On the strong laws of large numbers for weighted sums of random variables

Statistics and Probability Letters 118 (2016) 87–93

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

On the strong laws of large numbers for weighted sums of random variables Pingyan Chen a , Soo Hak Sung b,∗ a

Department of Mathematics, Jinan University, Guangzhou, 510630, PR China

b

Department of Applied Mathematics, Pai Chai University, Daejeon, 35345, South Korea

article

info

Article history: Received 19 April 2016 Received in revised form 20 June 2016 Accepted 22 June 2016 Available online 28 June 2016 MSC: 60F15

abstract Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable. Let {an , n ≥ 1} and {bn , n ≥ 1} be  sequences of real numbers with n 0 < bn ↑ ∞. Sufficient conditions are given under which i=1 ai Xi /bn → 0 almost surely. No conditions are imposed on the joint distributions of the {Xn }. Our results generalize and improve some known results of the strong law of large numbers for random variables. We also give two examples which show the sharpness of our results. © 2016 Elsevier B.V. All rights reserved.

Keywords: Strong law of large numbers Weighted sums Almost sure convergence

1. Introduction Let {Xn , n ≥ 1} be a sequence of random variables defined on a probability space (Ω , F , P ). Let {an , n ≥ 1} and {bn , n ≥ 1} be sequences of real numbers with 0 < bn ↑ ∞. Then {an Xn , n ≥ 1} is said to obey the general strong law of large numbers (SLLN) with norming constants {bn , n ≥ 1} if n 

ai X i

i=1

bn

→ 0 almost surely (a.s.).

(1.1)

When {Xn , n ≥ 1} is a sequence of independent and identically distributed (i.i.d.) random variables, the SLLNs of the form (1.1) have been established by many authors. The special cases of (1.1) are the Kolmogorov SLLN (an = 1, bn = n) and the Marcinkiewicz–Zygmund SLLN (an = 1, bn = n1/p , 0 < p < 2). Jamison et al. (1965) obtained sufficient conditions for (1.1) n when bn = a . For more general weights, Jajte (2003) established sufficient conditions for (1.1). On the other hand, i=1 i Sung (2011) gave sufficient conditions for (1.1) under the condition that {Xn , n ≥ 1} is a sequence of dependent random variables satisfying some moment inequalities. In this paper, we will focus on the random variables which have no conditions on the joint distributions of the {Xn }. It is not assumed that E |Xn | < ∞ for all n ≥ 1. When an = 1 for all n ≥ 1, Martikainen and Petrov (1980) proved a SLLN for random variables.

∗

Corresponding author. E-mail address: [email protected] (S.H. Sung).

http://dx.doi.org/10.1016/j.spl.2016.06.020 0167-7152/© 2016 Elsevier B.V. All rights reserved.

88

P. Chen, S.H. Sung / Statistics and Probability Letters 118 (2016) 87–93

Theorem 1.1 (Martikainen and Petrov, 1980). Let {X , Xn , n ≥ 1} be a sequence of identically distributed random variables. Let {bn , n ≥ 1} be a sequence of positive numbers with 0 < bn ↑ ∞. If bn

∞  1

= O(n)

(1.2)

P (|X | > bn ) < ∞,

(1.3)

i=n

bi

and ∞  n =1

then

n

i=1

Xi /bn → 0 a.s.

A sequence {Xn , n ≥ 1} of random variables is said to be stochastically dominated by a random variable X if there exists a constant 0 < D < ∞ such that P (|Xn | > x) ≤ DP (|X | > x) for all x > 0 and n ≥ 1. Adler and Rosalsky (1987) extended the SLLN of Martikainen and Petrov (1980) to weighted sums. Theorem 1.2 (Adler and Rosalsky, 1987). Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X . Let {an , n ≥ 1} and {bn , n ≥ 1} be sequences of real numbers satisfying 0 < bn ↑ ∞ and max 1≤i≤n

∞ bi  |ai |

|ai |

= O(n).

bi

i =n

(1.4)

