A supplement to the convergence rate in a theorem of Heyde

R E S E A R C H

Open Access

A supplement to the convergence rate in a

theorem of Heyde

Jianjun He

*

_{and Tingfan Xie}

*_{Correspondence: [email protected]}

Department of Mathematics, China Jiliang University, Hangzhou, 310018, China

Abstract

Let{X,Xn,n≥1}be a sequence of i.i.d. random variables with zero mean, set

Sn=

n

k=1Xk,EX2=

σ

2> 0, and

λ

(

) =

_∞

n=1P(|Sn| ≥n

). In this paper, the authors

discuss the rate of approximation of

σ

2_by

2

_λ

₍

_{) 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 rate; i.i.d. random variable; theorem of Heyde

1 Introduction and main results

Let{X,Xn,n≥}be a sequence of i.i.d. random variables, setSn=

n

k=Xk, andλ() =

∞

n=P(|Sn| ≥n). Heyde [] proved that

lim →

_{λ() =σ}_,

wheneverEX=σ<∞andEX= .

There are various extensions of this result: Chen [], Gut and Spˇatara [], Lanzinger and Stadtmüller []. Liu and Lin [] introduced a new kind of complete moment convergence; Klesov [] studied the rate of approximation ofσ byλ() and proved the following Theorem A.

Theorem A Let{X,Xn,n≥}be a sequence of i.i.d. random variables with zero mean, if

EX_=σ_{> , and E|X|}_{<∞, then}

λ() –σ=o/, as→.

Recently, He and Xie [] obtained Theorem B which improved Theorem A. Gut and Steinebach [] extended the results of Klesov [].

Theorem B Let{X,Xn,n≥}be a sequence of i.i.d. random variables, and <δ≤, if

EX= , EX=σ>  and E|X|+δ<∞,

then

λ() –σ= ⎧ ⎨ ⎩

O(), δ= ,

o(δ_{),  <δ< .}

LetGbe the set of functionsg(x) that are defined for all realxand satisfy the following conditions: (a)g(x) is nonnegative, even, nondecreasing in the intervalx> , andg(x)=  forx= ; (b)_g(x_x) is nondecreasing in the intervalx> .

Let G be the set of functions g(x)∈G satisfying the supplementary condition (c) limx→∞g(x

₎

xg(x)= . Obviously, the functiong(x) =|x|

δ_{with  <δ<  belongs toG}

and does

not belong toGifδ= . The purpose of this paper is to generalize Theorem B to the case

where the conditionE|X|+δ_{<∞is replaced by a more general conditionE|X|}_{g(X) <∞}

in which the functiong belongs to some subset ofG. DenoteTg(v) =EXg(X)I(|X|>v),

Tg(v) is a nonnegative nonincreasing function in the intervalv> , andlimv→∞Tg(v) = 

withEXg(X) <∞. Now we state our results as follows.

Theorem . Let{X,Xn;n≥}be a sequence of i.i.d. random variables with zero mean

and EX_=σ_{> , if EX}_{g(X) <∞for some function g(x)∈G, and}

∞

n=



ng(√n)<∞, (.)

then

λ() –σ=O/+o()h() +f()

, as→, (.)

where f() =

∞

n=[

]+



ng(√n), h() = [



] n= g(√n).

Theorem . Under the conditions of Theorem ., and g(x)∈G, then

λ() –σ=o()h() +f()

, as→. (.)

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 ofx.

2 Proofs of the main results

Before we prove the main results we state some lemmas. Lemma . is from [].(x) is the standard normal distribution function,(x) = √

π x –∞e–t

_/

dt.

Lemma . Let{X,Xn,n≥}be a sequence of i.i.d. standard normal distribution random

variables. Then

λ() =

∞

n=

 √

π

∞

√n

e–t/dt=  –



 +O

If{Xn,n≥}is a sequence of independent random variables with zero mean and finite

variance, and put EX

j =σj, Bn=nj=σj, Bikelis [] obtained the following inequality:

P

 √

Bn n

j=

Xj<x

–(x)

≤C

B–_n  +|x|–

n

j=

|u|>(+|x|)B/n

udVj(u)

+B–/_n  +|x|–

n

j=

|u|≤(+|x|)B/n

|u|dVj(u)

,

for every x, where Vj(x) =P(Xj<x)is the distribution function of the random variable Xj.

By applying the above inequality to the sequence of i.i.d. random variables with zero mean and variance , and letting|x|=√n, we have the following lemma.

Lemma . Let{X,Xn,n≥}be a sequence of i.i.d. random variables with zero mean and

EX= . Then for any given> , we have

P|Sn|>n

–√

π

_∞

√n

e–t/dt

≤C( +√n)–

|u|>(+√n)√n

udV(u)

+Cn–/( +√n)–

|u|≤(+√n)√n

|u|dV(u),

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

Proof of Theorem . Without loss of generality, we suppose thatσ_{= ,  << , and}

write

λ() =I+

∞

n=

 √

π

∞

√n

e–t/dt,

where

I=

∞

n=

P|Sn|>n

–√

π

∞

√n

e–t/dt

.

Applying Lemma ., we obtain

λ() =I+  –



 +O

,

then

λ() –  = –



 +

 ∞

n=

Rn+O

hereRn=P(|Sn|>n) – √_π ∞√ne–t _/

dt. By Lemma .,

|Rn| ≤Rn+Rn,

where

Rn=C( +

√

n)–

|u|>(+√n)√n

udV(u),

Rn=Cn–/( +

√

n)–

|u|≤(+√n)√n

|u|dV(u).

