Open Access

A supplement to the convergence rate in a theorem of Heyde

Journal of Inequalities and Applications20122012:195

https://doi.org/10.1186/1029-242X-2012-195

Received: 27 May 2012

Accepted: 20 August 2012

Published: 4 September 2012

Abstract

Let { X , X n , n 1 } be a sequence of i.i.d. random variables with zero mean, set S n = k = 1 n X k , E X 2 = σ 2 > 0 , and λ ( ϵ ) = n = 1 P ( | S n | n ϵ ) . In this paper, the authors discuss the rate of approximation of σ 2 by ϵ 2 λ ( ϵ ) under suitable conditions, improve the results of Klesov (Theory Probab. Math. Stat. 49:83-87, 1994), and extend the work He and Xie (Acta Math. Appl. Sin. 2012, doi:10.1007/s10255-012-0138-6).

MSC:60F15, 60G50.

Keywords

convergence ratei.i.d. random variabletheorem of Heyde

1 Introduction and main results

Let { X , X n , n 1 } be a sequence of i.i.d. random variables, set S n = k = 1 n X k , and λ ( ϵ ) = n = 1 P ( | S n | n ϵ ) . Heyde [1] proved that
lim ϵ 0 ϵ 2 λ ( ϵ ) = σ 2 ,

whenever E X 2 = σ 2 < and E X = 0 .

There are various extensions of this result: Chen [2], Gut and Spǎtara [3], Lanzinger and Stadtmüller [4]. Liu and Lin [5] introduced a new kind of complete moment convergence; Klesov [6] studied the rate of approximation of σ 2 by ϵ 2 λ ( ϵ ) and proved the following Theorem A.

Theorem A Let { X , X n , n 1 } be a sequence of i.i.d. random variables with zero mean, if E X 2 = σ 2 > 0 , and E | X | 3 < , then
ϵ 2 λ ( ϵ ) σ 2 = o ( ϵ 1 / 2 ) , as ϵ 0 .

Recently, He and Xie [7] obtained Theorem B which improved Theorem A. Gut and Steinebach [8] extended the results of Klesov [6].

Theorem B Let { X , X n , n 1 } be a sequence of i.i.d. random variables, and 0 < δ 1 , if
E X = 0 , E X 2 = σ 2 > 0 and E | X | 2 + δ < ,
then
ϵ 2 λ ( ϵ ) σ 2 = { O ( ϵ ) , δ = 1 , o ( ϵ δ ) , 0 < δ < 1 .

Let G be the set of functions g ( x ) that are defined for all real x and satisfy the following conditions: (a) g ( x ) is nonnegative, even, nondecreasing in the interval x > 0 , and g ( x ) 0 for x 0 ; (b) x g ( x ) is nondecreasing in the interval x > 0 .

Let G 0 be the set of functions g ( x ) G satisfying the supplementary condition (c) lim x g ( x 2 ) x g ( x ) = 0 . Obviously, the function g ( x ) = | x | δ with 0 < δ < 1 belongs to G 0 and does not belong to G 0 if δ = 1 . The purpose of this paper is to generalize Theorem B to the case where the condition E | X | 2 + δ < is replaced by a more general condition E | X | 2 g ( X ) < in which the function g belongs to some subset of G. Denote T g ( v ) = E X 2 g ( X ) I ( | X | > v ) , T g ( v ) is a nonnegative nonincreasing function in the interval v > 0 , and lim v T g ( v ) = 0 with E X 2 g ( X ) < . Now we state our results as follows.

Theorem 1.1 Let { X , X n ; n 1 } be a sequence of i.i.d. random variables with zero mean and E X 2 = σ 2 > 0 , if E X 2 g ( X ) < for some function g ( x ) G , and
n = 1 1 n g ( n ) < ,
(1.1)
then
ϵ 2 λ ( ϵ ) σ 2 = O ( ϵ 1 / 2 ) + o ( 1 ) ( h 1 ( ϵ ) + f 1 ( ϵ ) ) , as ϵ 0 ,
(1.2)

where f 1 ( ϵ ) = n = [ 1 ϵ 2 ] + 1 1 n g ( n ) , h 1 ( ϵ ) = ϵ 2 n = 1 [ 1 ϵ 2 ] 1 g ( n ) .

Theorem 1.2 Under the conditions of Theorem 1.1, and g ( x ) G 0 , then
ϵ 2 λ ( ϵ ) σ 2 = o ( 1 ) ( h 1 ( ϵ ) + f 1 ( ϵ ) ) , as ϵ 0 .
(1.3)

Throughout this paper, we suppose that C denotes a constant which only depends on some given numbers and may be different at each appearance, and that [ x ] denotes the integer part of x.