If ∞ 

P (|an X | > bn ) < ∞,

(1.5)

n =1

then the SLLN (1.1) holds. If an = 1 for all n ≥ 1, then conditions (1.4) and (1.5) are identical to conditions (1.2) and (1.3), respectively. The Marcinkiewicz–Zygmund SLLN states that if {X , Xn , n ≥ 1} is a sequence of i.i.d. random variables such that E |X |p < ∞ for some 0 < p < 2 and EX = 0 if 1 ≤ p < 2, then (1.1) holds with an = 1 and bn = n1/p . It is well known that the independence hypothesis is not necessary when 0 < p < 1. This fact can also be obtained from Theorem 1.1 noting that ∞  1

i1/p i=n

= O(n1−1/p ) if 0 < p < 1,

(1.6)

and n=1 P (|X | > n1/p ) < ∞ is equivalent to E |X |p < ∞. Rosalsky and Stoica (2010) obtained a SLLN for identically distributed random variables under a more general condition than (1.6).

∞

Theorem 1.3 (Rosalsky and Stoica, 2010). Let {X , Xn , n ≥ 1} be a sequence of identically distributed random variables with E |X |p < ∞ for some 0 < p < 1. Let {bn , n ≥ 1} be a sequence of positive numbers satisfying 0 < bn ↑ ∞ and ∞  1 i=n

bi





1

=O

1−p

bn

If (1.3) holds, then

n

i =1

.

Xi /bn → 0 a.s.

Recently, Liao and Rosalsky (2013) proved a SLLN for weighted sums of random variables. Theorem 1.4 (Liao and Rosalsky, 2013). Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X with E |X |p < ∞ for some 0 < p < 1. Let {an , n ≥ 1} and {bn , n ≥ 1} be sequences of real numbers satisfying 0 < bn ↑ ∞ and

| an | bn

 =O

1 n1/p



.

Then the SLLN (1.1) holds.

P. Chen, S.H. Sung / Statistics and Probability Letters 118 (2016) 87–93

89

In this paper, we establish some general SLLNs of the form (1.1), where the {Xn , n ≥ 1} is stochastically dominated by a random variable X . No conditions are imposed on the joint distributions of the {Xn }. From our results, we can easily obtain the SLLNs of Adler and Rosalsky (1987), Rosalsky and Stoica (2010), and Liao and Rosalsky (2013). We also give two examples which show the sharpness of our results. Throughout this paper, the symbol C denotes a positive constant which is not necessarily the same one in each appearance, I (A) denotes the indicator function of the event A, and f (x) ∼ g (x) means that f (x)/g (x) → 1 as x → ∞. It proves convenient to define log x = max{1, ln x} for x > 0, where ln x denotes the natural logarithm. 2. General SLLNs for random variables In this section, we present some general SLLNs for sequences of random variables. For a sequence of positive numbers {cn , n ≥ 1} satisfying 0 < cn → ∞, let N (x) be the distribution function of {cn }, i.e., N (x) = ♯{n : cn ≤ x},

x ≥ 0.

Then N (x) is a nondecreasing integer valued function with limx→∞ N (x) = ∞. Lemma 2.1. Let X be a random variable with the distribution function F and let {cn , n ≥ 1} be a sequence of positive numbers such that 0 < cn → ∞. Let N (x) be the distribution function of {cn , n ≥ 1}. Then the following statement holds. ∞



N (y)

 |x|

−∞

y2

y≥|x|

∞ 

dy dF (x) =

P (|X | > cn ) +

n=1

∞  1 n =1

cn

E |X |I (|X | ≤ cn ).

Proof. By the definition of N (x), we have ∞



N (y)

 |x|

−∞



∞

=

y≥|x|

∞  

∞  

∞  

∞

|x| (I (|x| > cn ) + I (|x| ≤ cn )) ∞

  |x| I (|x| > cn )

∞

∞ 

1 y≥|x|



I (|x| > cn ) +

−∞

n=1

=

I (cn ≤ y) dy dF (x)

−∞

n=1

=

y2 n=1

−∞

n=1

=

∞ 1 

 |x|

−∞

=

dy dF (x)

y2

y≥|x|

P (|X | > cn ) +

n=1

cn

∞  1 n =1

Hence the result is proved.

1

cn

y2



1 2 y≥|x| y

I (cn ≤ y) dy dF (x)

dy + I (|x| ≤ cn )



1 y≥cn

y2

 dy

dF (x)

 |x|I (|x| ≤ cn ) dF (x)

E |X |I (|X | ≤ cn ).



