On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions

Statistics and Probability Letters 83 (2013) 1963–1968

Contents lists available at SciVerse ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

On the strong law of large numbers for pairwise i.i.d. random variables with general moment conditions Soo Hak Sung ∗ Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea

article

abstract

info

Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑ and let {X , Xn , n ≥ 1} be a sequence of pairwise independent identically distributed random variables. In this paper, we obtain the strong law of large numbers and complete convergence forthe sequence {X , Xn , n ≥ 1}, which are equivalent to the general moment condition ∞ n=1 P (|X | > an ) < ∞. We obtain, as a corollary, the strong law of large numbers due to Kruglov [Kruglov, V.M., 2008. A strong law of large numbers for pairwise independent identically distributed random variables with infinite means. Statist. Probab. Lett. 78, 890–895]. © 2013 Elsevier B.V. All rights reserved.

Article history: Received 25 March 2013 Received in revised form 4 May 2013 Accepted 11 May 2013 Available online 18 May 2013 MSC: 60F15 Keywords: Strong law of large numbers Pairwise independent random variables General moment conditions Complete convergence

1. Introduction Let 0 < p < 2 and let {X , Xn , n ≥ 1} be a sequence of independent identically distributed (i.i.d.) random variables. Put n Sn = i=1 Xi for n ≥ 1. Then it is well known (see Baum and Katz, 1965) that the following statements are equivalent. E |X |p < ∞, Sn n1/p

where EX = 0 if p ≥ 1;

→ 0 a.s.;

∞  1  n =1

n

n =1

n

(1.2)

P |Sn | > n1/p ϵ < ∞

 ∞  1 P

(1.1)



max |Sk | > n1/p ϵ

1≤k≤n



for all ϵ > 0;

(1.3)

< ∞ for all ϵ > 0.

(1.4)

Now let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables. For p = 1, Etemadi (1981) proved that (1.1) and (1.2) are equivalent, and Chen et al. (2013) proved that (1.1) and (1.4) are equivalent. It is easy to prove that for 0 < p < 2, (1.1) implies (1.3). But it is not known whether the converse holds true. On the other hand, Kruglov (1994) proved that for a sequence of pairwise i.i.d. and non-negative random variables {X , Xn , n ≥ 1}, and any non-negative constant µ, the

∗

Tel.: +82 425205369. E-mail address: [email protected].

0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.05.009

1964

S.H. Sung / Statistics and Probability Letters 83 (2013) 1963–1968

following two statements are equivalent. EX < ∞ ∞  n =1

1 n

and EX = µ;

(1.5)

P (|Sn − nµ| > nϵ) < ∞ for all ϵ > 0.

(1.6)

Note that Kruglov’s result cannot be applied to the implication (1.3) ⇒ (1.1), since (1.3) does not imply

   n     + P  X − nb > nϵ < ∞ for some non-negative constant b.  i =1 i  n

∞  1 n =1

Furthermore, when 1 < p < 2, it is not known whether any two of (1.1)–(1.4) are equivalent. Recently, Sung (2013) showed that for 1 < p < 2, (1.2) holds under the stronger moment conditions E |X |p (log log(e + |X |))2(p−1) and EX = 0. In this paper, we obtain the strong law of large numbers and complete convergence  for a sequence {X , Xn , n ≥ 1} of ∞ pairwise i.i.d. random variables, which are equivalent to the general moment condition n=1 P (|X | > an ) < ∞, where {an , n ≥ 1} is a sequence of positive constants with an /n ↑. 2. The main results Throughout this section, let {an , n ≥ 1} be a sequence of positive constants with an /n ↑ and let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables. To prove our results, the following lemmas are needed. Lemma 2.1. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑. Then the following properties hold. (i) { an , n ≥ 1} is a strictly increasing sequence with an ↑ ∞. ∞ (ii) ∞ n=1 P (X > an ) < ∞ if and only if n=1 P (X > 2an ) < ∞.

(iii)

∞

n =1

P (X > an ) < ∞ if and only if

∞

n=1

P (X > α an ) < ∞ for any α > 0.

Proof. By the condition 0 < an /n ↑, we have an /n ≤ an+1 /(n + 1) and a1 ≤ an /n, which imply an < an+1 and a1 n ≤ an . Hence (i) holds. By (i), a2n < a2n+1 and so ∞ 

P (X > an ) ≤ P (X > a1 ) + 2

n =1

∞ 

P (X > a2n ).