2 Proofs of the main results

Before we prove the main results we state some lemmas. Lemma 2.1 is from [7]. Φ ( x ) is the standard normal distribution function, Φ ( x ) = 1 2 π x e t 2 / 2 d t .

Lemma 2.1 Let { X , X n , n 1 } be a sequence of i.i.d. standard normal distribution random variables. Then
ϵ 2 λ ( ϵ ) = ϵ 2 n = 1 2 2 π ϵ n e t 2 / 2 d t = 1 ϵ 2 2 + O ( ϵ 3 ) , as ϵ 0 .
(2.1)
If { X n , n 1 } is a sequence of independent random variables with zero mean and finite variance, and put E X j 2 = σ j 2 , B n = j = 1 n σ j 2 , Bikelis[9]obtained the following inequality:

for every x, where V j ( x ) = P ( X j < x ) is the distribution function of the random variable X j . By applying the above inequality to the sequence of i.i.d. random variables with zero mean and variance 1, and letting | x | = ϵ n , we have the following lemma.

Lemma 2.2 Let { X , X n , n 1 } be a sequence of i.i.d. random variables with zero mean and E X 2 = 1 . Then for any given ϵ > 0 , we have

where V ( x ) = P ( X < x ) is the distribution function of a random variable X.

Proof of Theorem 1.1 Without loss of generality, we suppose that σ 2 = 1 , 0 < ϵ < 1 , and write
ϵ 2 λ ( ϵ ) = I + ϵ 2 n = 1 2 2 π ϵ n e t 2 / 2 d t ,
where
I = ϵ 2 n = 1 ( P ( | S n | > n ϵ ) 2 2 π ϵ n e t 2 / 2 d t ) .
Applying Lemma 2.1, we obtain
ϵ 2 λ ( ϵ ) = I + 1 ϵ 2 2 + O ( ϵ 3 ) ,
then
ϵ 2 λ ( ϵ ) 1 = ϵ 2 2 + ϵ 2 n = 1 R n + O ( ϵ 3 ) ,
here R n = P ( | S n | > n ϵ ) 2 2 π ϵ n e t 2 / 2 d t . By Lemma 2.2,
| R n | R 1 n + R 2 n ,
where
We obtain
ϵ 2 λ ( ϵ ) 1 = ϵ 2 n = 1 R 1 n + ϵ 2 n = 1 R 2 n + O ( ϵ 2 ) .
(2.2)
Firstly, we estimate ϵ 2 n = 1 R 1 n . Note that
ϵ 2 n = 1 R 1 n = ϵ 2 n = 1 [ 1 ϵ 2 ] R 1 n + ϵ 2 n = [ 1 ϵ 2 ] + 1 R 1 n = : T 1 + T 2 .
Applying the condition E X 2 g ( X ) < , we have
lim n | u | > n 4 u 2 g ( u ) d V ( u ) = 0 .
Therefore, for any η > 0 , there is an integer N 0 such that | u | > n 4 u 2 g ( u ) d V ( u ) η , whenever n > N 0 . Hence
T 1 C ϵ 2 n = 1 N 0 | u | > n u 2 d V ( u ) + C ϵ 2 n = N 0 + 1 [ 1 ϵ 2 ] ( 1 + ϵ n ) 2 | u | > ( 1 + ϵ n ) n u 2 d V ( u ) C ϵ 2 N 0 + C ϵ 2 η n = N 0 + 1 [ 1 ϵ 2 ] 1 ( 1 + ϵ n ) 2 g ( n ( 1 + ϵ n ) ) C ϵ 2 ( N 0 + η n = 1 [ 1 ϵ 2 ] 1 g ( n ) ) = C h 1 ( ϵ ) ( N 0 n = 1 [ 1 ϵ 2 ] 1 g ( n ) + η ) C h 1 ( ϵ ) ( N 0 ϵ + η ) = o ( h 1 ( ϵ ) ) ,
(2.3)
where h 1 ( ϵ ) = ϵ 2 n = 1 [ 1 ϵ 2 ] 1 g ( n ) . For T 2 , noting that g ( x ) G , we have the following inequality:
T 2 C ϵ 2 n = [ 1 ϵ 2 ] + 1 1 n ϵ 2 | u | > n ( 1 + ϵ n ) u 2 d V ( u ) C n = [ 1 ϵ 2 ] + 1 1 n g ( n ( 1 + ϵ n ) ) | u | > n ( 1 + ϵ n ) u 2 g ( u ) d V ( u ) C n = [ 1 ϵ 2 ] + 1 1 n g ( n ) | u | > 1 ϵ u 2 g ( u ) d V ( u ) C T g ( 1 ϵ ) f 1 ( ϵ ) .
(2.4)
Next, we estimate the second term of (2.2). Note that
ϵ 2 n = 1 R 2 n = C ϵ 2 n = 1 n 1 / 2 ( 1 + ϵ n ) 3 | u | ( n ( 1 + ϵ n ) ) 1 / 2 | u | 3 d V ( u ) + C ϵ 2 n = 1 n 1 / 2 ( 1 + ϵ n ) 3 ( n ( 1 + ϵ n ) ) 1 / 2 < | u | < n ( 1 + ϵ n ) | u | 3 d V ( u ) = : J 1 + J 2 .
For J 1 , we can write
J 1 = C ϵ 2 ( n = 1 [ 1 ϵ 2 ] + n = [ 1 ϵ 2 ] + 1 ) n 1 / 2 ( 1 + ϵ n ) 3 | u | ( n ( 1 + ϵ n ) ) 1 / 2 | u | 3 d V ( u ) = : J 11 + J 12 .
Noting that x g ( x ) is nondecreasing in the interval x > 0 , we have
J 11 = C ϵ 2 n = 1 [ 1 ϵ 2 ] 1 n ( 1 + ϵ n ) 3 | u | ( n ( 1 + ϵ n ) ) 1 / 2 | u | 3 d V ( u ) C ϵ 2 n = 1 [ 1 ϵ 2 ] 1 n 1 / 4 ( 1 + ϵ n ) 5 / 2 g ( ( n ( 1 + ϵ n ) ) 1 / 2 ) | u | ( n ( 1 + ϵ n ) ) 1 / 2 u 2 g ( u ) d V ( u ) C ϵ 2 n = 1 [ 1 ϵ 2 ] 1 n 1 / 4 g ( n 1 / 4 ) = C h 2 ( ϵ ) ,
(2.5)