The following theorem provides sufficient conditions for the SLLN (1.1). Theorem 2.1. Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X with the distribution F . Let {an , n ≥ 1} and {bn , n ≥ 1} be sequences of real numbers satisfying 0 < bn ↑ ∞ and 0 < bn /|an | → ∞. Let N (x) be the distribution of {bn /|an |, n ≥ 1}. If



∞

N (y)

 |x|

−∞

y≥|x|

y2

dy dF (x) < ∞,

(2.1)

then the SLLN (1.1) holds. Proof. By Lemma 2.1, (2.1) is equivalent to the pair of conditions (1.5) and ∞  |an | n =1

bn

E |X |I (|X | ≤ bn /|an |) < ∞.

(2.2)

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P. Chen, S.H. Sung / Statistics and Probability Letters 118 (2016) 87–93

Observe that n 

n 

ai Xi

i =1

=

bn

n 

ai Xi I (|Xi | > bi /|ai |)

i =1

+

bn

ai Xi I (|Xi | ≤ bi /|ai |)

i=1

bn

:= I1 + I2 . Then we have by the stochastic domination condition and (1.5) that ∞ 

P (|Xn | > bn /|an |) ≤ D

n =1

∞ 

P (|X | > bn /|an |) < ∞.

n =1

The Borel–Cantelli lemma, together with the condition 0 < bn ↑ ∞, implies that I1 → 0 a.s. We also have by the stochastic domination condition, (1.5), and (2.2) that ∞  |an | n =1

bn

E |Xn |I (|Xn | ≤ bn /|an |) ≤

∞  |an | n =1

=D

bn

D {E |X |I (|X | ≤ bn /|an |) + bn /|an |P (|X | > bn /|an |)}

∞  |an | n =1

bn

E |X |I (|X | ≤ bn /|an |) + D

∞ 

P (|X | > bn /|an |)

n=1

< ∞, and so ∞  an n =1

Xn I (|Xn | ≤ bn /|an |) converges a.s.

bn

The Kronecker lemma implies that I2 → 0 a.s. Thus the result is proved.



The following remark is suggested by a reviewer. Remark 2.1. For a sequence of random variables {Yn , n ≥ 1}, Cantrell and Rosalsky (2003) observed that

 ∞  E

n =1

|Yn | |Yn | + bn



<∞

(2.3)

is equivalent to the pair of conditions ∞ 

P (|Yn | > bn ) < ∞

(2.4)

n =1

and ∞  1 n =1

bn

E |Yn |I (|Yn | ≤ bn ) < ∞.

(2.5)

They proved that i=1 Yi /bn → 0 a.s under the moment condition (2.3). Now we consider a sequence of random variables {Xn , n ≥ 1} which is stochastically dominated by a random variable X . Let Yn = an Xn , n ≥ 1. Then we see from the proof of Theorem 2.1 that the pair of conditions (1.5) and (2.2) implies (2.4) and (2.5). The converse is also true if the random variables are identically distributed. Since (2.1) is equivalent to the pair of conditions (1.5) and (2.2), (2.1) implies (2.3) and hence Theorem 2.1 follows from the SLLN of Cantrell and Rosalsky (2003).

n

Although Theorem 2.1 is a general SLLN, it cannot be applied to the case where the distribution function N (x) of {bn /|an |} cannot be estimated appropriately. We now give some sufficient conditions irrespective of the N (x) under which (1.1) holds. To do this, the following lemma is needed. Lemma 2.2. Let X be a random variable. Let {cn , n ≥ 1} be a sequence of positive numbers satisfying cn

∞  1

= O(n).

(2.6)

P (|X | > cn ) < ∞,

(2.7)

i=n

ci

If ∞  n =1

P. Chen, S.H. Sung / Statistics and Probability Letters 118 (2016) 87–93

91

then ∞  1 n =1

cn

E |X |I (|X | ≤ cn ) < ∞.

Proof. Let F (x) be the distribution function of X . Then ∞  1

J1 :=

=

E |X |I (|X | ≤ cn )

cn

n=1

 ∞  1 n =1



cn

∞

|x|I (|x| ≤ cn ) dF (x) −∞

∞

|x|

= −∞

1



dF (x).

c n: cn ≥|x| n

Now we estimate J2 :=



n: cn ≥|x|

M (x) = ♯{n : cn < x},

(2.8)

1/cn . To do this, we define M (x) by

x ≥ 0.

