**R E S E A R C H**

**Open Access**

## Density expansions of extremes from

## general error distribution with applications

### Chunqiao Li and Tingting Li

**_{Correspondence:}

[email protected] School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

**Abstract**

In this paper the higher-order expansions of density of normalized maximum with parent following general error distribution are established. The main results are applied to derive the higher-order expansions of the moments of extremes.

**MSC:** Primary 60G70; secondary 60F05

**Keywords:** density expansion; general error distribution; maximum; moment
expansion

**1 Introduction**

In extreme value theory, the quality of convergence of normalized partial maximum of a
sample has been studied in recent literature. For the convergence rate of distribution of
normalized maximum, we refer to Smith [], Leadbetter*et al.*[], de Haan and Resnick []
for general cases, and speciﬁc cases were studied by Hall [], Nair [], Peng*et al.*[] and Jia
and Li []. Nair [] derived the higher-order expansions of moments of normalized
maxi-mum with parent following normal distribution. Liao*et al.*[] and Jia*et al.*[] extended
Nair’s results to skew-normal distribution and general error distribution, respectively.

The main objective of this paper is to derive the higher-order expansions of density of
normalized maximum with parent following the general error distribution. To the best of
our knowledge, there are few studies on the rate of convergence of density of normalized
maximum except the work of de Haan and Resnick [] for local limit theorems and Omey
[] for rates of convergence of densities with regular variation with remainders excluding
the case we will study in this paper,*i.e., the general error distribution.*

Let{X*n*,*n*≥}be a sequence of independent and identically distributed (i.i.d.) random

variables with marginal cumulative distribution function (cdf )*Fv*following the general

error distribution (F*v*∼GED(v) for short), and let*Mn*=max≤*k*≤*nXk* denote its partial

maximum. The probability density function (pdf ) of theGED(v) is given by

*fv*(x) =

*v*exp(–(/)|x/*λ*|*v*_{)}

*λ*+/*v _{}*

_{(/v)},

*x*∈R,

where*v*> is the shape parameter,*λ*= [–/*v _{}*

_{(/v)/}

_{}_{(/v)]}/

_{and}

_{}_{(}

_{·}

_{) denotes the gamma}

function (Nelson []). Note that theGED() reduces to the standard normal distribution.

For theGED(v), the limiting distribution of maximum*Mn* and its associated

higher-order expansions are given by Peng*et al.*[] and Jia and Li []. Peng*et al.*[] showed
that

lim

*n*→∞P(M*n*≤*anx*+*bn*) =(x) =exp

–exp(–x), *x*∈R (.)

provided the norming constants*an*and*bn*satisfy the following equations:

–*Fv*(b*n*) =*n*–, *an*=*f*(b*n*), (.)

where*f*(x) = v–* _{λ}v_{x}*–

*v*

_{. In the sequel, let}

*gn*(x) =*nanFvn*–(a*nx*+*bn*)f*v*(a*nx*+*bn*) (.)

denote the density of normalized maximum, and

*n*

*gn*,;*x*

=*gn*(x) –(x) (.)

with(x) =*e*–*x*(x). By Proposition . in Resnick [],*n*(g*n*,)→ as*n*→ ∞. For

both applications and theoretical analysis, it is of interest to know the convergence rate of (.). This paper focuses on this topic and applies the main results to derive the high-order expansions of moments of extremes.

The paper is organized as follows. Section provides the main results and all proofs are deferred to Section . Auxiliary lemmas with proofs are given in Section .

**2 Main results**

In this section, we present the asymptotic expansions of density for the normalized max-imum formed by theGED(v) random variables and its applications to the higher-order expansions of moments of extremes.

**Theorem .** *Let Fv*(x)*denote the cdf of*GED(v)*with v*> ,*then for v*= ,*with norming*

*constants anand bngiven by*(.),*we have*

*bv _{n}bv_{n}n*

*gn*,;*x*

–*kv*(x)(x)

→*ωv*(x)(x) (.)

*as n*→ ∞,*where kv*(x)*andωv*(x)*are respectively given by*

*kv*(x) =*kv*(x) +*kv*(x), (.)

*ωv*(x) =

–*v*–*λ**vωv*(x) +*ωv*(x)e–*x*+*ωv*(x)e–*x*

(.)