where h 2 ( ϵ ) = ϵ 2 n = 1 [ 1 ϵ 2 ] 1 n 1 / 4 g ( n 1 / 4 ) .

Similarly, we can obtain
J 12 = C ϵ 2 n = [ 1 ϵ 2 ] + 1 1 n ( 1 + ϵ n ) 3 | u | ( n ( 1 + ϵ n ) ) 1 / 2 | u | 3 d V ( u ) C ϵ 2 n = [ 1 ϵ 2 ] + 1 1 n 1 / 4 ( 1 + ϵ n ) 5 / 2 g ( ( n ( 1 + ϵ n ) ) 1 / 2 ) | u | ( n ( 1 + ϵ n ) ) 1 / 2 u 2 g ( u ) d V ( u ) C ϵ 2 n = [ 1 ϵ 2 ] + 1 1 ϵ 5 / 2 n 3 / 2 g ( n 1 / 4 ) = C 1 ϵ f 2 ( ϵ ) ,
(2.6)

where f 2 ( ϵ ) = n = [ 1 ϵ 2 ] + 1 1 n 3 / 2 g ( n 1 / 4 ) .

For J 2 , we write
J 2 = C ϵ 2 ( n = 1 [ 1 ϵ 2 ] + n = [ 1 ϵ 2 ] + 1 ) n 1 / 2 ( 1 + ϵ n ) 3 ( n ( 1 + ϵ n ) ) 1 / 2 < | u | < n ( 1 + ϵ n ) | u | 3 d V ( u ) = : J 21 + J 22 .
Using the properties of g ( x ) by simple calculation, it follows that
J 21 = C ϵ 2 n = 1 [ 1 ϵ 2 ] n 1 / 2 ( 1 + ϵ n ) 3 ( n ( 1 + ϵ n ) ) 1 / 2 < | u | < n ( 1 + ϵ n ) | u | 3 d V ( u ) C ϵ 2 ( n = 1 N 0 + n = N 0 + 1 [ 1 ϵ 2 ] ) 1 ( 1 + ϵ n ) 2 g ( n ( 1 + ϵ n ) ) × ( n ( 1 + ϵ n ) ) 1 / 2 < | u | < n ( 1 + ϵ n ) u 2 g ( u ) d V ( u ) C ϵ 2 ( n = 1 N 0 + n = N 0 + 1 [ 1 ϵ 2 ] ) 1 g ( n ) | u | > n 1 / 4 u 2 g ( u ) d V ( u ) C ϵ 2 ( N 0 + η n = 1 [ 1 ϵ 2 ] 1 g ( n ) ) = o ( h 1 ( ϵ ) ) ,
(2.7)
and
J 22 C ϵ 2 n = [ 1 ϵ 2 ] + 1 n 1 2 ( 1 + ϵ n ) 3 ( n ( 1 + ϵ n ) ) 1 / 2 < | u | < n ( 1 + ϵ n ) | u | 3 d V ( u ) C n = [ 1 ϵ 2 ] + 1 1 n g ( n ) ( n ( 1 + ϵ n ) ) 1 / 2 < | u | < n ( 1 + ϵ n ) u 2 g ( u ) d V ( u ) C T g ( 1 ϵ ) n = [ 1 ϵ 2 ] + 1 1 n g ( n ) C T g ( 1 ϵ ) f 1 ( ϵ ) .
(2.8)
From (2.2) to (2.8), we conclude that
ϵ 2 λ ( ϵ ) 1 C 1 ϵ f 2 ( ϵ ) + C T g ( 1 ϵ ) f 1 ( ϵ ) + o ( 1 ) h 1 ( ϵ ) + C h 2 ( ϵ ) .
(2.9)
Since
1 ϵ f 2 ( ϵ ) C ϵ n = [ 1 ϵ 2 ] + 1 1 n 3 / 2 C ϵ ,