Note that M (x) is slightly different from the distribution of {cn }. Let n0 = inf{n : cn ≥ |x|}. Then c1 < |x|, . . . , cn0 −1 < |x| and so M (|x|) ≥ n0 − 1. It follows by (2.6) that J2 ≤

∞  1

≤C

cn

n=n0

n0 cn0

(by (2.6))

n0

(by the definition of n0 ) |x| M (|x|) + 1 ≤C . |x| ≤C

(2.9)

Substituting (2.9) into (2.8), we get that ∞



M (|x|) + 1 dF (x) = CE [M (|X |)] + C .

J1 ≤ C −∞

On the other hand, we also get by (2.7) that

∞>

∞ 

P (|X | > cn ) = E

 ∞ 

n=1

 I (|X | > cn )

= E [M (|X |)].

n=1

Hence we obtain that J1 < ∞.



Remark 2.2. From condition (2.6), we can directly obtain that cn → ∞. Under the stronger condition 0 < cn ↑ ∞, Lemma 2.2 can be easily proved by the standard method. Theorem 2.2. Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X . Let {an , n ≥ 1} and {bn , n ≥ 1} be sequences of real numbers satisfying 0 < bn ↑ ∞ and ∞ bn  |ai |

|an |

i=n

bi

= O(n).

(2.10)

If (1.5) holds, then the SLLN (1.1) holds. Proof. The proof is similar to that of Theorem 2.1. It suffices to show that (2.2) holds. But, (2.2) follows from Lemma 2.2 with cn replaced by bn /|an |.  Remark 2.3. Theorem 2.2 improves Theorem 1.2, since (2.10) is weaker than (1.4). Remark 2.4. Using Theorem 2.2, we can prove Theorem 1.4. If |an |/bn = O(1/n1/p ) for some 0 < p < 1, then there exists a constant C > 0 such that |an |/bn ≤ C /n1/p for all n ≥ 1. Set a∗n = Cbn /n1/p , n ≥ 1. Then |an | ≤ a∗n and ∞ bn  a∗i

a∗n i=n bi

= O(n).

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P. Chen, S.H. Sung / Statistics and Probability Letters 118 (2016) 87–93

∗ p Furthermore, we have that n=1 P (|X | > bn /an ) < ∞, since E |X | < ∞. Hence, we obtain by Theorem 2.2 that ∗ ai |Xi |/bn → 0 a.s., and so the SLLN (1.1) holds.

∞

n

i =1

Corollary 2.1. Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X . Let L : (0, ∞) → (0, ∞) be a slowly varying function. If ∞ 

P (|X | > nα /L(n)) < ∞

for some α > 1,

n =1

then

n

i=1

L(i)Xi /nα → 0 a.s.

Proof. Let an = L(n) and bn = nα , n ≥ 1. Then the hypotheses of Theorem 2.2 are easily satisfied. Hence the result follows from Theorem 2.2.  The following theorem is a version of Theorem 1.3 for weighted sums. Theorem 2.3. Let {Xn , n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X with E |X |p < ∞ for some 0 < p < 1. Let {an , n ≥ 1} and {bn , n ≥ 1} be sequences of real numbers satisfying 0 < bn ↑ ∞ and ∞  |ai |

bi

i=n

 =O

bn

p−1 

|an |

.

(2.11)

If (1.5) holds, then the SLLN (1.1) holds. Proof. Define a sequence {a∗n , n ≥ 1} by

∗

an =

   |an |,

if

bn    1/p ,

if

n

bn

≤ n1/p ,

|an | bn

> n1/p .

|an |

Then |an | ≤ a∗n for all n ≥ 1. Hence, it is enough to show that n 

a∗i |Xi |

i =1

bn

→ 0 a.s.

(2.12)

To prove (2.12), we will apply Theorem 2.2 to sequences {a∗n } and {bn }. By (2.11), ∞ bn  a∗i

a∗n i=n bi

≤

bn a∗n

≤C

=

 ∞  |ai |

bn

bi

i =n



a∗n

 bn    C   |an |   1/p   Cn

+

∞  1 i =n

i1/p

p−1

bn

+

|an | 

bn





n n1/p

p−1

|an |  p−1 bn

+

+

|an |



n n1/p n

if



n1/p

if

bn

|an | bn

|an |

≤ n1/p , > n1/p ,

≤ Cn. Thus condition (2.10) of Theorem 2.2 holds. By (1.5) and E |X |p < ∞, ∞ 

P (|X | > bn /a∗n ) ≤

n =1

∞ 

P (|X | > bn /|an |) +

n =1

≤

∞ 

∞ 

P (|X | > n1/p )

n =1

P (|X | > bn /|an |) + E |X |p

n =1

< ∞. Hence (2.12) holds by Theorem 2.2.



