A note to the convergence rates in precise asymptotics

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R E S E A R C H

Open Access

A note to the convergence rates in precise

asymptotics

Jianjun He

*

*_{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

r,p(

) = _∞

n=1nr/p–2P(|Sn| ≥n1/p

). In this paper, the

author discusses the rate of approximation of_rp_–_pE|N|2(r–p)/(2–p)by

2(r–p)/(2–p)

_λ

r,p(

)

under suitable moment conditions, whereNis normal with zero mean and variance

σ

2_{> 0, which improves the results of Gut and Steinebach (J. Math. Anal. Appl.}

390:1-14, 2012) and extends the work He and Xie (Acta Math. Appl. Sin. 29:179-186, 2013). Specially, for the caser= 2 andp= 1

β+1,

β

> – 1

2, the author discusses the rate of

approximation of₂_βσ₊₁2 by

2

_λ

2,1/(β+1)(

) under the conditionEX2I(|X|>t) =O(t–δl(t))

for some

δ

> 0, wherel(t) is a slowly varying function at inﬁnity.

MSC: 60F15; 60G50

Keywords: convergence rate; precise asymptotics; slowly varying function

1 Introduction

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

n

k=Xkandλr,p() =

∞

n=nr/p–P(|Sn| ≥n/p). Heyde [] proved that

lim

→

_λ

,() =σ,

wheneverEX=  andEX=σ<∞. Klesov [] studied the rate of the approximation of σ byλ,() under the conditionE|X|<∞. He and Xie [] improved the results of Klesov []. Gut and Steinebach [] extended the results of Klesov [] and obtained the following TheoremA. Gut and Steinebach [] studied the general idea of proving precise asymptotics.

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

 <p< ,r≥.

() IfEX₌_σ_{> }_and_E_|_X_|q_<_∞_{for some}_r_<_q_≤_,_then

(r–p)/(–p)λr,p() –

p r–pE|N|

(r–p)/(–p)₌_op((qq––)(p)(–r–pp))_.

() IfEX₌_σ_{> }_and_E_|_X_|q_<_∞_{for some}_q_≥__with_q_>r–p

–p ,then

(r–p)/(–p)λr,p() –

p r–pE|N|

(r–p)/(–p)₌_o(–p)(pp(r+–pq)–pq)_,

whereNis normal with meanand varianceσ_{> .}

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The purpose of this paper is to strengthen TheoremAand extend the theorem of He and Xie [] under suitable moment conditions. In addition, we shall discuss the rate at which λ,/(β+)() converges to σ



β+under the conditionT(t) =O(t

–δ_l₍_t_{)) for some}_δ_{> , where}

T(t) =EX_I_(|_X_|_>_t_),_l₍_t_{) is a slowly varying function at inﬁnity. Throughout this paper,} C represents a positive constant, though its value may change from one appearance to the next, and [x] denotes the integer part ofx.(x) is the standard normal distribution function,(x) =√

π

x

–∞e–t

_/

dt,ϕ(x) =(x).

2 Main results

From Gut and Steinebach [], it is easy to obtain the following lemma.

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

with zero mean and varianceσ_{> .}_Set_{ <}_p_{< }_{and r}_≥_,_then

(r–p)/(–p)

∞

n=

n(r/p)–P|Sn| ≥n/p

– p

r–pE|N|

(r–p)/(–p)

= ⎧ ⎨ ⎩

O((r–p)/(–p)), r< p,

O(p/(–p)_), _r_≥__p_. (.)

Lemma .(Binghamet al.[]) Let l(t)be a slowly varying function.We have

()for anyη> ,

lim

t→∞t

η

l(t) =∞, lim

t→∞t

–η

l(t) = ;

()if <δ< ,then

t a

s–δl(s)ds∼ 

 –δt

–δ

l(t), t→ ∞;

()ifδ> ,then

∞

t

s–δl(s)ds∼–   –δt

–δ

l(t), t→ ∞;

()ifδ= ,then L(t) =_t∞l(_ss)ds,m(t) =_at l(_ss)ds are slowly varying functions;and

lim

t→∞

l(t)

L(t)= , tlim→∞

l(t)

m(t)= .

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

and <p< ,r≥.

()If EX₌_σ_{> }_{and E}_|_X_|_<_∞_{for some r}_{< ,}_then

(r–p)/(–p)λr,p() –

p r–pE|N|

(r–p)/(–p)₌ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O((r–p)/(–p)), ≤r<_p,

O(p/(–p)log), r=_p,

O(p/(–p)), _p<r< .

