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A characterization of Chover-type law of iterated logarithm

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 XC T L I L(α,β)) if limsup n S n n 1 / α ( log log n ) 1 = e β 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 XC 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 XC 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., XC T L I L(α,1/α)) is given; that is, XC T L I L(α,1/α) if and only if inf b : E | X | α ( log ( e | X | ) ) < =1/α where EX=0 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 S n = i = 1 n X i ,n1. Write L x= log(ex), x ≥ 0.

When X has a symmetric stable distribution with exponent α(0,2), i.e., E e itX = e | t | α for t(−,), Chover 1966 proved that

limsup n S n n 1 / α ( log log n ) 1 = e 1 / α almost surely (a.s.).
(1.1)

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 {| S n / n 1 / α | ( log log n ) 1 ;n2}, 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 XC T L I L(α,β)) if

limsup n S n n 1 / α ( log log n ) 1 = e β a.s.
(1.2)

From the classical Chover LIL and Definition 1.1, we see that XC 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 XC T L I L(α,β). The main results are stated in Section 2. We obtain sets of necessary and sufficient conditions for XC 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

XCTLIL(2,β),i.e., limsup n S n n ( log log n ) 1 = e β a.s.
(2.1)

if and only if

EX=0andinf b > 0 : E X 2 ( L | X | ) 2 b < =β.
(2.2)

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

limsup n S n n ( log log n ) 1 1a.s.
(2.3)

if and only if

EX=0andinf b > 0 : E X 2 ( L | X | ) 2 b < =0.
(2.4)

In either case, we have

XCTLIL(2,0),i.e., limsup n S n n ( log log n ) 1 =1a.s.
(2.5)

Remark 2.1

Let c be a constant. Note that

limsup n nc n ( log log n ) 1 = 0 if c = 0 , if c 0 .

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

X CTLIL ( α , β ) , i.e., limsup n S n n 1 / α ( log log n ) 1 = e β a.s.

if and only if

EX = 0 and inf b : E | X | α ( L | X | ) < = β.

Theorem 2.4.

Let − <β <. Let {X,X n ;n ≥ 1} be a sequence of i.i.d. real-valued random variables. Then

X CTLIL ( 1 , β ) , i.e., limsup n S n n ( log log n ) 1 = e β a.s.

if and only if

inf b : E | X | ( L | X | ) b < = β if β > 0 , either E | X | < and EX 0 or inf b : E | X | ( L | X | ) b < = 0 if β = 0 , EX = 0 and inf b : E | X | ( L | X | ) b < = β if β < 0 .

In particular, E|X|< and EX0 imply that

lim n S n n ( log log n ) 1 = 1 a.s.

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

X CTLIL ( α , β ) , i.e., limsup n S n n 1 / α ( log log n ) 1 = e β a.s.

if and only if

inf b : E | X | α ( L | X | ) < = β.

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., XC T L I L(α,1/α)) is obtained as follows. For 0 < α≤ 2, we have

X CTLIL ( α , 1 / α ) , i.e., limsup n S n n 1 / α ( log log n ) 1 = e 1 / α a.s.

if and only if

inf b : E | X | α ( L | X | ) < = 1 / α where EX = 0 whenever 1 < α 2 .

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

lim n S n n 1 / α ( log log n ) 1 = 0 a.s.

if and only if

inf b : E | X | α ( L | X | ) < = if 0 < α < 1 , EX = 0 and inf b : E | X | α ( L | X | ) < = if 1 α < 2 , X = 0 a.s. if α = 2 .

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

limsup n S n n 1 / α ( log log n ) 1 = a.s.

if and only if

inf b : E | X | α ( L | X | ) < = if 0 < α 1 , eithor EX 0 or inf b : E | X | α ( L | X | ) < = if 1 < α 2 .

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 E( X 2 )= and EX=0 whenever E|X|<. Then

limsup n S n n 1 / α ( log log n ) 1 = e β a.s.

where

β = inf b : E | X | α ( L | X | ) < [ , ] .