and
h 2 ( ϵ ) = ϵ 2 n = 1 [ 1 ϵ 2 ] 1 n 4 g ( n 4 ) C ϵ 2 n = 1 [ 1 ϵ 2 ] 1 n 4 C ϵ ,
by (2.9), we have
ϵ 2 λ ( ϵ ) 1 = O ( ϵ 1 / 2 ) + o ( 1 ) ( f 1 ( ϵ ) + h 1 ( ϵ ) ) .

This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2 By the conditions g ( x ) G 0 , and lim x g ( x 2 ) x g ( x ) = 0 , for any η > 0 , there is an integer N 1 such that g ( n ) n 4 g ( n 4 ) η , whenever n > N 1 . We have
h 2 ( ϵ ) ϵ 2 n = 1 N 1 1 n 4 g ( n 4 ) + ϵ 2 n = N 1 [ 1 ϵ 2 ] η g ( n ) C ϵ 2 N 1 + ϵ 2 n = N 1 + 1 [ 1 ϵ 2 ] η g ( n ) C ϵ 2 N 1 + ϵ 2 n = 1 [ 1 ϵ 2 ] η g ( n ) = o ( 1 ) h 1 ( ϵ ) ,
(2.10)
and
1 ϵ f 2 ( ϵ ) 1 ϵ n = [ 1 ϵ 2 ] + 1 η n 5 / 4 g ( n ) n = [ 1 ϵ 2 ] + 1 η n g ( n ) = o ( 1 ) f 1 ( ϵ ) .
(2.11)
By (2.9)-(2.11), note that T g ( 1 ϵ ) = o ( 1 ) , as ϵ 0 , we have
ϵ 2 λ ( ϵ ) σ 2 = o ( 1 ) ( h 1 ( ϵ ) + f 1 ( ϵ ) ) , as  ϵ 0 .

This completes the proof of Theorem 1.2. □

Remark 2.1 If g ( x ) = | x | δ , 0 < δ < 1 , then f 1 ( ϵ ) = O ( ϵ δ ) , h 1 ( ϵ ) = O ( ϵ δ ) . By Theorem 1.2, we get
ϵ 2 λ ( ϵ ) σ 2 = o ( ϵ δ ) , as  ϵ 0 .
Remark 2.2 If g ( x ) = | x | , δ = 1 , then 1 ϵ f 2 ( ϵ ) = O ( ϵ ) , f 1 ( ϵ ) = O ( ϵ ) , h 1 ( ϵ ) = O ( ϵ ) , h 2 ( ϵ ) = O ( ϵ ) . By (2.9), we get
ϵ 2 λ ( ϵ ) σ 2 = O ( ϵ ) , as  ϵ 0 .

Declarations

Acknowledgements

The authors are very grateful to the referees and editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper.

Authors’ Affiliations

(1)
Department of Mathematics, China Jiliang University

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Copyright

© He and Xie; licensee Springer. 2012

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