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()If EX₌_σ_{> }_{and E}_|_X_|+δ_<_∞_{for some}_{ <}_δ_{< ,}_r_{<  +}_δ_,_then

(r–p)/(–p)λr,p() –

p r–pE|N|

(r–p)/(–p)

= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O((r–p)/(–p)), ≤r< ( +δ/)p,

O(pδ/(–p)_log

), r= ( +δ/)p,

o(pδ/(–p)), ( +δ/)p<r<  +δ.

(.)

()If EX₌_σ_{> }_{and E}_|_X_|q_<_∞_{for some q}_≥__{with q}_>r–p

–p ,then

(r–p)/(–p)λr,p() –

p r–pE|N|

(r–p)/(–p)₌ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O((r–p)/(–p)), ≤r<_p,

O(p/(–p)_log

), r= p/,

O(p/(–p)), r> p/,

(.)

where N is normal with meanand varianceσ_{> .}

Remark . Clearly, Theorem  and Theorem  in He and Xie [] are special cases of

Theorem ., by takingr=  andp= .

Remark . If  <p< ,r≥, we havemin((_–r–_pp),_–pδ_p) >_(+p_δδ_–(r_p–_)(–p)_p₎ forr<  +δ=q≤ andmin((_–r–_pp),_–p_p) > _(–_p(₎₍r_p–_+p)_qp_–_pq₎ for someq≥ withq> _–r–_pp. So, the results of Theo-rem . are stronger than those of TheoTheo-rem A.

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

and let T(t) =O(t–δ_l₍_t₎₎_{for some}_δ_{> ,}_{where l}₍_t₎_{is a slowly varying function at inﬁnity}_.

Set EX=σ> andβ> –_. ()Ifδ> ,then

λ,/(β+)() –

σ β+ =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O(), –_<β< –_,

O(log), β= –_,

O(/(β+)), β> –_.

(.)

()If <δ< ,then

λ,/(β+)() –

σ β+ =

⎧ ⎨ ⎩

O(), –_<β< –_+δ_,

O(δ/(β+)_l₍–/(β+)_)), _β_≥_– +

δ

.

(.)

()Ifδ= ,then

λ,/(β+)() –

σ β+ =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O(+_–/(β+)l(_tt)dt), –_<β< –_,

O(_{( +}–



l(t)

t dt)log



), β= –

 ,

O(/(β+)( +_–(β+)/(β+)l(_tt)dt)), β> – .

(.)

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δ= . But the conditionT(t) =O(t–δ_{) is weaker than the condition}_E_|_X_|+δ_<_{∞. If}_l₍_t₎_→_

ast→ ∞, then the result of Theorem . is the same as that of Theorem . for  <δ< .

Remark . Forδ> , the conditionE|X|+δ_<_∞_{is neither suﬃcient nor necessary for}

the conditionT(t) =O(t–δ_l₍_t_{)). Here are some suitable examples.}

Example  LetXbe a random variable with densityf(x) =_|C_x_|(++δδ_lnln|x_|_x|)_|I(|x|>e), whereCis

a normalizing constant, and  <δ< , thenEX=  andT(t) = _tδC_ln_tI(t>e),l(t) = lnt is a

slowly varying function at inﬁnity. ButE|X|+δ₌_C

|x|>e

+δln|x|

|x|ln_|_x_|dx=∞.

Example  LetX be a random variable with density f(x) = C_|(δln|x|+|x|(ln|x|–))

x|δ+_ln_|_x_|_e|x|/ln|x| I(|x|>e), where  <δ < , then EX=  andT(t) = C

tδ_et/ln(t)I(t>e), h(t) = _et/lnt, E|X|+

δ _<_∞_{. But}

h(t) = _et/lnt is not a slowly varying function at inﬁnity.

In fact, we have the following result.

Theorem . Suppose X is a real random variable andδ> .Then E|X|+δ_<_∞_{if and}

only if tδ_T₍_t₎_→__and∞

t s

δ–_T₍_s₎_ds_→__{as t}_{→ ∞.}

Remark . Iftδ_T₍_t_{) is bounded as}_t_{→ ∞}_{for some}_δ_{> , then we have}_E_|_X_|+α_<_∞_for

everyα∈(,δ) from Theorem ..