P. Chen, S.H. Sung / Statistics and Probability Letters 118 (2016) 87–93

93

Remark 2.5. If an ≡ 1 and {X , Xn , n ≥ 1} is a sequence of identically distributed random variables with E |X |p < ∞ for some 0 < p < 1, then Theorem 2.3 reduces to Theorem 1.3. 3. Some examples In this section, we give two interesting examples. The first satisfies the hypotheses of Theorem 2.1, but not those of Theorem 2.2. Example 3.1. Let {X , Xn , n ≥ 1} be a sequence of identically distributed random variables with E [|X |/(log |X |)α ] < ∞ for some α > 0. Let an = 1/ log n and bn = n(log n)α , n ≥ 1. Let F (x) and N (x) be the distributions of X and {bn /an }, respectively. Then N (x) ∼ x/(log x)α+1 . It follows that



∞

N (y)

 |x| y≥|x|

−∞

y2

dy dF (x) ≤ C



∞



y

|x| −∞

y≥|x|

y2

(log y)α+1

dy dF (x)

≤ CE [|X |/(log |X |)α ] < ∞. Thus by Theorem 2.1, the SLLN (1.1) holds. That is, n 

1 X log i i

i =1

n(log n)α

→ 0 a.s.

However, ∞ bn  |ai |

|an |

i=n

bi

= n(log n)α+1

∞ 

1

i(log i)α+1 i=n

∼

1

α

n log n

and so condition (2.10) of Theorem 2.2 is not satisfied. The following example satisfies the hypotheses of Theorem 2.2. Example 3.2. Let α > 1 and let {X , Xn , n ≥ 1} be a sequence of identically distributed random variables with P (|X | > nα /L(n)) = O



1 n(log n)2



,

(3.1)

where L : (0, ∞) → (0, ∞) is a slowly varying function. Let an = L(n) and bn = nα , n ≥ 1. Then the hypotheses of Theorem 2.2 are easily satisfied. By Theorem 2.2 or Corollary 2.1, n 

L(i)Xi

i=1

nα

→ 0 a.s.

Remark 3.1. If bn /an = nα /L(n) ↑ ∞, then it is easy to show that (1.5) implies (2.2). Since (2.1) is equivalent to the pair of conditions (1.5) and (2.2), (3.1) implies (2.1). Hence, in this case, Example 3.2 can be applied to Theorem 2.1. Acknowledgments The authors thank the referee for a very careful reading and useful comments which helped in improving the presentation. The research of Pingyan Chen is supported by the National Natural Science Foundation of China (No. 11271161). The research of Soo Hak Sung is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2058041). References Adler, A., Rosalsky, A., 1987. Some general strong laws for weighted sums of stochastically dominated random variables. Stoch. Anal. Appl. 5, 1–16. Cantrell, A., Rosalsky, A., 2003. Some strong laws of large numbers for Banach space valued summands irrespective of their joint distributions. Stoch. Anal. Appl. 21, 79–95. Jajte, R., 2003. On the strong law of large numbers. Ann. Probab. 31, 409–412. Jamison, B., Orey, S., Pruitt, W., 1965. Convergence of weighted averages of independent random variables. Z. Wahrsch. Gebiete 4, 40–44. Liao, Y., Rosalsky, A., 2013. Some strong laws for weighted sums of stochastically dominated Banach space valued random elements irrespective of their joint distributions. Stoch. Anal. Appl. 31, 427–439. Martikainen, A.I., Petrov, V.V., 1980. On a theorem of feller. Theory Probab. Appl. 25, 191–193. Rosalsky, A., Stoica, G., 2010. On the strong law of large numbers for identically distributed random variables irrespective of their joint distributions. Statist. Probab. Lett. 80, 1265–1270. Sung, S.H., 2011. On the strong law of large numbers for weighted sums of random variables. Comput. Math. Appl. 62, 4277–4287.