*with*

*kv*(x) = –

–*v*–*λvx*+ x,

*kv*(x) =

–*v*–*λv* + x+*x*+ x*e*–*x*,

*ωv*(x) = – –

–*v*–*x*–

– v–*x*+

*ωv*(x) = –

*vx*–

*vx*

_{–}

– v–*x*–

–*v*–*x*,

*ωv*(x) =

–*v*–*x*+ x.

**Remark .** If we choose the norming constants*an*and*bn*such that

–*Fv*(b*n*)

*fv*(b*n*) ∼

*λv*

*v* *b*
–*v*

*n* , *an*=*f*(b*n*) (.)

with*v*= , then

*bv _{n}bv_{n}n*

*gn*,;*x*

–*k*¯*v*(x)(x)

→ ¯*ωv*(x)(x) (.)

as*n*→ ∞, where*k*¯*v*(x) and*ω*¯*v*(x) are respectively given by

¯
*kv*(x) =

–*v*–*λv*–x+ + x+*x**e*–*x* (.)

and

¯

*ωv*(x) =*λ**v*

–*v*–*ω*¯*v*(x) +*ω*¯*v*(x)e–*x*+*ω*¯*v*(x)e–*x*

(.)

with

¯

*ωv*(x) =

(v–_{– )}

*x*

_{+} –*v*–
*x*

_{,}

¯

*ωv*(x) =

*v*–– + *v*–– *x*+ v–– *x*+

v–– *x*+

*v*–– *x*,

¯

*ωv*(x) =

–*v*– +*x*+*x*

.

For the case of*v*= , we have the following results.

**Theorem .** *For v*= ,*with norming constants an*= –/*and bn*= –/log(n/),*we have*

*nnn*

*gn*,;*x*

–*k(x)*(x)→

*ω*(x) +*k*
(x)

(x) (.)

*as n*→ ∞,*where k(x)andω*(x)*are respectively given by*

*k*(x) = –
*e*

–*x*_{,} _{ω}

(x) = –
*e*

–*x*_{.} _{(.)}

To end this section, we apply the higher-order expansions of densities to derive the
asymptotic expansions of the moments of extremes. Methods used here are diﬀerent from
those in Nair [] and Jia*et al.*[].

In the sequel, for nonnegative integers*r, let*

*mr*(n) =

*x*∈R*x*

*r _{g}*

*n*(x)*dx,* *mr*=

*x*∈R*x*

*r _{}*

_{(x)}

_{dx}**Theorem .** *Let*{X*n*,*n*≥}*be an iid sequence with marginal distribution Fv*∼GED(v),

*then*

(i) *forv*= ,*with norming constantsanandbngiven by*(.),*we have*

*bv*
*n*

*bv*
*n*

*mr*(n) –*mr*

+ –*v*–*λvr(mr*++ m*r*)

→r*λ**v* –*v*– –*v*–(r+ ) + *mr*+

–*v*–(r+ ) + *mr*+

+

–*v*–(r– ) +

–*v*–*mr*+

(.)

*asn*→ ∞;

(ii) *forv*= ,*with norming constantsan*= –/*andbn*= –/log(n/),*we have*

*n* *nmr*(n) –*mr*

+ (–)*rr*

(*r*–)_{()}

→(–)*r*– *r*

(*r*–)() – (*r*–)() (.)

*asn*→ ∞,*where*(*r*–)_{(t)}_{denote the}_{(r}_{– )}_{th derivative of the gamma function at}*x*=*t*.

**3 Auxiliary lemmas**

In this section we provide auxiliary lemmas which are needed to prove the main results.

**Lemma .** *Let Fv*(x)*and fv*(x)*respectively denote the cdf and pdf of*GED(v)*with v*= ,*for*

*large x,we have*

–*Fv*(x)

=*fv*(x)

*λv*

*v* *x*

–*v*_{ + }* _{v}*–

_{– }

*–*

_{λ}v_{x}*v*

_{+ }

*–*

_{v}_{– }

*–*

_{v}_{– }

**

_{λ}*v*–

_{x}*v*

_{+}

*–*

_{O}_{x}*v*

_{.}

_{(.)}

*Furthermore,with the norming constants anand bngiven by*(.),*we have*

(i) *forv* = ,

*bv _{n}bv_{n}F_{v}n*(a

*nx*+

*bn*) –(x)

–*k*˜*v*(x)(x)

→

˜

*ωv*(x) +

˜
*k*

*v*(x)

(x) (.)

