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A note on the bilateral inequality for a sequence of random variables
Statistics and Probability Letters 82 (2012) 871–875
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A note on the bilateral inequality for a sequence of random variables Jicheng Liu School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China
article
abstract
info
Article history: Received 6 October 2011 Received in revised form 3 February 2012 Accepted 5 February 2012 Available online 15 February 2012
Two bilateral inequalities based on the Borel–Cantelli lemma and a non-negative sequence of bounded random variables were respectively obtained by Xie (2008, 2009). However, we observe that the upper bounds in the above cited references are greater than or equal to 1, so the upper bounds of these bilateral inequalities always hold true. In this note, we will extend the lower bound results on the assumptions that the random variables are neither non-negative nor bounded, which could be considered as a version of the Borel–Cantelli lemma with a random weight sequence. As an application, we also discuss the example given in Xie (2008) and Hu et al. (2009), and the best result is easily obtained for this example by taking the appropriate weight sequence. © 2012 Elsevier B.V. All rights reserved.
MSC: 60F15 Keywords: A sequence of random variables The bilateral inequality
1. Introduction Recently, Xie (2008) proved a bilateral inequality based on the Borel–Cantelli lemma; then Xie (2009) proved the following general bilateral inequality based on a bounded non-negative sequence of random variables: Theorem 1. Let b > 0, (Ω , F , P ) be a probability space, and {Xn } be a sequence of non-negative bounded random variables ∞ satisfying the conditions Xn ≤ b and n=1 E (Xn ) = +∞. Then for p > 1 or 0 < p < 1, P (lim sup{Xn ̸= 0}) ≥ lim sup Tn (p);
(1)
n→∞
and for p < 0, P (lim sup{Xn ̸= 0}) ≤ lim inf Tn (p),
(2)
n→∞
∞ where lim sup{Xn ̸= 0} = ∩∞ n=1 ∪k=n {Xk ̸= 0},
Tn (p) = and Sn =
n
k=1
E
Sn
p 1−1 p
E (Sn )
Xk .
However, by the Hölder inequality, we observe that Tn (p) ≤ 1 for p > 0 and p ̸= 1, and Tn (p) ≥ 1 for p < 0. Hence the upper bound in (2) always holds true. Let {Xn } be a sequence of non-negative random variables on a probability space (Ω , F , P ). Notice that lim sup{Xn ̸= 0} ⊃
∞
Xk = ∞
k=1
⊃ lim sup Xn ̸= 0 . n→∞
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.02.004
872
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
If k=1 Xk = ∞ = 0. If n=1 E (Xn ) < ∞, then P bounded assumption on the random variables.
∞
∞
∞
n =1
E (Xn ) = ∞, we can establish the following result without the
Theorem 2. Let (Ω , F , P ) be a probability space, and {Xn } be a sequence of non-negative random variables. Assume that ∞ E ( X ) = ∞. Then for all p > 1, n n =1 P
∞
Xn = ∞
≥ lim sup Tn (p),
(3)
n→∞
n=1
where Tn (p) has the same meaning as in Theorem 1. If the random variables {Xn } could be negative, then
lim sup{Xn ̸= 0} ⊃
lim sup n→∞
n
Xk = +∞
∪ lim inf n→∞
k=1
n
Xk = −∞
k=1
⊃ lim sup Xn ̸= 0 ∪ lim inf Xn ̸= 0 . n→∞
n→∞
Let Sn = k=1 Xk . Note that limn→∞ E (Sn ) = n=1 E (Xn ) and limn→∞ E (Sn ) = ∞ imply limn→∞ E (|Sn |) = ∞. The condition limn→∞ E (Sn ) < ∞ does not imply that both events above have zero probability. But we can prove the following result.
∞
n
Theorem 3. Let (Ω , F , P ) be a probability space, and {Xn } be a sequence of random variables. Assume that limn→∞ E (|Sn |) = ∞. Then for all p > 1 or 0 < p < 1, P (lim sup{Xn ̸= 0}) ≥ lim sup Tn′ (p)
(4)
n→∞
∞ where lim sup{Xn ̸= 0} = ∩∞ n=1 ∪k=n {Xk ̸= 0} and
Tn (p) = ′
E
|Sn | E (|Sn |)
p 1−1 p
≤ 1.