(2.1)

n =1

Since an /n ≤ a2n /2n, we have 2an ≤ a2n and so ∞ 

P (X > an ) ≥

n =1

∞ 

P (X > 2an ) ≥

n =1

∞ 

P (X > a2n ).

(2.2)

n=1

Combining (2.1) and (2.2) gives (ii). ∞ ∞ Finally, we prove (iii). If we apply (ii) with an replaced by 2an , then n=1 P (X > 2an ) < ∞ if and only if n=1 P (X > 4an ) ∞ ∞ < ∞. Combining this with (ii) gives n=1 P (X > an ) < ∞ if and only if n=1 P (X > 4an ) < ∞. If we use this method k − 2 times more, then ∞ 

P (X > an ) < ∞ ⇐⇒

n=1

P (X > 2k an ) < ∞.

n=1

−k

Replacing an by 2 ∞ 

∞ 

an , from the above statement, we have

P (X > an ) < ∞ ⇐⇒

n=1

∞ 

P (X > 2−k an ) < ∞.

n=1

If α > 1, then α ≤ 2 for some k and so n=1 P (X > an ) ≥ (X > α an ) ≥ n=1 P (X > 2k an ). If 0 < α < 1, then n=1 P ∞ ∞ −k −k α ≥ 2 for some k and so n=1 P (X > an ) ≤ n=1 P (X > α an ) ≤ ∞ an ). Hence (iii) holds.  n =1 P ( X > 2

∞

k

∞

∞

Lemma 2.2. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑. Let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables. Then Xn an

→ 0 a.s. if and only if

∞ 

P (|X | > an ) < ∞.

n=1

Proof. Note that Xn /an → 0 a.s. if and only if P (|Xn |/an > ϵ i.o.) = 0 for all ϵ > 0. By the Borel–Cantelli lemma and the second Borel–Cantelli lemma for pairwise independent events (for ∞example, see Theorem 4.2.5 in Chung (1974) or Theorem 2.18.5 in Gut (2005)), P (|Xn |/an > ϵ i.o.) = 0 is equivalent to n=1 P (|Xn |/an > ϵ) < ∞. The latter is also equivalent to ∞  n=1 P (|Xn | > an ) < ∞ by Lemma 2.1. Hence the result is proved.

S.H. Sung / Statistics and Probability Letters 83 (2013) 1963–1968

1965

Lemma 2.3. If {an , n ≥ 1} is a sequence of positive constants with an /n ↑, then 1 an

nE |X |I (|X | ≤ an ) ≤

∞ 

P (|X | > an ),

n =0

where a0 = 0 (this convention will be used throughout the paper). Proof. Since 0 < an /n ↑, we obtain ai ≤ n−1 an i for 1 ≤ i ≤ n. It follows that 1 −1 a− n nE |X |I (|X | ≤ an ) = an n

n 

E |X |I (ai−1 < |X | ≤ ai )

i =1 1 ≤ a− n n

n 

ai P (ai−1 < |X | ≤ ai )

i =1 n 

≤

iP (ai−1 < |X | ≤ ai )

i=1

∞ 

≤

iP (ai−1 < |X | ≤ ai )

i=1

=

∞ 

P (|X | > ai ).

i=0

Hence the result is proved.



If the condition 0 < an /n ↑ in Lemma 2.3 is replaced by the stronger condition 0 < an /n ↑ ∞, then the result can be improved. Lemma 2.4. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑ ∞. If (|X | ≤ an )/an → 0 as n → ∞.

∞

n=1

P (|X | > an ) < ∞, then nE |X |I

Proof. As noted in the proof of Lemma 2.3, ai ≤ n−1 an i for 1 ≤ i ≤ n. By this and an /n ↑ ∞, we get 1 a− n nE |X |I (|X | ≤ an ) =

n 

1 −1 a− n nE |X |I (|X | ≤ aN ) + an n

E |X |I (ai−1 < |X | ≤ ai )

i =N +1 n 

1 −1 a− n naN + an n

≤

ai P (ai−1 < |X | ≤ ai )

i=N +1 1 a− n naN +

≤

n 

iP (ai−1 < |X | ≤ ai )

i =N +1

∞ 

−→ n→∞

since

∞

i=1

iP (ai−1 < |X | ≤ ai ) → 0

as N → ∞,

i=N +1

iP (ai−1 < |X | ≤ ai ) =

∞

i=0

P (|X | > ai ) < ∞.