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

lim n c n =∞.
(3.1)

Then we have

(i) There exists a constant −<β< such that

limsup n a n 1 / c n = e β
(3.2)

if and only if

limsup n a n e bc n = 0 for all b > β , for all b < β ;
(3.3)

(ii) There exists a constant −<β< such that

liminf n a n 1 / c n = e β

if and only if

liminf n a n e bc n = 0 for all b > β , for all b < β ;

(iii) There exists a constant −<β< such that

lim n a n 1 / c n = e β

if and only if

lim n a n e bc n = 0 for all b > β , for all b < β.

Proof We prove the sufficiency part of Part (i) first. It follows from (3.3) that the set

n 1 ; a n e bc n > 1 has finitely many elements if b > β , has infinitely many elements if b < β.

Note that

n 1 ; a n 1 / c n > e b = n 1 ; a n e bc n > 1 .

We thus conclude that

limsup n a n 1 / c n e b for all b > β , e b for all b < β

which ensures (3.2).

We now prove the necessity part of Part (i). For all bβ, let h=(b+β)/2. Then

β < h < b if b > β , b < h < β if b < β.

It follows from (3.2) that the set

n 1 ; a n 1 / c n > e h has finitely many elements if b > β , has infinitely many elements if b < β.

Note that

n 1 ; a n e hc n > 1 = n 1 ; a n 1 / c n > e h .

We thus have that

limsup n a n e hc n 1 if b > β , 1 if b < β.
(3.4)

Note that (3.1) implies that

lim n e hc n e bc n = lim n e ( h b ) c n = 0 if b > β , if b < β.

Thus it follows from (3.4) that

limsup n a n e bc n = lim n e hc n e bc n limsup n a n e hc n = 0 if b > β , if b < β ,

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

lim n S n n ( log n ) b = 0 a.s.

if and only if

EX = 0 , E X 2 ( L | X | ) 2 b < , and lim x LLx ( Lx ) 2 b H ( x ) = 0 ,

where H(x)=E X 2 I { | X | x } , 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

lim n S n n 1 / α ( log n ) b = 0 a.s.

if and only if

E | X | α ( L | X | ) <

where EX=0 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

E X 2 ( L | X | ) 2 b <for allb>β.
(4.1)

We thus see that for all b>β,

H ( x ) = E X 2 I { | X | x } E X 2 ( L | X | ) 2 h ( Lx ) 2 h I { | X | x } ( Lx ) 2 h E X 2 ( L | X | ) 2 h ,

where h=(b+β)/2. Since β<h<b, it follows from (4.1) that

LLx ( Lx ) 2 b H ( x ) LLx ( Lx ) 2 ( b h ) E X 2 ( L | X | ) 2 h 0 as x ∞.

We thus conclude from (2.2) that

EX = 0 , E X 2 ( L | X | ) 2 b < , and lim x LLx ( Lx ) 2 b H ( x ) = 0 for all b > β > 0

which, by applying Lemma 3.2, ensures that

lim n S n n ( log n ) b =0a.s. for allb>β.
(4.2)

Since β>0, the second part of (2.2) implies that

E X 2 ( L | X | ) 2 b = for all b < β

which ensures that

limsup n S n n ( log n ) b =a.s. for allb<β.
(4.3)

Let A n = S n / n and c n =L L n, n ≥ 1. It then follows from (4.2) and (4.3) that

limsup n A n e bc n = limsup n S n n ( log n ) b = 0 a.s. for all b > β , a.s. for all b < β.
(4.4)

By Lemma 3.1, we see that (4.4) is equivalent to

limsup n A n 1 / c n = e β a.s. ,

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

EX=0andE X 2 ( L | X | ) 2 b <for allb>β.
(4.5)

Since 0 < β<, it follows from (4.5) that

β 1 = Δ inf b > 0 : E X 2 ( L | X | ) 2 b < β.