Remark 4. Theorem 3 is not a simple application of Theorem 1 for the absolute value sequence of random variables. Note that for p > 1,
Sn Tn (p) ≥ E E (S ′
n
p 1−1 p . )
Let {Xn } and {wn } be sequences of random variables. Since wn could be zero (or even negative), we have
{wn Xn ̸= 0} ⊂ {Xn ̸= 0}. ∞
By Theorem 3, the condition
k=1
E (wk Xk ) = ∞ implies that
p 1−1 p n wk Xk k=1 . P (lim sup{Xn = ̸ 0}) ≥ Tn′ (p) ≥ Tn′′ (p) = n E E (w X ) k k k=1
When p is even, Tn (p) may be represented in a multi-summation form. For example, we can represent Tn′′ (2) as follows: ′′
n
i,j=1 n
Tn′′ (2) =
i,j=1
−1
E (wi wj Xi Xj )
E (wi Xi )E (wj Xj )
,
which could be considered as a version of the Borel–Cantelli lemma with a random weight sequence, and extend the result for the weighted Borel–Cantelli lemma given in Feng et al. (2009).
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
873
Example 5. Let (Ω , F , P ) be a probability space, A, B ∈ F and P (A ∪ B) > 0. For all positive integers n, let A3n−2 = A, A3n−1 = A3n = B. Applying Tn′′ (2) with Xn = 1An and the weight sequence wn as 1, 1, −1A , 1, 1, −1A , 1, 1, −1A , . . . , we have ∞
E (wn Xn ) = lim n(P (A ∪ B)) = ∞ n→∞
n =1
and n2 (P (A) + P (B) − P (AB))2
P (lim sup An ) ≥ lim sup
n2 (P (A) + P (B) − P (AB)) = P (A ∪ B) = P (lim sup An ) . n→∞
In fact, lim sup An = A ∪ B. The best result is easily obtained for this example by taking the appropriate weight sequence wn . The same example is considered by Xie (2008) and Hu et al. (2009) in a very complicated way, under the additional conditions P (A ∪ B) = 1 and P (AB) = 0. At the same time, we note that the best result for this example could not be obtained by using the Borel–Cantelli lemma with a determined weight sequence in Feng et al. (2009). 2. Proof of the theorems To prove our theorems, we need to prepare with the following lemma. Lemma 6. Let X be a real-valued integrable random variable. Then for any r ≤ 1 and p > 1, we have
P
X E (X )
>r
≥ (1 − r )
p p−1
p 1−1 p X E . − r E (X )
For any p > 0 and p ̸= 1, we have P (X ̸= 0) ≥
E
|X | E (|X |)
p 1−1 p
.
Proof. It suffices to prove the first inequality for E (X ) = 1. Note that for any r, X − r ≤ (X − r )1{X >r } . By taking expectations and using the Hölder inequality, we obtain for r ≤ 1 and p > 1, 1 − r ≤ E ((X − r )1{X >r } ) ≤ (E (|X − r |p ))1/p (P (X > r ))(p−1)/p , which yields the first inequality. Similarly, by the Hölder inequality, we have for 0 < p < 1, E (|X |) = E (|X |1{X ̸=0} ) ≥ (E (|X |p ))1/p (P (X ̸= 0))(p−1)/p and for p > 1, E (|X |) = E (|X |1{X ̸=0} ) ≤ (E (|X |p ))1/p (P (X ̸= 0))(p−1)/p . Both cases imply the second inequality.
k=1 Xk . By E (Sn ) → ∞ and the first part of Lemma 6, for all r ∈ (0, 1) and p > 1 we have lim Sm = ∞ = lim P lim Sm > rE (Sn )
Proof of Theorem 2. Let Sn = P
m→∞
n
n→∞
m→∞
≥ lim sup P (Sn > rE (Sn )) n→∞
≥ (1 − r )
p p−1
p 1−1 p Sn lim sup E − r , E ( S ) n→∞ n
where the first inequality is due to the easy inclusion relation: for all n,
lim Sm > rE (Sn ) ⊃ (Sn > rE (Sn )) .
m→∞
874
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
By the Minkowski inequality, we have for p > 1,
1−p p p 1−1 p p 1p Sn Sn E . − r ≥ E + r E (Sn ) E (Sn ) And therefore,
1−p p p 1p p Sn P lim Sm = ∞ ≥ (1 − r ) p−1 lim sup E + r . m→∞ E (Sn ) n→∞
As r ∈ (0, 1) is arbitrary, we complete the proof of the theorem by letting r tend to zero in the above inequality.