Remark . LetXbe a random variable with zero mean. If there exist positive constants

CandCsuch thatCl(t)≤tδT(t)≤Cl(t) for suﬃciently largetand someδ> , where

l(t) is a slowly varying function at inﬁnity, then from Lemma .() and Theorem ., we have

E|X|+δ<∞ ⇔ ∞

t

l(s)

s ds→ ast→ ∞.

3 Proofs of the main results

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

P|Sn| ≥n/p

=  –n(–p)/p+Rn, (.)

where

Rn=P

Sn≤–n/p

––n/p–/+n/p–/–PSn≤n/p

.

From (.), we have

(r–p)/(–p)λr,p() –

p r–pE|N|

(r–p)/(–p)

= (r–p)/(–p)

∞

n=

nr/p– –n(–p)/p– p

r–pE|N|

(r–p)/(–p)

+(r–p)/(–p)

∞

n=

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By Lemma ., in order to prove Theorem ., we only need to estimate (r–p)/(–p)_× ∞

n=nr/p–Rn.

() On account of a non-uniform estimate of the central limit theorem by Nagaev [], for everyx∈R,

P

Sn

√

n<x

–(x)≤ CE|X|  √

n( +|x|). (.)

By (.),|Rn| ≤ CE|X|



√

n(+n(–p)/p₎.

(a) Ifr< p/, then

(r–p)/(–p)

∞

n=

nr/p–Rn≤C(r–p)/(–p)

∞

n=

nr/p–/=O(r–p)/(–p). (.)

(b) If p/ <r< , then

(r–p)/(–p)

∞

n=

nr/p–Rn

≤C(r–p)/(–p)

∞

n=

nr/p– √

n( +n(–p)/p₎

≤C(r–p)/(–p)

_[–p/(–p)_]

n=

nr/p– √

n +

– ∞

n=[–p/(–p)_]+

nr/p–/–(–p)/p

=Op/(–p). (.)

(c) Ifr= p/, then

(r–p)/(–p)

∞

n=

nr/p–Rn≤Cp/(–p)

_[–p/(–p)_]

n= 

n+

– ∞

n=[–p/(–p)]+

n––(–p)/p

=O

p/(–p)log

. (.)

From (.), (.), (.), (.) and (.), we obtain (.). This completes the proof of part (). () By the inequality in Osipov and Petrov [], there exists a bounded and decreasing functionψ(u) on the interval (,∞) such thatlimu→∞ψ(u) =  and

P

 √

nσSn<x

–(x)≤ψ( √

n( +|x|)) nδ/_{( +}_|_x_|₎+δ.

Letx=n(–p)/p, we have|Rn| ≤ψ(

√

n(+n(–p)/p))

nδ/_(+_n(–p)/p₎+δ, so that:

(a) If  <r< ( +δ/)p, then

(r–p)/(–p)

∞

n=

nr/p–Rn≤(r–p)/(–p)

∞

n=

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(b) If ( +δ/)p<r<  +δ, then by noticing thatlimu→∞ψ(u) =  for anyη> , there exists a natural numberNsuch thatψ(√n) <ηwhenevern>N. We conclude that

(r–p)/(–p)

∞

n=

nr/p–Rn

≤C(r–p)/(–p)

∞

n=

nr/p–ψ(√n( +n(–p)/p))

nδ/_{( +}_n(–p)/p₎+δ

≤C(r–p)/(–p) _N



n=

nr/p––δ/ψ(√n) +η

[–p/(–p)]

n=N+

nr/p––δ/

+C(r–p)/(–p)––δψ–p/(–p)

∞

n=[–p/(–p)_]+

nr/p––δ/–(/p–/)(+δ)

≤(r–p)/(–p)N_r/p––δ/+Cηpδ/(–p)+Cψ–p/(–p)pδ/(–p)

=opδ/(–p). (.)

(c) Ifr= ( +δ/)p, then

(r–p)/(–p)

∞

n=

nr/p–Rn=O

pδ/(–p)log

. (.)

By (.) and combining with (.), (.), (.) and (.), we obtain (.), which completes the proof of part ().

() We make use of the following large deviation estimate in Petrov []:

P

 √

nσSn<x

–(x)≤√ C

n( +|x|)q, x> .

So,|Rn| ≤√_n_(+_nC(–p)/p₎q. Hence we have the following.

(a) Ifr< p/, then

(r–p)/(–p)

∞

n=

nr/p–Rn≤(r–p)/(–p)

∞

n=

nr/p–/=O(r–p)/(–p). (.)

