.9) is equivalent to
i.e., (2.5) holds, so does (2.3).
Using the same argument used in the proof of the necessity part of Theorem 2.1, we conclude from (2.3) that
Clearly
Thus (2.4) holds. The proof of Theorem 2.2 is therefore complete. □
5 Example
In this section, we provide the following examples to illustrate our main results. By applying Theorems 2.3-2.5, we rededuce the classical Chover LIL in the first example.
Example 5.1.
Let 0 < α≤ 2. Let X be a symmetric real-valued stable random variable with exponent α. Clearly, whenever 1<α≤ 2.
For 0 < α<2, we have
it follows that
and hence that
Thus, by Theorems 2.3-2.5, X∈C T L I L(α,1/α) (i.e., the classical Chover LIL follows).
However, for α=2, we have . Hence, by Theorems 2.1 and 2.2, we see that X∉C T L I L(2,1/2) but X∈C T L I L(2,0).
From our second example, we will see that X∈C T L I L(α,β) for some certain α and β even if the distribution of X is not in the domain of attraction of the stable distribution with exponent α.
Example 5.2.
Let 0 < α≤ 2. Let d n = exp(2n), n ≥ 1. Given −∞<λ<∞. Let X be a symmetric i.i.d. real-valued random variable such that
where
Then
Thus the distribution of X is not in the domain of attraction of the stable distribution with exponent α. Also whenever either 1<α≤ 2 or α=1 and λ<0. It is easy to see that
and hence that
Thus, by Theorems 2.1-2.5, we have
-
(1)
If α = 2 and 0 < λ<∞, then X∈C T L I L(2,λ/2).
-
(2)
If α=2 and −∞<λ≤ 0, then X∈C T L I L(2,0).
-
(3)
If 0 < α<2, then X∈C T L I L(α,λ/α).
Our third example shows that X may satisfy the other Chover-type LIL studied by Chen and Hu 2012 when X∉C T L I L(α,β).
Example 5.3.
Define the density function f(x) of X by
where p≠0, 0 < γ<1, c=c(α,p,γ) is a positive constant such that . On simplification one can show that for any −∞<b<∞
From Theorem 2.1-2.5, we have
-
(1)
If α = 2 and p < 0, then X∈C T L I L(2,0).
-
(2)
If α = 2 and p > 0, then X∉C T L I L(2,β) for any 0≤ β <∞.
-
(3)
If 0 < α < 2, then X ∉C T L I L(α,β) for any −∞ <β <∞. However, for 0 < α < 2, by Theorem 3.1 in Chen and Hu 2012, we have
where B(x) is the inverse function of xα/ exp(p(logx)γ), x ≥e.
References
Chen P: Limiting behavior of weighted sums with stable distributions. Statist Probab Lett 2002, 60: 367-375. 10.1016/S0167-7152(02)00286-9
Chen P, Hu T-C: Limiting behavior for random elements with heavy tail. Taiwanese J Math 2012, 16: 217-236.
Chover J: A law of the iterated logarithm for stable summands. Proc Amer Math Soc 1966, 17: 441-443. 10.1090/S0002-9939-1966-0189096-2
Einmahl U, Li D: Some results on two-sided LIL behavior. Ann Probab 2005, 33: 1601-1624. 10.1214/009117905000000198
Feller W: A limit theoerm for random variables with infinite moments. Amer J Math 1946, 68: 257-262. 10.2307/2371837
Heyde CC: A note concerning behaviour of iterated logarithm type. Proc Amer Math Soc 1969, 23: 85-90. 10.1090/S0002-9939-1969-0251772-3
Kuelbs J, Kurtz T: Berry-esseen estimates in Hilbert space and an application to the law of the iterated logarithm. Ann Probab 1974, 2: 387-407. 10.1214/aop/1176996655
Pakshirajan RP, Vasudeva R: A law of the iterated logarithm for stable summands. Tran Amer Math Soc 1977, 232: 33-42.
Peng L, Qi Y: Chover-type laws of the iterated logarithm for weighted sums. Statist Probab Lett 2003, 65: 401-410. 10.1016/j.spl.2003.08.009
Qi Y, Cheng P: On the law of the iterated logarithm for the partial sum in the domain of attraction of stable distribution. Chin Ann Math 1996, 17(A):195-206. (in Chinese)
Scheffler H-P: A law of the iterated logarithm for heavy-tailed random vectors. Probab Theory Relat Fields 2000, 116: 257-271. 10.1007/PL00008729
Vasudeva R: Chover’s law of the iterated logarithm and weak convergence. Acta Math Hung 1984, 44: 215-221. 10.1007/BF01950273
Acknowledgements
The authors are grateful to the referees for very carefully reading the manuscript. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Pingyan Chen was partially supported by the National Natural Science Foundation of China (No. 11271161).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DL and PC contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Li, D., Chen, P. A characterization of Chover-type law of iterated logarithm. SpringerPlus 3, 386 (2014). https://doi.org/10.1186/2193-1801-3-386
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/2193-1801-3-386
Keywords
- (α,β)-Chover-type law of the iterated logarithm
- Sums of i.i.d. random variables
- Symmetric stable distribution with exponent α