Proof of Theorem 3. Fix the natural number m and note that
{Sn − Sm ̸= 0} ⊂ {∪nk=m+1 {Xk ̸= 0}}. By the second part of Lemma 6, for p > 0 and p ̸= 1 we have
∞
P
{Xk ̸= 0}
n
= lim P n→∞
k=m+1
{Xk ̸= 0}
k=m+1
≥ lim P (Sn − Sm ̸= 0) n→∞
n→∞
1−1 p
E (|Sn − Sm |p )
≥ lim sup
(E (|Sn − Sm |))p
=: I .
For p > 1, by the Minkowski inequality,
(E (|Sn − Sm |p ))1/p ≤ (E (|Sn |p ))1/p + (E (|Sm |p ))1/p and E (|Sn − Sm |) ≥ |E (|Sn |) − E (|Sm |)|, we obtain
(E (|Sn |p ))1/p + (E (|Sm |p ))1/p I ≥ lim sup |E (|Sn |) − E (|Sm |)| n→∞
1−p p
.
By the Hölder inequality, we have E (|Sn |) ≤ (E (|Sn |p ))1/p for p > 1. The condition E (|Sn |) → ∞ implies E (|Sn |p ) → p ∞, EE(|(|SSm||p )) → 0 and (E (|ES |S|pm))| 1/p → 0. Hence n
n
I ≥ lim sup
1 + (E (|Sm|p ))1/p n
E (|Sn |) (E (|Sn |p ))1/p −
n→∞
1−p p
(E (|S |p ))1/p
E (|Sm |) (E (|Sn |p ))1/p
= lim sup Tn′ (p). n→∞
Similarly, for 0 < p < 1, by the inequalities E (|Sn − Sm |p ) ≥ |E (|Sn |p ) − E (|Sm |p )| and E (|Sn − Sm |) ≤ E (|Sn |) + E (|Sm |), we obtain
I ≥ lim sup n→∞
|E (|Sn |p ) − E (|Sm |p )| (E (|Sn |) + E (|Sm |))p
1−1 p
.
By the Hölder inequality, we have E (|Sn |) ≥ (E (|Sn |p ))1/p for 0 < p < 1. By the condition E (|Sn |) → ∞, we have (E (|Sm |p ))1/p E (|Sn |)
→ 0 and
E |Sm | E (|Sn |)
→ 0. Hence
1−1 p E (|Sn |p ) E (|Sm |p ) − (E (|Sn |))p (E (|Sn |))p I ≥ lim sup = lim sup Tn′ (p). p E (|Sm |) n→∞ n→∞ 1 + E (|S |) n
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
875
Combining the above two cases, we obtain for all p > 0 and p ̸= 1,
P (lim sup{Xn ̸= 0}) = lim P m→∞
Now we have finished the proof.
∞ k=m+1
{Xk ̸= 0} ≥ lim sup Tn′ (p). n→∞
Acknowledgments The author is very grateful to the Co-Editor-in-Chief and an anonymous referee for their valuable suggestions and comments. This work was supported by the NSF of China (Nos. 10901065 and 10871215). References Feng, C., Li, P., Shen, J., 2009. On the Borel–Cantelli lemma and its generalization. C. R. Acad. Sci. Paris, Sér. I 347, 1313–1316. Hu, S.H., Wang, X.J., Li, X.Q., Zhang, Y.Y., 2009. Comments on the paper: a bilateral inequality on the Borel–Cantelli lemma. Statist. Probab. Lett. 79, 889–893. Xie, Y.Q., 2008. A bilateral inequality on the Borel–Cantelli lemma. Statist. Probab. Lett. 79, 1577–1580. Xie, Y.Q., 2009. A bilateral inequality on nonnegative bounded random sequence. Statist. Probab. Lett. 78, 2052–2057.
Further reading Chandra, T.K., 2008. The Borel–Cantelli lemma under dependence conditions. Statist. Probab. Lett. 78, 390–395. Kallenberg, O., 2002. Foundations of Modern Probability, second ed. Springer, New York. Kochen, S.P., Stone, C.J., 1964. A note on the Borel–Cantelli lemma. Illinois J. Math. 8, 248–251. Petrov, V.V., 2002. A note on the Borel–Cantelli lemma. Statist. Probab. Lett. 58, 283–286. Petrov, V.V., 2004. A generalization of the Borel–Cantelli lemma. Statist. Probab. Lett. 67, 233–239. Yan, J., 2006. A new proof of a generalized Borel–Cantelli lemma. In: In Memoriam Paul-André Meyer: Seminar on Probability Theory XXXIX. In: Lecture Notes in Mathematics, vol. 1874. Springer-Verlag, Berlin, pp. 77–79.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A note on the bilateral inequality for a sequence of random variables Jicheng Liu School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, PR China
article
abstract
info
Article history: Received 6 October 2011 Received in revised form 3 February 2012 Accepted 5 February 2012 Available online 15 February 2012
Two bilateral inequalities based on the Borel–Cantelli lemma and a non-negative sequence of bounded random variables were respectively obtained by Xie (2008, 2009). However, we observe that the upper bounds in the above cited references are greater than or equal to 1, so the upper bounds of these bilateral inequalities always hold true. In this note, we will extend the lower bound results on the assumptions that the random variables are neither non-negative nor bounded, which could be considered as a version of the Borel–Cantelli lemma with a random weight sequence. As an application, we also discuss the example given in Xie (2008) and Hu et al. (2009), and the best result is easily obtained for this example by taking the appropriate weight sequence. © 2012 Elsevier B.V. All rights reserved.