*asn*→ ∞,*wherek*˜*v*(x)*andω*˜*v*(x)*are respectively given by*

˜
*kv*(x) =

–*v*–*λvx*+ x*e*–*x*, (.)

˜

*ωv*(x) =

*v*–– *λ**v*

x+ x+

–*v*–*x*+

–*v*–*x*

*e*–*x*; (.)

(ii) *forv*= ,*with norming constantsan*= –/*andbn*= –/log(n/),*we have*

*nnF*_{}*n*(a*nx*+*bn*) –(x)

–*k*(x)(x)

→

*ω*(x) +

*k*
(x)

(x) (.)

*Proof* See Lemma and Theorem in Jia and Li [].

**Lemma .** *Let Fv*(x)*denote the cdf of*GED(v)*with v*> ,*then*

(i) *forv* = ,*with norming constants given by*(.),*we have*

*F _{v}n*–(a

*nx*+

*bn*) =

+*k*˜*v*(x)b–*nv*+

˜

*ωv*(x) +

˜
*k*

*v*(x)

*b*–_{n}*v* +*o()*(x) (.)

*asn*→ ∞,*wherek*˜*v*(x)*andω*˜*v*(x)*are respectively given by*(.)*and*(.);

(ii) *forv*= ,*with norming constantsan*= –/*andbn*= –/log(n/),*we have*

*F*_{}*n*–(a*nx*+*bn*) =

+
*nk*(x) +

*n*

*ω*(x) +

*k*
(x)

+*o()*(x) (.)

*asn*→ ∞,*wherek(x)andω*(x)*are given by*(.).

*Proof* (i) It follows from (.) and (.) that

*F _{v}n*(a

*nx*+

*bn*) =

*b*–

*nv*

*b*–_{n}v

˜

*ωv*(x) +

˜
*k*

*v*(x)

+*o()*+*k*˜*v*(x)

(x) +(x)

=

+*k*˜*v*(x)b–*nv*+

˜

*ωv*(x) +

˜
*k*

*v*(x)

*b*–* _{n}v* +

*o()*(x). (.)

Noting that

*F _{v}n*(a

*nx*+

*bn*)→(x) =exp

–exp(–x),

by taking logarithms, we have

*n* –*Fv*(a*nx*+*bn*)

→*e*–*x*.

Thus

–*Fv*(a*nx*+*bn*)∼*n*–*e*–*x*=*o*

*b*–_{n}*v*

since*bn*∼/*vλ*(log*n)*/*v*, which implies

*Fv*(a*nx*+*bn*)

= + –*Fv*(a*nx*+*bn*)

+*o()*= +*ob*–* _{n}v*. (.)

The desired result (.) follows by (.) and (.). The proof of (ii) is similar and details

are omitted here.

**Lemma .** *Let fv*(x)*denote the pdf of*GED(v)*with v* = ,*then*

*fv*(x) =

–*Fv*(x)

*v*

*λvx*

*for large x,and*

*Cn*(x) :=

*anfv*(a*nx*+*bn*)

–*Fv*(a*nx*+*bn*)

= + –*v*–*λv*(x+ )b* _{n}*–

*v*+

*λ*

*v* –

*v*–– –

*x*

*v*+

– v–
*x*

*b*–_{n}*v*

+*Ob*–* _{n}v* (.)

*as n*→ ∞.

*Proof* The desired results follow directly by (.).

**Lemma .** *Let*

*Hv*(b*n*;*x) =*

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

*ex*–

*and the norming constants anand bnbe given by*(.),*then*

*bv _{n}bv_{n}Hv*(b

*n*;

*x) –kv*(x)

→*ω _{v}*◦(x) (.)

*as n*→ ∞,*where kv*(x)*is given by*(.)*andω*◦*v*(x)*is given by*

*ω*◦* _{v}*(x) = –

*v*–

*λ*

*v*

x+ x+

–*v*–*x*+

–*v*–*x*

.

*Proof* Let

*Bn*(x) =

+ (*v*–_{– )}_{λ}v_{(}_{a}

*nx*+*bn*)–*v*+ (*v*–– )(*v*–– )*λ**v*(*anx*+*bn*)–*v*+*O*((*anx*+*bn*)–*v*)

+ (*v*–_{– )}_{λ}v_{(}_{b}

*n*)–*v*+ (*v*–– )(*v*–– )*λ**v*(*bn*)–*v*+*O*((*bn*)–*v*)

,

it is easy to check thatlim*n*→∞*Bn*(x) = and

*Bn*(x) – =

+*o()*–*v*–– *λ**vb _{n}*–

*vx*+

*Ob*–

_{n}*v*.