We now present two results on the complete convergence for pairwise i.i.d. random variables. Theorem 2.1. Let {a n , n ≥ 1} be a sequence of positive constants with an /n ↑. Let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. ∞ random variables. If n=1 P (|X | > an ) < ∞, then

   n     I := n P  (X − EXi I (|Xi | ≤ an )) > an ϵ < ∞ for all ϵ > 0.  i=1 i  n =1 ∞ 

−1

Proof. Note at the outset that the condition an /n ↑ implies ∞  1

a2n n =i

≤

∞ i2  1

a2i n=i

n2

≤

2i a2i

.

(2.3)

1966

S.H. Sung / Statistics and Probability Letters 83 (2013) 1963–1968

Then I ≤

∞ 

n

−1

n 

n =1

≤

i=1

∞ 

   n     P (|Xi | > an ) + n P  (X I (|Xi | ≤ an ) − EXi I (|Xi | ≤ an )) > an ϵ  i=1 i  n =1 ∞ 

P (|X | > an ) + ϵ −2

n =1

≤

∞ 

−1

2 n−1 a− n

n =1

∞ 

P (|X | > an ) + ϵ −2

n =1

∞ 

n 

Var (Xi I (|Xi | ≤ an ))

i=1 2 2 a− n E |X | I (|X | ≤ an )

n =1

:= I1 + ϵ −2 I2 . Since I1 < ∞ by the assumption, it remains to show that I2 < ∞. By (2.3), we obtain I2 =

∞ 

2 a− n

n 

n =1

=

∞ 

E |X |2 I (ai−1 < |X | ≤ ai )

i =1

E |X |2 I (ai−1 < |X | ≤ ai )

∞  n =i

i =1

≤2

2 a− n

∞ 

2 E |X |2 I (ai−1 < |X | ≤ ai )ia− i

i=1

≤2

∞ 

P (ai−1 < |X | ≤ ai )i

i=1

=2

∞ 

P (|X | > ai ).

(2.4)

i=0

Hence I2 < ∞.



If the condition 0 < an /n ↑ in Theorem 2.1 is replaced by the stronger condition 0 < an /n ↑ ∞, then the following result can be obtained. Theorem 2.2. Let {an , n  ≥ 1} be a sequence of positive constants with an /n ↑ ∞. Let {X , Xn , n ≥ 1} be a sequence of pairwise ∞ i.i.d. random variables. If n=1 P (|X | > an ) < ∞, then J :=

∞ 

n− 1 P



max |Sk | > an ϵ

1≤k≤n

n =1



< ∞ for all ϵ > 0.

Proof. By Lemma 2.4, nE |X |I (|X | ≤ an )/an < ϵ/2 for all large n. Then

   k     J ≤ n P (|Xi | > an ) + n P max  Xi I (|Xi | ≤ an ) > an ϵ  1≤k≤n  n =1 i=1 n =1 i=1   ∞ ∞ n    ≤ P (|X | > an ) + n− 1 P |Xi |I (|Xi | ≤ an ) > an ϵ ∞ 

n=1

≤

∞  n=1

−1

∞ 

n 

n =1

P (|X | > an ) + C

∞ 



−1

i =1



 n  n− 1 P (|Xi |I (|Xi | ≤ an ) − E |Xi |I (|Xi | ≤ an )) > an ϵ/2 ,

n =1

i=1

∞ where C > 0 is a positive constant. From the proof of Theorem 2.1, we obtain that J ≤ (1 + ϵ −2 8C ) n=0 P (|X | > an ) < ∞.  We next present some results on the strong law of large numbers for pairwise i.i.d. random variables. Theorem 2.3. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑. Let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables. Then the following statements are equivalent. (i) nn=1 P (|X | > an ) < ∞. (ii) i=1 (Xi − EXi I (|Xi | ≤ ai ))/an → 0 a.s.