MSC: 60F15 Keywords: A sequence of random variables The bilateral inequality
1. Introduction Recently, Xie (2008) proved a bilateral inequality based on the Borel–Cantelli lemma; then Xie (2009) proved the following general bilateral inequality based on a bounded non-negative sequence of random variables: Theorem 1. Let b > 0, (Ω , F , P ) be a probability space, and {Xn } be a sequence of non-negative bounded random variables ∞ satisfying the conditions Xn ≤ b and n=1 E (Xn ) = +∞. Then for p > 1 or 0 < p < 1, P (lim sup{Xn ̸= 0}) ≥ lim sup Tn (p);
(1)
n→∞
and for p < 0, P (lim sup{Xn ̸= 0}) ≤ lim inf Tn (p),
(2)
n→∞
∞ where lim sup{Xn ̸= 0} = ∩∞ n=1 ∪k=n {Xk ̸= 0},
Tn (p) = and Sn =
n
k=1
E
Sn
p 1−1 p
E (Sn )
Xk .
However, by the Hölder inequality, we observe that Tn (p) ≤ 1 for p > 0 and p ̸= 1, and Tn (p) ≥ 1 for p < 0. Hence the upper bound in (2) always holds true. Let {Xn } be a sequence of non-negative random variables on a probability space (Ω , F , P ). Notice that lim sup{Xn ̸= 0} ⊃
∞
Xk = ∞
k=1
⊃ lim sup Xn ̸= 0 . n→∞
E-mail address: [email protected]. 0167-7152/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2012.02.004
872
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
If k=1 Xk = ∞ = 0. If n=1 E (Xn ) < ∞, then P bounded assumption on the random variables.
∞
∞
∞
n =1
E (Xn ) = ∞, we can establish the following result without the
Theorem 2. Let (Ω , F , P ) be a probability space, and {Xn } be a sequence of non-negative random variables. Assume that ∞ E ( X ) = ∞. Then for all p > 1, n n =1 P
∞
Xn = ∞
≥ lim sup Tn (p),
(3)
n→∞
n=1
where Tn (p) has the same meaning as in Theorem 1. If the random variables {Xn } could be negative, then
lim sup{Xn ̸= 0} ⊃
lim sup n→∞
n
Xk = +∞
∪ lim inf n→∞
k=1
n
Xk = −∞
k=1
⊃ lim sup Xn ̸= 0 ∪ lim inf Xn ̸= 0 . n→∞
n→∞
Let Sn = k=1 Xk . Note that limn→∞ E (Sn ) = n=1 E (Xn ) and limn→∞ E (Sn ) = ∞ imply limn→∞ E (|Sn |) = ∞. The condition limn→∞ E (Sn ) < ∞ does not imply that both events above have zero probability. But we can prove the following result.
∞
n
Theorem 3. Let (Ω , F , P ) be a probability space, and {Xn } be a sequence of random variables. Assume that limn→∞ E (|Sn |) = ∞. Then for all p > 1 or 0 < p < 1, P (lim sup{Xn ̸= 0}) ≥ lim sup Tn′ (p)
(4)
n→∞
∞ where lim sup{Xn ̸= 0} = ∩∞ n=1 ∪k=n {Xk ̸= 0} and
Tn (p) = ′
E
|Sn | E (|Sn |)
p 1−1 p
≤ 1.