Hence,

lim
*n*→∞*b*

*v*
*n*

*Bn*(x) –

= (.)

and

lim
*n*→∞*b*

*v*
*n*

*Bv*(x) –

= –*v*–– *λ**vx.* (.)

By (.), we have

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

*ex*

=*Bn*(x)exp –

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt*

=*Bn*(x)
–
*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–
*λv* –

*dt*
+
*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt*

_{}

+*o()*. (.)

Combining (.)-(.) together, we have

lim
*n*→∞*b*

*v*

*nHv*(b*n*;*x)*

= lim
*n*→∞*b*

*v*
*n*

*Bn*(x)

–

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt* +*o()*–

= lim
*n*→∞

*bv _{n}Bn*(x) –

–*bv _{n}Bn*(x)

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–
*λv* –

*dt*

= –lim
*n*→∞

*x*

*bv _{n}*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt*

= – –*v*–*λvx*+ x

=*kv*(x).

Hence,

lim
*n*→∞*b*

*v*
*n*

*bv*

*nHv*(b*n*;*x) –kv*(x)

= lim
*n*→∞ *b*

*v*
*n*

*Bn*(x) –

–*Bn*(x)b*vn*

×

*x*

*bv _{n}*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt*+*kv*(x)

+

+*o()Bn*(x)b*nv*

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt*

= –*v*–– *λ**vx*+*λ*

*v*_{( –}* _{v}*–

_{)}

*v* *x*

_{–}

–*v*– – v–*λ**vx*

+( –*v*
–_{)}

*λ*

*v _{x}*

_{+ x}

=*ω*◦* _{v}*(x).

The proof is complete.

**Lemma .** *Let Cn*(x)*be given by*(.)*and Dn*(x)*be denoted by*

*Dn*(x) = +*k*˜*v*(x)b–*nv*+

˜

*ωv*(x) +

˜
*k** _{v}*(x)

*b*–_{n}*v* +*o()*.

*For v* = *and*–dlog*bn*<*x*<*cb*

*v*

*n* *with* <*c,d*< ,*we have*

_{C}_{n}_{(x)D}_{n}_{(x)}_{< ,}

*bv _{n}Cn*(x)D

*n*(x) – ≤ +

*bv _{n}bv_{n}Cn*(x)D

*n*(x) –

–*kv*(x)

≤ + +*v*–*λ**v* +
*v*|x|+

+ v–*x*+

*v*|x|+

+ v–*x*

+

+ v–|*x*|+ +*v*
–

*x*

*e*–*x*+

+*v*–*x*+ |*x*|*e*–*x*

*for large n,where kv*(x)*and kv*(x)*are given by*(.).

*Proof* The desired results follow from Lemmas . and ..

The following Mills’ inequalities are from theGED(v) in Jia*et al.*[], which will be used
later.

**Lemma .** *Let Fv*(x)*and fv*(x)*denote the cdf and pdf of*GED(v),*respectively.Then*

(i) *forv*> *and allx*> ,*we have*

*λv*

*v* *x*
–*v*

+(v– )*λ*

*v*

*v* *x*

–*v*

–

< –*Fv*(x)
*fv*(x)

<*λ*

*v*

*v* *x*

–*v*_{;} _{(.)}

(ii) *for* <*v*< *and allx*>*λ*[(/v– )]/*v*,*we have*

*λv*

*v* *x*

–*v*_{<} –*Fv*(x)

*fv*(x)

<*λ*

*v*

*v* *x*
–*v*

+(v– )*λ*

*v*

*v* *x*

–*v*

–

. (.)

**Lemma .** *Let the norming constant bnbe given by*(.),*for any constant* <*c*< *and*

*arbitrary nonnegative integers i,j and k,we have*

lim
*n*→∞*b*

*i*
*n*

∞

*cb*
*v*

*n*

|*x*|*je*–*kx*(x)*dx*= , (.)

lim
*n*→∞*b*

*i*
*n*

_{∞}

*cb*
*v*

*n*

|x|*j _{g}*

*n*(x)dx= (.)

*if v* = .

