Second-order asymptotics for convolution of distributions with light tails

Statistics and Probability Letters xx (xxxx) xxx–xxx

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Second-order asymptotics for convolution of distributions with light tails Q1

Zuoxiang Peng a , Xin Liao b,∗ a

School of Mathematics and Statistics, Southwest University, 400715 Chongqing, China

b

Business School, University of Shanghai for Science and Technology, 200093 Shanghai, China

article

info

Article history: Received 22 May 2015 Received in revised form 14 July 2015 Accepted 17 July 2015 Available online xxxx

abstract In this paper, asymptotic behaviors of convolutions of distributions belonging to two classes of distributions with light tails are considered. The precise second-order tail asymptotics of the convolutions are derived under the condition of second-order regular variation. © 2015 Elsevier B.V. All rights reserved.

MSC: primary 62E20 60G50 secondary 60F15 60F05 Keywords: Convolution Light tail Second-order approximation Second-order regular variation

1. Introduction

1

Let X and Y be two independent and nonnegative random variables with distribution F and G, respectively. The distribution of the sum X + Y , written as F ∗ G, is called the convolution of F and G. A distribution F on [0, ∞) is said to belong to the class Lα for some α ≥ 0, if its right tail satisfies F (t ) = 1 − F (t ) > 0 for all t > 0 and the relation lim

F (t − u)

t →∞

F (t )

= eαu

(1.1)

holds for all u. For α = 0, the class L0 reduces to the well-known class of long-tailed distributions. Clearly, the class Lα is related to the class RV−α of regularly varying functions with exponent −α by the fact that F ∈ Lα

∗

if and only if F (ln t ) ∈ RV−α .

Corresponding author. E-mail address: [email protected] (X. Liao).

http://dx.doi.org/10.1016/j.spl.2015.07.027 0167-7152/© 2015 Elsevier B.V. All rights reserved.

2 3 4 5

6

7 8

9

2

1 2

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

Applying Karamata’s representation theorem for regularly varying functions (see Bingham et al., 1987 and Cline, 1986), we know that F ∈ Lα if and only if F (t ) can be expressed as t

 

F (t ) = c (t ) exp −

3

 α(y)dy

(1.2)

0 4 5 6

with limt →∞ c (t ) = c > 0 and limt →∞ α(t ) = α . An important subclass of Lα is the class of convolution equivalent distributions Sα . We say that F belongs to the class Sα if F ∈ Lα and the limit lim

7

t →∞ 8 9 10 11 12

15

F (t )

= 2mF (α)

(1.3)

exist and is finite, where the constant mF (α) = 0 eα u dF (u); see Chistyakov (1964) and Chover et al. (1973a,b), Cline (1987). The class S := S0 is called the class of subexponential distributions. Properties of those mentioned classes have been extensively investigated by many researchers, see, e.g., Embrechts and Goldie (1980), Cline (1986, 1987) and Foss et al. (2009). For F , G ∈ Lα , Cline (1986) investigated the related relationship between F ∗ G and its components F and G. Especially for the following two classes of distributions:

∞

F (t ) = e−α t +χ (t ) ,

13 14

F ∗ F (t )

χ (t ) ∈ RVρ , 0 < ρ < 1

(1.4)

and F (t ) = b(t )e−α t ,

G(t ) = c (t )e−α t ,

b(t ) ∈ RVβ , c (t ) ∈ RVγ , β, γ ∈ R,

(1.5)

32

he derived the first-order tail asymptotics of F ∗ G. Motivated by works in Cline (1986), in this paper we are interested in the second-order tail asymptotic expansions of convolutions of distributions given by (1.4) and (1.5). Clearly, such distributions F and G not only belong to Lα , but also are Weibull-type distributions. In recent literature, more and more researchers focus on the second-order asymptotic behaviors for the sake of understanding precisely the tail behaviors of risks, ruin probabilities and random summation. Hua and Joe (2011) and Mao and Hu (2013) derived the second-order approximations of the conditional tail expectation CTEp and the risk concentrations based on the risk measures of Value-at-Risk and CTEp , respectively. Yang and Wang (2013) investigated the asymptotic behavior for the (in)finite-time ruin probabilities in a discrete-time insurance risk model. For a renewal risk model, the second-order behavior of ruin probabilities was given by Lin (2012). Mao and Ng (2015) considered the second-order properties of tail probabilities of sums and randomly weighted sums. For more studies on the higher-order asymptotics behaviors on risks, sums and ruin probabilities, we refer to Degen et al. (2010), Hashorva et al. (2014) and Lin (2014). The main goal of this paper is to investigate the second-order asymptotics of convolutions of light tail distributions F , G which satisfied (1.4) and (1.5). In order to get the desired results, we assume that χ (t ), b(t ), c (t ) in (1.4) and (1.5) are second-order regularly varying functions. The rest of this paper is organized as follows. In Section 2, some preliminary concepts and results of second-order regularly varying functions are presented, which will be used to prove the main results. The main results and some illustrating examples are given in Section 3. All proofs are deferred to Section 4.