Remark 4. Theorem 3 is not a simple application of Theorem 1 for the absolute value sequence of random variables. Note that for p > 1,
Sn Tn (p) ≥ E E (S ′
n
p 1−1 p . )
Let {Xn } and {wn } be sequences of random variables. Since wn could be zero (or even negative), we have
{wn Xn ̸= 0} ⊂ {Xn ̸= 0}. ∞
By Theorem 3, the condition
k=1
E (wk Xk ) = ∞ implies that
p 1−1 p n wk Xk k=1 . P (lim sup{Xn = ̸ 0}) ≥ Tn′ (p) ≥ Tn′′ (p) = n E E (w X ) k k k=1
When p is even, Tn (p) may be represented in a multi-summation form. For example, we can represent Tn′′ (2) as follows: ′′
n
i,j=1 n
Tn′′ (2) =
i,j=1
−1
E (wi wj Xi Xj )
E (wi Xi )E (wj Xj )
,
which could be considered as a version of the Borel–Cantelli lemma with a random weight sequence, and extend the result for the weighted Borel–Cantelli lemma given in Feng et al. (2009).
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
873
Example 5. Let (Ω , F , P ) be a probability space, A, B ∈ F and P (A ∪ B) > 0. For all positive integers n, let A3n−2 = A, A3n−1 = A3n = B. Applying Tn′′ (2) with Xn = 1An and the weight sequence wn as 1, 1, −1A , 1, 1, −1A , 1, 1, −1A , . . . , we have ∞
E (wn Xn ) = lim n(P (A ∪ B)) = ∞ n→∞
n =1
and n2 (P (A) + P (B) − P (AB))2
P (lim sup An ) ≥ lim sup
n2 (P (A) + P (B) − P (AB)) = P (A ∪ B) = P (lim sup An ) . n→∞
In fact, lim sup An = A ∪ B. The best result is easily obtained for this example by taking the appropriate weight sequence wn . The same example is considered by Xie (2008) and Hu et al. (2009) in a very complicated way, under the additional conditions P (A ∪ B) = 1 and P (AB) = 0. At the same time, we note that the best result for this example could not be obtained by using the Borel–Cantelli lemma with a determined weight sequence in Feng et al. (2009). 2. Proof of the theorems To prove our theorems, we need to prepare with the following lemma. Lemma 6. Let X be a real-valued integrable random variable. Then for any r ≤ 1 and p > 1, we have
P
X E (X )
>r
≥ (1 − r )
p p−1
p 1−1 p X E . − r E (X )
For any p > 0 and p ̸= 1, we have P (X ̸= 0) ≥
E
|X | E (|X |)
p 1−1 p
.
Proof. It suffices to prove the first inequality for E (X ) = 1. Note that for any r, X − r ≤ (X − r )1{X >r } . By taking expectations and using the Hölder inequality, we obtain for r ≤ 1 and p > 1, 1 − r ≤ E ((X − r )1{X >r } ) ≤ (E (|X − r |p ))1/p (P (X > r ))(p−1)/p , which yields the first inequality. Similarly, by the Hölder inequality, we have for 0 < p < 1, E (|X |) = E (|X |1{X ̸=0} ) ≥ (E (|X |p ))1/p (P (X ̸= 0))(p−1)/p and for p > 1, E (|X |) = E (|X |1{X ̸=0} ) ≤ (E (|X |p ))1/p (P (X ̸= 0))(p−1)/p . Both cases imply the second inequality.
k=1 Xk . By E (Sn ) → ∞ and the first part of Lemma 6, for all r ∈ (0, 1) and p > 1 we have lim Sm = ∞ = lim P lim Sm > rE (Sn )
Proof of Theorem 2. Let Sn = P
m→∞
n
n→∞
m→∞
≥ lim sup P (Sn > rE (Sn )) n→∞
≥ (1 − r )
p p−1
p 1−1 p Sn lim sup E − r , E ( S ) n→∞ n
where the first inequality is due to the easy inclusion relation: for all n,
lim Sm > rE (Sn ) ⊃ (Sn > rE (Sn )) .
m→∞
874
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
By the Minkowski inequality, we have for p > 1,
1−p p p 1−1 p p 1p Sn Sn E . − r ≥ E + r E (Sn ) E (Sn ) And therefore,
1−p p p 1p p Sn P lim Sm = ∞ ≥ (1 − r ) p−1 lim sup E + r . m→∞ E (Sn ) n→∞
As r ∈ (0, 1) is arbitrary, we complete the proof of the theorem by letting r tend to zero in the above inequality.