*Proof* By arguments similar to Lemma . in Jia*et al.*[], we can get (.). The rest is to
prove (.). By (.) and (.), and Lemma ., we have

*bi _{n}*

_{∞}

*cb*
*v*

*n*

|x|*j _{g}*

*n*(x)

*dx*

=*bi _{n}*

∞

*cb*
*v*

*n*

|x|*j _{na}*

*nFvn*–(a*nx*+*bn*)f*v*(a*nx*+*bn*)dx

=*bi _{n}*

∞

*cb*
*v*

*n*

|*x*|*j* –*Fv*(a*nx*+*bn*)
–*Fv*(b*n*)

*exCn*(x)D*n*(x)(x)*dx*

≤b*i _{n}*

∞

*cb*
*v*

*n*

|x|*j _{e}*–

*x*

_{}_{(x)}

_{dx}→

**Lemma .** *Assume that the shape parameter v* = ,*then for any constant* <*d*< *and*
*arbitrary nonnegative integers i,j and k,we have*

lim
*n*→∞*b*

*i*
*n*

–*d*log*bn*

–∞ |*x*|

*j _{e}*–

*kx*

_{}_{(x)}

_{dx}_{= ,}

_{(.)}

lim
*n*→∞*b*

*i*
*n*

–*d*log*bn*

–∞ |x|

*j _{g}*

*n*(x)*dx*= . (.)

*Proof* By arguments similar to that of Lemma ., we have

*bi _{n}*

–*d*log*bn*

–∞

|x|*j _{e}*–

*kx*

_{}_{(x)dx}

≤*bi _{n}*exp

–*b*

*d*
*n*

∞

*xjekx*exp

–*e*

*x*

*dx*

→

as*n*→ ∞since_{}∞*xj _{e}kx*

_{exp}

_{(–}

*ex*

)*dx*<∞.

For assertion (.), we only consider the case of*v*> since the proof of the case of
<*v*< is similar. Rewrite

*bi _{n}*

–*d*log*bn*

–∞

|*x*|*jgn*(x)*dx*=*bin*

–*vλ*–_{}*vbvn*

–∞

|*x*|*jgn*(x)*dx*+*bin*

–*vλ*–*vb*

*v*–

*n*

–*vλ*–*vbvn*

|*x*|*jgn*(x)*dx*

+*bi _{n}*

–*d*log*bn*

–*vλ*–*vb*

*v*–

*n*

|x|*j _{g}*

*n*(x)*dx*

=*In*+*IIn*+*IIIn*.

First note that_{R}|*x*|*j _{f}*

*v*(x)*dx*<∞and the symmetry of *fv*implies*Fv*(–x) +*Fv*(x) = . By

using (.) and (.) we have

*In* < *jvλ*–*vbin*+*j*+*v*

–*Fv*(b*n*)

*n*–

< *jvλ*–*vbi _{n}*+

*j*+

*v*exp

(n– )

*c*– (v– )log*bn*–

(b*n*)*v*

*λv*

→ (.)

as*n*→ ∞, where*c*= ( –*v*– /v)log + (v– )log*λ*–log(/v).

To show*IIn*→ and*IIIn*→, we consider the case of*v*> ﬁrst. By using the following

inequalities

–*vx*< ( –*x)v*< –*vx*+*v(v*– )
*x*

_{,} _{ <}_{x}_{<}

,*v*> , (.)

we can get

*nanfv*

*bn*–*b*

–*v*

*n*

< exp

*v*
*λvb*

*v*–

*n*

and

–*Fv*(b*n*–*b*

–*v*

*n* )

–*Fv*(b*n*)

>
*λv*

*v* (b*n*–*b*

–*v*

*n* )–*v*( +(*v*–)*λ*
*v*

*v* (b*n*–*b*

–*v*

*n* )–*v*)–*fv*(b*n*–*b*

–*v*

*n* )

*λv*

*v* *b*–*nvfv*(b*n*)

> ( –*b*
–+_{}*v*

*n* )–*v*

+(*v*–)_{v}*λv*(b*n*–*b*

–*v*

*n* )–*v*
exp

*v*
*λvb*

*v*–

*n* –

*v(v*– )
*λv* *b*

–
*n*
>
exp
*v*
*λvb*

*v*–

*n*

for large*n, which implies that*

*IIn* < *bin*

v*λ*–*vbv*
*n*

*j*

*nanfv*

–a*n*

*vλ*–*vb*

*v*–

*n*

+*bn*

*Fn*–

*v*

–a*n*

*vλ*–*vb*

*v*–

*n*

+*bn*

v*λ*–*vbv*
*n*

< *bi _{n}*

v*λ*–*v _{b}v*

*n*

*j*+
*nanfv*

–a*n*

*vλ*–*v _{b}v*–

*n*

+*bn*

exp–(n– ) –*Fv*

*bn*–*b*

–*v*

*n*

= b*i _{n}*

v*λ*–*vbv*
*n*

*j*+_{λ}_{}+_{v}_{}_{(}

*v*)

*v* exp

*bv*
*n*+*vb*

*v*–

*n*

*λv* –

exp

*v*
*λvb*

*v*–

*n*

→ (.)

as*n*→ ∞.