∞

S.H. Sung / Statistics and Probability Letters 83 (2013) 1963–1968

1967

Proof. First we prove that (i) ⇒ (ii). Let Yn = Xn I (|Xn | ≤ an ) for n ≥ 1. Then we have by Lemma 2.3 that 1 a− n

n 

1 E |Yi − EYi | ≤ 2a− n

i=1

n 

1 E |Yi | ≤ 2na− n E |X |I (|X | ≤ an ) ≤ 2

i=1

∞ 

P (|X | > an ) < ∞.

n=0

We also have by (2.4) in the proof of Theorem 2.1 that ∞ 

2 a− n Var (Yn ) ≤

∞ 

n =1

2 2 a− n E |X | I (|X | ≤ an ) ≤ 2

n =1

∞ 

P (|X | > an ) < ∞.

n =0

By Corollary 1 in Chandra and Goswami (1992), n 1 

an i = 1

(Xi I (|Xi | ≤ ai ) − EXi I (|Xi | ≤ ai )) → 0 a.s.

(2.5)

By the Borel–Cantelli lemma, (i) implies n 1 

an i = 1

Xi I (|Xi | > ai ) → 0 a.s.

(2.6)

Hence (ii) holds by (2.5) and (2.6). n We next prove that (ii) ⇒ (i). Assume that i=1 (Xi − EXi I (|Xi | ≤ ai ))/an → 0 a.s. Then we have Xn − EXn I (|Xn | ≤ an ) an

=

n 1 

an i = 1

(Xi − EXi I (|Xi | ≤ ai )) −

an−1

1

n −1 

an an−1 i=1

(Xi − EXi I (|Xi | ≤ ai )) → 0 a.s.

We also have EXn I (|Xn | ≤ an )/an → 0 as n → ∞, since E |X |I (|X | ≤ an ) an

≤

aN an

+ P (|X | > aN ) −→ P (|X | > aN ) → 0 as N → ∞. n→∞

Hence Xn /an → 0 a.s., which is equivalent to (i) by Lemma 2.2.



Remark 2.1. For a sequence {X , Xn , n ≥ 1} of pairwise i.i.d. random  ∞variables with E |X | < ∞, Etemadi (1981) proved n that n ( X − EX )/ n → 0 a.s. Note that E | X | < ∞ is equivalent to P (| X | > n ) < ∞ , and E | X | < ∞ implies i i i =1 n=1 i=1 EXi I (|Xi | > i)/n → 0. Hence Etemadi’s strong law of large numbers follows from Theorem 2.3 with an = n. If the condition 0 < an /n ↑ in Theorem 2.3 is replaced by the stronger condition 0 < an /n ↑ ∞, then we have the following theorem. Theorem 2.4. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑ ∞. Let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables. Then the following statements are equivalent. (i) n=1 P (|X | > an ) < ∞. (ii) S n /an → 0 a.s. n (iii) i=1 |Xi |/an → 0 a.s.

∞

Proof. The proof that  (ii) ⇒ (i) is essentially the same as that given in Theorem 2.3 and be omitted. To prove the will ∞ n converse, assume that n=1 P (|X | > an ) < ∞. By Theorem 2.3, it suffices to prove that i=1 EXi I (|Xi | ≤ ai )/an → 0 as n → ∞. But this follows by Lemma 2.4. ∞ We have proved that (i) and (ii) are equivalent. By Lemma 2.1, (i) is equivalent to n=1 P (|X | > an ϵ) < ∞ for all ϵ > 0. ∞ ∞ The latter is also equivalent to the pair of expressions n=1 P (X + > an ϵ) < ∞, ∀ϵ > 0, and n=1 P (X − > an ϵ) < ∞, ∞  + ∀ϵ > 0, which are equivalent to n=1 P (X > an ) < ∞ and ∞ P (X − > an ) < ∞ by Lemma 2.1. On the other hand, n=1 n n + (iii) is equivalent to the pair of expressions i=1 Xi /an → 0 a.s. and i=1 Xi− /an → 0 a.s. Hence the equivalence of (i) and (iii) follows from the equivalence of (i) and (ii).  If an additional condition a2n /an = O(1) is imposed on the sequence {an , n ≥ 1} in Theorem 2.4, then we have the following result. Theorem 2.5. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑ ∞ and a2n /an = O(1). Let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables. Then the following statements are equivalent. (i) n=1 P (|X | > an ) < ∞. (ii)  Sn /an → 0 a.s. n (iii) =1 |Xi |/an → 0 a.s.  i∞ −1 (iv) P max1≤k≤n |Sk | > an ϵ < ∞ for all ϵ > 0. n =1 n