Proof of Theorem 3. Fix the natural number m and note that
{Sn − Sm ̸= 0} ⊂ {∪nk=m+1 {Xk ̸= 0}}. By the second part of Lemma 6, for p > 0 and p ̸= 1 we have
∞
P
{Xk ̸= 0}
n
= lim P n→∞
k=m+1
{Xk ̸= 0}
k=m+1
≥ lim P (Sn − Sm ̸= 0) n→∞
n→∞
1−1 p
E (|Sn − Sm |p )
≥ lim sup
(E (|Sn − Sm |))p
=: I .
For p > 1, by the Minkowski inequality,
(E (|Sn − Sm |p ))1/p ≤ (E (|Sn |p ))1/p + (E (|Sm |p ))1/p and E (|Sn − Sm |) ≥ |E (|Sn |) − E (|Sm |)|, we obtain
(E (|Sn |p ))1/p + (E (|Sm |p ))1/p I ≥ lim sup |E (|Sn |) − E (|Sm |)| n→∞
1−p p
.
By the Hölder inequality, we have E (|Sn |) ≤ (E (|Sn |p ))1/p for p > 1. The condition E (|Sn |) → ∞ implies E (|Sn |p ) → p ∞, EE(|(|SSm||p )) → 0 and (E (|ES |S|pm))| 1/p → 0. Hence n
n
I ≥ lim sup
1 + (E (|Sm|p ))1/p n
E (|Sn |) (E (|Sn |p ))1/p −
n→∞
1−p p
(E (|S |p ))1/p
E (|Sm |) (E (|Sn |p ))1/p
= lim sup Tn′ (p). n→∞
Similarly, for 0 < p < 1, by the inequalities E (|Sn − Sm |p ) ≥ |E (|Sn |p ) − E (|Sm |p )| and E (|Sn − Sm |) ≤ E (|Sn |) + E (|Sm |), we obtain
I ≥ lim sup n→∞
|E (|Sn |p ) − E (|Sm |p )| (E (|Sn |) + E (|Sm |))p
1−1 p
.
By the Hölder inequality, we have E (|Sn |) ≥ (E (|Sn |p ))1/p for 0 < p < 1. By the condition E (|Sn |) → ∞, we have (E (|Sm |p ))1/p E (|Sn |)
→ 0 and
E |Sm | E (|Sn |)
→ 0. Hence
1−1 p E (|Sn |p ) E (|Sm |p ) − (E (|Sn |))p (E (|Sn |))p I ≥ lim sup = lim sup Tn′ (p). p E (|Sm |) n→∞ n→∞ 1 + E (|S |) n
J. Liu / Statistics and Probability Letters 82 (2012) 871–875
875
Combining the above two cases, we obtain for all p > 0 and p ̸= 1,
P (lim sup{Xn ̸= 0}) = lim P m→∞
Now we have finished the proof.
∞ k=m+1
{Xk ̸= 0} ≥ lim sup Tn′ (p). n→∞
Acknowledgments The author is very grateful to the Co-Editor-in-Chief and an anonymous referee for their valuable suggestions and comments. This work was supported by the NSF of China (Nos. 10901065 and 10871215). References Feng, C., Li, P., Shen, J., 2009. On the Borel–Cantelli lemma and its generalization. C. R. Acad. Sci. Paris, Sér. I 347, 1313–1316. Hu, S.H., Wang, X.J., Li, X.Q., Zhang, Y.Y., 2009. Comments on the paper: a bilateral inequality on the Borel–Cantelli lemma. Statist. Probab. Lett. 79, 889–893. Xie, Y.Q., 2008. A bilateral inequality on the Borel–Cantelli lemma. Statist. Probab. Lett. 79, 1577–1580. Xie, Y.Q., 2009. A bilateral inequality on nonnegative bounded random sequence. Statist. Probab. Lett. 78, 2052–2057.
Further reading Chandra, T.K., 2008. The Borel–Cantelli lemma under dependence conditions. Statist. Probab. Lett. 78, 390–395. Kallenberg, O., 2002. Foundations of Modern Probability, second ed. Springer, New York. Kochen, S.P., Stone, C.J., 1964. A note on the Borel–Cantelli lemma. Illinois J. Math. 8, 248–251. Petrov, V.V., 2002. A note on the Borel–Cantelli lemma. Statist. Probab. Lett. 58, 283–286. Petrov, V.V., 2004. A generalization of the Borel–Cantelli lemma. Statist. Probab. Lett. 67, 233–239. Yan, J., 2006. A new proof of a generalized Borel–Cantelli lemma. In: In Memoriam Paul-André Meyer: Seminar on Probability Theory XXXIX. In: Lecture Notes in Mathematics, vol. 1874. Springer-Verlag, Berlin, pp. 77–79.