Similarly, for –*vλ*–*vb*

*v*–

*n*

<*x*< –dlog*bn*, we have

*nanfv*(a*nx*+*bn*)

<

+(v– )*λ*

*v*

*v* *b*

–*v*
*n*

*fv*(a*nx*+*bn*)

*fv*(b*n*)

<

+(v– )*λ*

*v*
*v* *b*
–*v*
*n*
exp
–*b*
*v*
*n*

*λv**λ*
*v _{b}*–

*v*

*n* *x*

< e–*x*

and

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

>
*λv*

*v* (a*nx*+*bn*)

–*v*_{( +}(*v*–)*λv*

*v* (a*nx*+*bn*)

–*v*_{)}–_{f}

*v*(a*nx*+*bn*)

*λv*

*v* *b*–*nvfv*(b*n*)

> ( +
*λv*

*v* *b*–*nvx)*–*v*

+(*v*–)_{v}*λv*(a*nx*+*bn*)–*v*
exp

–x–*v*–
*v* *λ*

*v _{b}*–

*v*

*nx*

>
*e*

for large*n*by using (.). Then

*IIIn* < b*in*

–*d*log*bn*

–*vλ*–*vb*

*v*–

*n*

|*x*|*je*–*x*exp

–
*e*

–*x*

*dx*

< b*i _{n}*exp

–*e*

*d*log*bn*

–*d*log*bn*

–*vλ*–*vb*

*v*–

*n*

|x|*j _{e}*–

*x*

_{exp}

–
*e*

–*x*

*dx*

→ (.)

as*n*→ ∞.

Combining with (.)-(.), the assertion (.) is derived for*v*> . Similar proofs for
the case of <*v*≤ and details are omitted here. The proof is complete.

**Lemma .** *Letα*=min(,*v)as v* = .*For large n and*–dlog*bn*<*x*<*cb*
*v*

*n*,*both xrbvnn*(g*n*,
;*x)and xr _{b}v*

*n*[b*vnn*(g*n*,;*x) –kv*(x)(x)]*are bounded by integrable functions *

*indepen-dent of n,with r*> , <*c*< *and* <*d*<*α*,*where anand bnare given by*(.),*and kv*(x)

*is given by*(.).

*Proof* We only consider the case of*v*> . For the case of <*v*< , the proofs are similar
and details are omitted here. Rewrite

*bv _{n}n*

*gn*,;*x*

=*bv _{n}Hv*(b

*n*;

*x)e*–

*x*(x) +

*bvn*

*Cn*(x)D*n*(x) –

*e*–*x*(x),

where*Cn*(x) is given by (.),*Hv*(b*n*;*x) andDn*(x) are respectively deﬁned in Lemma .

and Lemma .. Note that_{–}∞_{∞}*xk _{e}*–

*tx*

_{exp}

_{(–e}–

*x*

_{)}

_{dx}_{= (–)}

*k*(

_{}*k*)

_{(t) is ﬁnite for}

_{t}_{> and }non-negative integers

*k. Lemma . shows thatbv*

*n*(C*n*(x)D*n*(x) – )e–*x*(x) is bounded by

inte-grable function independent of*n. The rest is to prove thatbv*

*nHv*(b*n*;*x) is bounded bym(x),*

where*m(x) is a polynomial onx. Rewrite*

*bv _{n}Hv*(b

*n*;

*x) =In*(x) +

*Jn*(x), (.)

where

*In*(x) =*bvn*

*Bn*(x) –

,

*Jn*(x) =*bvnBn*(x)

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt.*

For –dlog*bn*<*x*<*cb*
*v*

*n*, from Lemma . it follows that

*In*(x)< (.)

and

*Jn*(x)< +

*v*–

–* _{v}_{λ}*–

*v*

_{–}

*|x|+*

_{d}

–*v*–*λvx*+

–*v*– – v–*λ**v*|x|

. (.)