If β1<β then, using the argument in the proof of the sufficiency part, we have that

lim n S n n ( log n ) b =0a.s. for allb> β 1 .
(4.6)

Hence, by Lemma 3.1, (4.6) implies that

limsup n S n n ( log log n ) 1 e β 1 < e β a.s.

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

lim n S n n ( log n ) b =0a.s. for allb>0.
(4.7)

Since X is a non-degenerate random variable, by the classical Hartman-Wintner-Strassen LIL, we have that

limsup n | S n | 2 n log log n > 0 a.s.

which implies that

limsup n | S n | n ( log n ) b =a.s. for allb<0.
(4.8)

Let A n = S n / n and c n =L L n, n ≥ 1. It then follows from (4.7) and (4.8) that

limsup n A n e bc n = limsup n S n n ( log n ) b = 0 a.s. for all b > 0 , a.s. for all b < 0 .
(4.9)

By Lemma 3.1, we see that (4.9) is equivalent to

limsup n A n 1 / c n = e 0 = 1 a.s. ,

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

EX = 0 and inf b > 0 : E X 2 ( L | X | ) 2 b < 0 .

Clearly

inf b > 0 : E X 2 ( L | X | ) 2 b < 0 .

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, EX=0 whenever 1<α≤ 2.

For 0 < α<2, we have

P ( | X | > x ) sin πα / 2 Γ ( α ) π | x | α as x ,

it follows that

E | X | α ( L | X | ) < if b > 1 / α , = if b 1 / α

and hence that

inf b : E | X | α ( L | X | ) < = 1 / α.

Thus, by Theorems 2.3-2.5, XC T L I L(α,1/α) (i.e., the classical Chover LIL follows).

However, for α=2, we have E X 2 =1. Hence, by Theorems 2.1 and 2.2, we see that XC T L I L(2,1/2) but XC T L I L(2,0).

From our second example, we will see that XC 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

P X = d n = P X = d n = c 2 log λ d n d n α = c 2 2 d n α , n 1

where

c = c ( α , λ ) = n = 1 log λ d n d n α 1 > 0 .

Then

limsup x x α log λ x P ( | X | x ) = c > 0 and liminf x x α log λ x P ( | X | x ) = 0 .

Thus the distribution of X is not in the domain of attraction of the stable distribution with exponent α. Also EX=0 whenever either 1<α≤ 2 or α=1 and λ<0. It is easy to see that

E | X | α ( L | X | ) = c 2 n = 1 2 λ n < if b > λ / α , = if b λ / α

and hence that

inf b : E | X | α ( L | X | ) < = λ / α.

Thus, by Theorems 2.1-2.5, we have

  1. (1)

    If α = 2 and 0 < λ<, then XC T L I L(2,λ/2).

  2. (2)

    If α=2 and −<λ≤ 0, then XC T L I L(2,0).

  3. (3)

    If 0 < α<2, then XC 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 XC T L I L(α,β).

Example 5.3.

Define the density function f(x) of X by

f ( x ) = 0 if 0 | x | < e , c | x | α + 1 exp p ( log | x | ) γ if | x | e ,

where p≠0, 0 < γ<1, c=c(α,p,γ) is a positive constant such that f(x)dx=1. On simplification one can show that for any −<b<

E | X | α ( L | X | ) = 2 c e exp p ( log x ) γ x ( log x ) dx < if p < 0 , = if p > 0 .

From Theorem 2.1-2.5, we have

  1. (1)

    If α = 2 and p < 0, then XC T L I L(2,0).

  2. (2)

    If α = 2 and p > 0, then XC T L I L(2,β) for any 0≤ β <.

  3. (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

    limsup n S n B ( n ) ( log log n ) 1 = e 1 / α a.s. ,

where B(x) is the inverse function of xα/ exp(p(logx)γ), xe.

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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).

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DL and PC contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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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

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Keywords

  • (α,β)-Chover-type law of the iterated logarithm
  • Sums of i.i.d. random variables
  • Symmetric stable distribution with exponent α