We obtain

λ() –  =

∞

n=

Rn+ ∞

n=

Rn+O

. (.)

Firstly, we estimate∞_n=Rn. Note that



∞

n=

Rn= [

]

n=

Rn+ ∞

n=[

]+

Rn=:T+T.

Applying the conditionEXg(X) <∞, we have

lim n→∞

|u|>√n

ug(u)dV(u) = .

Therefore, for anyη> , there is an integerNsuch that _|u|>√_nug(u)dV(u)≤η,

when-evern>N. Hence

T≤C N

n=

|u|>√n

udV(u) +C

[

]

n=N+

( +√n)–

|u|>(+√n)√n

udV(u)

≤CN+Cη [

]

n=N+



( +√n)_g(√_{n( +}√_n))

≤C

N+η [

]

n=



g(√n)

=Ch()

N

[

] n= g(√n)

+η

≤Ch()(N+η)

=oh()

whereh() =

[

]

n= g(√n). ForT, noting thatg(x)∈G, we have the following inequality:

T≤C ∞

n=[

]+ 

n

|u|>√n(+√n)

udV(u)

≤C

∞

n=[

]+



ng(√n( +√n))

|u|>√n(+√n)

ug(u)dV(u)

≤C

∞

n=[

]+ 

ng(√n)

|u|>

ug(u)dV(u)

≤CTg



f(). (.)

Next, we estimate the second term of (.). Note that



∞

n=

Rn =C ∞

n=

n–/( +√n)–

|u|≤(√n(+√n))/|u| _dV(u)

+C

∞

n=

n–/( +√n)–

(√n(+√n))/_<|u|<√_n(+√_n)|

u|dV(u)

=:J+J.

ForJ, we can write

J =C

[

]

n=

+

∞

n=[

]+

n–/( +√n)–

|u|≤(√n(+√n))/|u| _dV(u)

=:J+J.

Noting that_g(x_x)is nondecreasing in the intervalx> , we have

J=C [

]

n=

 √

n( +√n)

|u|≤(√n(+√n))/|u| _dV(u)

≤C

[

]

n=



n/_{( +}√_n)/_g((√_{n( +}√_n))/₎

|u|≤(√n(+√n))/u

_g(u)dV(u)

≤C

[

]

n=



n/_g(n/₎

=Ch(), (.)

whereh() =

[

]

Similarly, we can obtain

J=C ∞

n=[

]+

 √

n( +√n)

|u|≤(√n(+√n))/|

u|dV(u)

≤C

∞

n=[

]+



n/_{( +}√_n)/_g((√_{n( +}√_n))/₎

|u|≤(√n(+√n))/u

_g(u)dV(u)

≤C

∞

n=[

]+  /_n/_g(n/₎

=C√

f(), (.)

wheref() =∞_n=[

]+

 n/_g(n/₎.

ForJ, we write

J =C

[

]

n=

+

∞

n=[

]+

n–/( +√n)–

(√n(+√n))/_<|u|<√_n(+√_n)|u| _dV(u)

=:J+J.

Using the properties ofg(x) by simple calculation, it follows that

J=C [

]

n=

n–/( +√n)–

(√n(+√n))/_<|u|<√_n(+√_n)|u| _dV(u)

≤C

_N

n=

+

[

]

n=N+



( +√n)_g(√_{n( +}√_n))

×

(√n(+√n))/_<|u|<√_n(+√_n)u

_g(u)dV(u)

≤C

_N

n=

+

[

]

n=N+



g(√n)

|u|>n/u

_g(u)dV(u)

≤C

N+η [

]

n=



g(√n)

=oh()

, (.)

and

J≤C ∞

n=[

]+

n–_{( +}√_n)–

(√n(+√n))/_<|u|<√_n(+√_n)|u| _dV(u)

≤C

∞

n=[ 

]+ 

ng(√n)

(√n(+√n))/_<|u|<√_n(+√_n)u

≤CTg

 √

∞

n=[

]+ 

ng(√n)

≤CTg

 √

f(). (.)

From (.) to (.), we conclude that

λ() – ≤C√

f() +CTg

 √

f() +o()h() +Ch(). (.)

Since

 √

f()≤

C

√

∞

n=[

]+



n/ ≤C

√ ,

and

h() = [

]

n=





√

ng(√_n)≤C 

[

]

n=





√

n≤C

√ ,

by (.), we have

λ() –  =O/+o()f() +h()

.

This completes the proof of Theorem ..

Proof of Theorem . By the conditionsg(x)∈G, andlimx→∞g(x ₎

xg(x)= , for anyη> , there

is an integerNsuch that g( √

n)  √

ng(√_n)≤η, whenevern>N. We have

h()≤ N

n=





√

ng(√_n)+ 

[

]

n=N

η

g(√n)

≤CN+ [

]

n=N+

η

g(√n)

≤CN+ [ 

]

n=

η

g(√n)

=o()h(), (.)

and

 √

f()≤  √

∞

n=[

]+ η

n/_g(√_n)

≤

∞

n=[

]+ η

By (.)-(.), note thatTg(√) =o(), as→, we have

λ() –σ=o()h() +f()

, as→.

This completes the proof of Theorem ..

Remark . Ifg(x) =|x|δ_{,  <δ< , thenf}

() =O(δ),h() =O(δ). By Theorem ., we

get

λ() –σ=oδ, as→.

Remark . Ifg(x) =|x|,δ= , then√

f() =O(),f() =O(),h() =O(),h() =O().

By (.), we get

λ() –σ=O(), as→.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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.

Received: 27 May 2012 Accepted: 20 August 2012 Published: 4 September 2012

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