Rewrite

*bv _{n}bv_{n}n*

*gn*,;*x*

–*kv*(x)(x)

=*bv _{n}*

*bv*

_{n}

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

*exCn*(x)D*n*(x) –

–*kv*(x)

(x)

=*bv _{n}*

*bv*(x)D

_{n}Cn*n*(x)

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

*ex*–

+*bv _{n}Cn*(x)D

*n*(x) –

–*kv*(x) +*kv*(x)

(x)
=*bv _{n}bv_{n}Cn*(x)D

*n*(x)

*Hv*(b*n*;*x) –b*–*nvkv*(x)

+*bv _{n}Cn*(x)D

*n*(x) – –

*b*–

*nvkv*(x)

+*Cn*(x)D*n*(x) –

*kv*(x)

(x).

By Lemma ., we only need to estimate the bound of *bv _{n}*[b

*v*(x)D

_{n}Cn*n*(x)(H

*v*(b

*n*;

*x) –*

*b*–*v*

*nkv*(x))]. Rewrite

*bv _{n}bv_{n}Cn*(x)D

*n*(x)

*Hv*(b*n*;*x) –b*–*nvkv*(x)

=*Hn*(x) –*Kn*(x) +*Ln*(x),

where

*Hn*(x) =*b**nv*

*Bn*(x) –

,

*Kn*(x) =*b**nv*

*Bn*(x)

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–
*λv* –

*dt*+*b*–* _{n}vkv*(x)

,

*Ln*(x) =

+*o()Bn*(x)b*nv*

*x*

(v– )a*n*

*bn*+*ant*

+*van*(b*n*+*ant)*

*v*–

*λv* –

*dt*

.

For <*x*<*cb*

*v*

*n*, by using –*vy*< ( +*y)*–*v*< for*v*> and*y*> , we have

*Hn*(x)< + *λ**v*

+ *λvx*

due to Lemma .. If –dlog*bn*<*x*< , by using +*vy*< ( +*y)v*< for*v*> and – <*y*< ,

Lemma . shows that

* _{H}_{n}*(x)< +

*λ*

*v* +

*λv*|x|

for large*n. Similarly,*

*Kn*(x)<

–*v*–_{} *v*
*λv*–*d*

–*λvx*+

–*v*– – v–*λ**v*|*x*|, (.)

*Ln*(x)< +

*v*–

–* _{v}_{λ}*–

*v*

_{–}

*|x|+*

_{d}

–*v*–*λvx*+

–*v*– – v–*λ**v*|x|

(.)

as –dlog*bn*<*x*<*cb*
*v*

*n*. Hence, we derive the desired result by combining (.), (.) and

(.) together.

For*v*= , note that theGED() is the Laplace distribution with pdf given by

*f(x) = *–_{exp}_{–}|x|_{,} * _{x}*∈R

_{,}

and its distributional tail can be written as

–*F(x) = *–* _{f(x) = }*–

_{exp}

_{–}

_{exp}

–

*x*

*f*(t)*dt*

, *x*>

with*f*(t) = –_{}_{. For the Laplace distribution, similar to the case of}_{v}_{> , we have the }
fol-lowing two results.

**Lemma .** *For* <*d*< *and an arbitrary nonnegative real number j,we have*

lim
*n*→∞*n*

–*db*

*n*

–∞ |*x*|

*j _{e}*–

*kx*

_{}_{(x)}

_{dx}_{= ,}

_{k}_{= , , . . . ,}

_{(.)}

lim
*n*→∞*n*

–*db*

*n*

–∞

|x|*j _{F}n*

(a*nx*+*bn*)*dx*= . (.)

**Lemma .** *For x*> –db

*n*,*both xrn((F**n*(a*nx*+*bn*))–(x))*and xrn[n((F**n*(a*nx*+*bn*))–
(x)) +

*e*

–*x _{}*

_{(x)]}

_{are bounded by integrable functions independent of n,}_{where r}_{> }

_{and} <*d*< .