(b) Ifr> p/, then_pr–_–q_–_ppq< –. By noting thatq>r_––_pp, we obtain

(r–p)/(–p)

∞

n=

nr/p–Rn≤C(r–p)/(–p)

∞

n=

nr/p– ( +n(–p)/p₎q√_n

≤C(r–p)/(–p)

[–p/(–p)]

n=

nr/p––/

+C(r–p)/(–p)–q

∞

n=[–p/(–p)_]+

nr/p––/–(–p)q/p

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(c) Ifr= p/, then

(r–p)/(–p)

∞

n=

nr/p–Rn≤Cp/(–p)

[–p/(–p)]

n= 

n+C

p/(–p)–q

∞

n=[–p/(–p)_]+

n––(–p)q/p

=O

p/(–p)log

. (.)

By (.), from (.), (.), (.) and (.), we have (.), which completes the proof of

part ().

Proof of Theorem. We write

λ,/(β+)() –  β+ 

=

 √

π

∞

n=

nβ

_∞

nβ+/e

–t/_dt_–  β+ 

+

_[–/(β+)_]

n= +

∞

n=[–/(β+)_]+

nβ

P|Sn| ≥nβ+

–√

π ∞

nβ+/e

–t/_dt

=:I+I+I. (.)

First, according to Lemma ., we have

I= ⎧ ⎨ ⎩

O(), –_<β<_,

O(/(β+)), β≥_. (.)

ForI, applying Lemma . of Xie and He [], and lettingx= y=nβ+, we obtain

P|Sn| ≥nβ+

≤nP

|X| ≥ 

n

β+

+ e–n–β–. (.)

Observing the following fact

 √

π ∞

nβ+/e

–t/_dt_{= }_{ –}_nβ+≤ϕ(n

β+/₎

nβ+/ =O

–n–β–/, (.)

from (.) and (.), we have

I≤ ∞

n=[–/(β+)_]+

nβP|Sn| ≥nβ+

+

∞

n=[–/(β+)_]+

nβ

√ π

_∞

nβ+/e

–t/_dt

≤

∞

n=[–/(β+)]+

nβ+P

|X|>n

β+ 

+C–

∞

n=[–/(β+)]+

n–β–

+C–

∞

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≤

∞

n=[–/(β+)_]+ nβ+

|x|≥_nβ+ dF(x) +O

+O

≤

∞

n=[–/(β+)]+

nβ+ ∞

k=n



kβ+≤x<(k+)β+

dF(x) +O

≤

∞

k=[–/(β+)_]+

k

n=

nβ+



kβ+≤x<(k+)β+

dF(x) +O

≤C

∞

k=[–/(β+)_]+ kβ+



kβ+≤x<(k+)β+

dF(x) +O

≤C

∞

k=[–/(β+)]+



kβ+≤x<(k+)β+

xdF(x) +O

≤C

x≥

–(β+)/(β+)

xdF(x) +O

=CT–(β+)/(β+)+O.

Using the assumption onT(t) and Lemma .(), we can obtain

I= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O(), –_<β≤min_(δ,)–_,

O(/(β+)_), _β_≥_– ,δ≥,

O(δ/(β+)_l₍–/(β+)_)), _β_≥_– +

δ

,  <δ< .

(.)

ForI, by Bikelis’s inequality (see []), we have

I≤

[–/(β+) ]

n=

Cnβ

( +nβ+/₎√_n

(+nβ+/)√n



T(v)dv

≤

[–/(β+) ]

n=

nβ–/

(+nβ+/)√n



T(v)dv

+–

[–/(β+) ]

n=[–/(β+)_]+ n–β–

(+nβ+/)√n



T(v)dv.

Now, the proof of Theorem . will be divided into the following cases. Case  ofδ> .

Noting thatT(t)≤EX_{= , let}_δ

be a real number such that  <δ<δ, by Lemma .(),

limt→∞tδ–δ_l₍_t_{) = . Therefore, there is a real number}_T

>  such that|_tlδ–δ(t)_|<  whenever t>T. Then

∞



T(t)dt≤





T(t)dt+ ∞



T(t)dt≤C+ ∞

T



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We have

I≤C

[–/(β+) ]

n=

nβ–/+C–

[–/(β+) ]+

n=[–/(β+)_]+ n–β–

= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O(), –_<β≤–_,

O(_log

), β= –

 ,

O(/(β+)_), _β_{> –} .

(.)

From (.), (.), (.) and (.), we obtain (.). Case  of  <δ< .