∞

1968

S.H. Sung / Statistics and Probability Letters 83 (2013) 1963–1968

Proof. The implication (i) ⇒ (iv) follows from Theorem 2.2. The implication (iv) ⇒ (ii) follows from Lemma 2.4 in Sung (2013). Since (i)–(iii) are equivalent by Theorem 2.4, the result is proved.  Remark It is not known whether the following condition (v) is equivalent to any one of (i)–(iv) in Theorem 2.5. 2.2. ∞ (v) n=1 n−1 P (|Sn | > an ϵ) < ∞ for all ϵ > 0. Corollary 2.1. Let {an , n ≥ 1} be a sequence of positive constants with an /n ↑. Let {X , Xn , n ≥ 1} be a sequence of pairwise i.i.d. random variables with E |X | = ∞. Then n 

|Xi |

i =1

lim

an

n→∞

lim sup n→∞

|Sn | an

∞ 

= 0 a.s. if and only if

P (|X | > an ) < ∞,

(2.7)

n=1

∞ 

= ∞ a.s. if and only if

P (|X | > an ) = ∞.

(2.8)

n=1

Proof. Assume that n=1 P (|X | > an ) < ∞. Since E |X | = ∞, we have an /n ↑ ∞. Hence the ‘if’ part of (2.7) follows by n Theorem 2.4. To prove the converse, assume that limn→∞ i=1 |Xi |/an = 0 a.s. Then Xn /an → 0 a.s. and so the converse holds by Lemma 2.2. ∞ The ‘only if’ part of (2.8) follows directly from (2.7). To prove the converse, assume that n=1 P (|X | > an ) = ∞. Then ∞ n=1 P (|Xn | > α an ) = ∞ for any α > 0 by Lemma 2.1. Thus P (|Xn | > α an i.o.) = 1 by the second Borel–Cantelli lemma for pairwise independent events. Since

∞

|Xn | an

=

|Sn − Sn−1 | an

≤

|Sn | an

+

|Sn−1 | an−1

,

we have that P (|Xn | > α an i.o.) ≤ P (|Sn | > α an /2 i.o.). Hence P (|Sn | > α an /2 i.o.) = 1 and so lim supn→∞ |Sn |/an ≥ α/2 a.s. Since α > 0 is arbitrary, we have lim supn→∞ |Sn |/an = ∞ a.s.  Remark 2.3. Note that lim supn→∞ |Sn |/an = ∞ a.s. is equivalent to P (|Sn | > α an i.o.) = 1 for any α > 0. Hence Corollary 2.1 improves the corresponding result of Kruglov (2008). Acknowledgment The author would like to thank the referee for helpful comments and suggestions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131). References Baum, L.E., Katz, M., 1965. Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123. Chandra, T.K., Goswami, A., 1992. Cesàro uniform integrability and the strong law of large numbers. Sankhya¯ A 54, 215–231. Chen, P., Bai, P., Sung, S.H., 2013. On complete convergence and the strong law of large numbers for pairwise independent random variables (submitted for publication). Chung, K.L., 1974. A Course in Probability Theory, second ed. Academic Press, New York. Etemadi, N., 1981. An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 55, 119–122. Gut, A., 2005. Probability: A Graduate Course. Springer, New York. Kruglov, V.M., 1994. Strong law of large numbers. In: Zolotarev, V.M., Kruglov, V.M., Korolev, V. Yu. (Eds.), Stability Problems for Stochastic Models. TVP/VSP, Moscow, Utrecht, pp. 139–150. Kruglov, V.M., 2008. A strong law of large numbers for pairwise independent identically distributed random variables with infinite means. Statist. Probab. Lett. 78, 890–895. Sung, S.H., 2013. Marcinkiewicz–Zygmund type strong law of large numbers for pairwise i.i.d. random variables. J. Theoret. Probab. http://dx.doi.org/10. 1007/s10959-012-0417-4 (in press).