**4 Proofs of the main results**

*Proof of Theorem*. From Lemma . and Lemma . it follows that

*Cn*(x)D*n*(x)

= +*kv*(x)b–*nv*+

–*v*–*λ**v* – –
*vx*+

– v–*x*+

–
*vx*+

– v–*x*

+( – v
–_{)}

*x*

_{–} –*v*–
*x*

*e*–*x*+( –*v*
–_{)}

*x*+ x*e*–*x*

*b*–* _{n}v*+

*Ob*–

*, (.)*

_{n}vwhere*kv*(x) is given by (.). Note that (.) shows

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

*ex*= +*kv*(x)b–*nv*+*ω*◦*v*(x)b–*n* *v*+*O*

*b*–* _{n}v*. (.)

Hence,

*n*

*gn*,;*x*

=

–*Fv*(a*nx*+*bn*)

–*Fv*(b*n*)

*exCn*(x)D*n*(x) –

(x)

= –*v*–*λv* –*x*+*x*+ x*e*–*xb*–* _{n}v*+ –

*v*–

*λ*

*v*

– – –*v*–*x*

–( – v
–_{)}

*x*

_{+} –*v*–
*x*

_{+}

–
*vx*–

*vx*

_{–}( – v–)

*x*

_{–}( –*v*–)

*x*

+ –*v*
–

*x*+ x*e*–*x*

*b*–* _{n}v*+

*Ob*–

*(x) =*

_{n}v*kv*(x)b–

*nv*+

–*v*–*λ**vωv*(x) +*ωv*(x)e–*x*+*ωv*(x)e–*x*

*b*–* _{n}v*+

*Ob*–

*(x) =*

_{n}v*kv*(x)b–

*nv*+

*ωv*(x)b–

*nv*+

*O*

*b*–_{n}*v*(x)

implying that

lim
*n*→∞*b*

*v*
*n*

*bv*
*nn*

*gn*,;*x*

–*kv*(x)(x)

=*ωv*(x)(x).

The proof is complete.

*Proof of Theorem*. For*v* = , by Lemmas .-. and the dominated convergence
theo-rem, we have

*bv _{n}mr*(n) –

*mr*

=*bv _{n}*

–*d*log*bn*

–∞ *x*

*r _{}*

*n*

*gn*,;*x*

*dx*+*bv _{n}*

*cb*
*v*

*n*

–*d*log*bn*

*xrn*

*gn*,;*x*

*dx*

+*bv _{n}*

∞

*cb*
*v*

*n*

*xrn*

*gn*,;*x*

*dx*

→

∞

–∞

*xrkv*(x)(x)dx

= – –*v*–*λvr(mr*+*mr*+)

and

*bv _{n}bv_{n}mr*(n) –

*mr*

+ –*v*–*λvr(mr*+*mr*+)

=

∞

–∞*x*

*r _{b}v*

*n*

*bv _{n}n*

*gn*,;*x*

–*kv*(x)(x)

*dx*

=

–*d*log*bn*

–∞

*xrb*_{n}vn

*gn*,;*x*

*dx*–

–*d*log*bn*

–∞

*xrbv _{n}kv*(x)(x)

*dx*

–

∞

*cb*
*v*

*n*

*xrbv _{n}kv*(x)(x)dx+

*cb*
*v*

*n*

–*d*log*bn*

*xrbv _{n}bv_{n}n*

*gn*,;*x*

–*kv*(x)(x)

*dx*

+

∞

*cb*
*v*

*n*

*xrb*_{n}vn

*gn*,;*x*

*dx*

→ ∞
–∞*x*

*r _{ω}*

*v*(x)(x)*dx*

= r*λ**v* –*v*– –*v*–(r+ ) + *mr*+

–*v*–(r+ ) + *mr*+

+

–*v*–(r– ) +

–*v*–*mr*+

For the case of*v*= , note that

_{∞}

–∞*x*

*k _{e}*–

*mx*

_{}_{(x)}

_{dx}_{= (–)}

*k*(

_{}*k*)

_{(m).}

Combining with Lemmas . and . and the dominated convergence theorem, we can derive the desired results.

The proof is complete.

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

CL drafted the manuscript and TL revised the whole paper critically. Both authors were involved in the proof of the main results of the paper. All authors read and approved the ﬁnal manuscript.

**Acknowledgements**

We would like to appreciate the reviewers for reading the paper and making helpful comments that improved the original paper. This work was supported by the National Natural Science Foundation of China (11171275), the Natural Science Foundation Project of CQ (cstc2012jjA00029), the Fundamental Research Funds for the Central Universities (XDJK2013C021) and the Doctoral Grant of Southwest University (SWU113012).

Received: 12 May 2015 Accepted: 1 November 2015
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