(a) If –_<β< –_+δ

, then ∞

n=nβ–/<∞and ∞

 t

β+–δ_l₍_t₎_dt_<_∞_{. Making use of}

Lemma .()-(), we have

I≤C

[–/(β+) ]

n=

nβ–/

 +

√n



T(t)dt

+C–

[–/(β+) ]

n=[–/(β+)_]+ n–β–

 +

nβ+



T(t)dt

≤C

[–/(β+) ]

n=

nβ–/(√n)–δl(√n) +O

+C/(β+)+C–

[–/(β+) ]

n=[–/(β+)_]+

n–β–nβ+–δlnβ+

≤C

–/(β+)



xβ–/₍√_x₎–δ_l₍√_x₎_dx

+C–δ ∞

–/(β+)x

–β–_l__xβ+_x(β+)(–δ)_dx₊_O

≤C

–/(β+)



tβ+–δ_l₍_t₎_dt₊_C

∞

–/(β+) l(t)

t+δdt+O



≤C+Cδ/(β+)l–/(β+)

=O. (.)

(b) Ifβ≥– +

δ

, then we have

I≤C–(β+)/(β+)

 +

–/(β+)



T(t)dt

+C/(β+)+Cδ/(β+)l–/(β+)

≤C/(β+)( +–(–δ)/(β+)l–/(β+)+Cδ/(β+)l–/(β+)

(10)

Therefore

I= ⎧ ⎨ ⎩

O(), –_<β≤–_+δ

,

O(δ/(β+)_l₍/(β+)_)), _β_≥_– +

δ

.

(.)

Combining the estimate with (.) and (.), by (.), this implies that (.) follows. Case  ofδ= .

(a) If –_<β< –_, then∞_n_=nβ–__<_{∞. We have}

I≤C

 +

–/(β+)



T(t)dt

[–/(β+) ]

n=

nβ–/

+C–  + – β+ β+ 

T(t)dt

[–/(β+) ]

n=[–/(β+)_]+ n–β–

≤C

 +

–/(β+)



l(t)

t dt

+/(β+)  + – β+ β+ 

l(t)

t dt

≤C

 +

–/(β+)



l(t)

t dt

. (.)

(b) Ifβ> –_, then we have

I≤C

– β+(β+/)

 +

–/(β+)



l(t)

t dt

+C/(β+)

 +

–(β+)/(β+)



l(t)

t dt

≤C/(β+)

 +

–/(β+)



l(t)

t dt

+/(β+)

 +

–(β+)/(β+)



l(t)

t dt

≤C/(β+)

 +

–(β+)/(β+)



l(t)

t dt

. (.)

(c) Ifβ= –

, then we have

I≤Clog   + – 

l(t)

t dt

+C

 +

–



l(t)

t dt

≤Clog

 + –



l(t)

t dt

so that

I= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

O(_{( +}–/(β+) 

l(t)

t dt)), –



<β≤–  ,

O(( +_–l(_tt)dt)log

), β= –

 ,

O(/(β+)_{( +}–(β+)/(β+) 

l(t)

t dt)), β> –

 .

(.)

Combining the estimate with (.) and (.), by (.), this implies that (.) follows, and

(11)

Proof of Theorem. SetT(t) =E|X|+δI(|X|>t). First, note that

E|X|+δ_I_|_X_|_>_t₌

|x|>t

|x|+δ_dF₍_x₎

=

|x|>t

x

|x| t

δyδ–dy

dF(x) +tδ

|x|>t

xdF(x)

= ∞

t

δyδ–

|x|>y

xdF(x)

dy+tδ_T₍_t₎

=δ ∞

t

sδ–T(s)ds+tδT(t).

We have

T(t) =δ ∞

t

sδ–T(s)ds+tδT(t).

Since_t∞sδ–_T₍_s₎_ds_≥_,_tδ_T₍_t₎_≥_{, we have}

T(t)→ ⇔ tδT(t)→ and ∞

t

sδ–T(s)ds→ ast→ ∞.

Next, it is easy to get

E|X|+δ_<_{∞ ⇔} _T

(t)→ ast→ ∞.

From the above facts, the proof of Theorem . is complete.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank the referee for helpful comments.

Received: 3 February 2013 Accepted: 18 July 2013 Published: 11 August 2013 References

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doi:10.1186/1029-242X-2013-378

Cite this article as:He:A note to the convergence rates in precise asymptotics.Journal of Inequalities and Applications