33

2. Preliminaries

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

34

For the analysis of tail asymptotics of convolutions of F and G satisfying (1.4) and (1.5), the theory of regular variation

35

Q2 on survival functions plays an important role. We refer to de Haan and Ferreira (2006) and Resnick (1987) for standard

36

37 38

39

references on regular variation. Definition 1. A measurable function χ : R+ → R that is eventually positive is regularly varying at ∞ with some α ∈ R (written χ ∈ RVα ) if for any u > 0, lim

t →∞ 40 41 42 43

44 45

46

χ(tu) = uα . χ (t )

(2.1)

We call α the index of variation. If α = 0, χ (t ) is said to be slowly varying at ∞. As we all known, Karamata’s theorem described the effect of integration on a regularly varying function. When an α varying function is differentiated, the associated property was investigated by Proposition 0.7 of Resnick (1987) which is cited as follows. ′ Lemma  x ′ 1 (Resnick, 1987, Proposition 0.7). Suppose χ : R+ → R+ is absolutely continuous with density χ so that χ (x) = χ ( t ) dt. If 0

lim

t →∞

t χ ′ (t )

χ(t )

= α,

(2.2)

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

3

then χ ∈ RVα . Conversely, if χ ∈ RVα , α ∈ R, and χ ′ is monotone then (2.2) holds and if α ̸= 0, then (sgn α)χ ′ (x) ∈ RVα−1 . The following definition of the second-order regular variation comes from de Haan and Ferreira (2006) and Geluk et al. (1997). Definition 2. A measurable function χ : R+ → R that is eventually positive is said to be of second-order regular variation with the first-order parameter α ∈ R and the second-order parameter ρ ≤ 0, denoted by χ ∈ 2RVα,ρ , if there exists some ultimately positive or negative function A(t ) with limt →∞ A(t ) = 0 such that χ(tx) χ(t )

− xα

= xα

xρ − 1

,

x > 0.

1 2 3 4

5 6 7

(2.3)

8

Here, x ρ−1 is interpreted as log x when ρ = 0, A(t ) is referred as auxiliary function of χ , and ρ governs the speed of convergence in (2.1).

10

Next result concerns the Drees-type inequalities for RV functions and 2RV functions which establish uniform inequalities.

11

lim

A(t )

t →∞

ρ

ρ

Lemma 2 (de Haan and Ferreira, 2006, Theorem B.1.10, Theorem 2.3.9; Drees, 1998). If χ ∈ RVα with α ∈ R, for each ϵ, δ > 0, there is a t0 = t0 (ϵ, δ) such that for t , tx ≥ t0 ,

   χ (tx)   δ −δ  α α  − x  χ (t )  ≤ ε x max x , x .

(2.4)

Further, if χ ∈ 2RVα,ρ with auxiliary function A(t ) and ρ ≤ 0, then for any ε, δ > 0, there exist an auxiliary function A1 (t ), A1 (t ) ∼ A(t ) as t → ∞, and t0 = t0 (ε, δ) > 0 such that for all t , tx > t0 ,

  χ(tx) α   ρ    χ(t ) − x α x − 1 −x  ≤ ε xα+ρ max xδ , x−δ .   A1 (t ) ρ 

(2.5)

To end this section, we provide the following nice representation of 2RVα,ρ with ρ < 0 given by Hua and Joe (2011). Lemma 3 (Hua and Joe, 2011). Let α ∈ R, ρ < 0 and |A(t )| ∈ RVρ . Then χ ∈ 2RVα,ρ with auxiliary function A(t ) if and only if there exists a constant a > 0 such that



χ (t ) = at α 1 +

1

ρ

 A(t ) + o (A(t ))

9

12 13

14

15 16

17

18

19 20

21

as t → ∞.

22

3. Main results

23

In this section, we provide the main results. For F (t ) = e−α t +χ (t ) given by (1.4), the second-order tail asymptotic of F ∗ F are presented in the following theorem by assuming that χ (t ) ∈ 2RVρ,ρ1 , 0 < ρ < 1, ρ + ρ1 < 0. Theorem 1. Let F (t ) = e−α t +χ(t ) with α > 0 and χ (t ) is eventually differentiable such that χ ′ (t ) is nonincreasing eventually. Assume that χ (t ) ∈ 2RVρ,ρ1 with 0 < ρ < 1, ρ + ρ1 < 0 and auxiliary function A1 (t ). Then, for large t we have 1

χ2

t  2

tF

2

F ∗ F (t )

=

t  2

√ π 2ρ−5 α π(ρ − 2)(ρ − 3) − 3 2 ρ(1 − ρ) (ρ(1 − ρ)) 2 χ (t )    1 ρπ χ (t ) χ (t ) 1 − ρ + o |A1 (t )|χ (t ) + + . 2 1−ρ t χ (t ) t α

24 25

26 27



28

(3.1)

Remark 1. For F (t ) = e−α t +χ (t ) , we have mF (α) = ∞ which implies that F ∈ Lα but F ̸∈ Sα . 1 2

30

Example 1. Let risk X has distribution tail F (x) = e−α x+χ(x) with χ (t ) = α bt 1 + (t ) + o(A1 (t )) , b > 0, |A1 (t )| ∈ RV−2 . Then by Lemma 3, χ (t ) ∈ 2RV 1 ,−2 , and the second-order expansion (3.1) leads to



1 A 2 1

29



31 32

2

3

F ∗ F (t ) = t 4 F for large t.

2

  t

2

1

24

 απ  12 b

1

− 2− 4

  π  12   1  15 − 12 1+ t + o t− 2 αb 8α b2

33

34

4

1 2 3

4 5

6

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

Theorem 2 provides the second-order expansions of F ∗ G where F and G are given by (1.5) with b(t ) ∈ 2RVβ,ρ2 , c (t ) ∈ 2RVγ ,ρ3 , β, γ ∈ R, ρ2 , ρ3 ≤ 0. These results are more complex than results derived in Theorem 1. In order to state the main result Theorem 2 clearly, we first present the following proposition. Proposition 1. Let F , G ∈ Lα , α > 0. Assume that b(t ) = eα t F (t ) ∈ 2RVβ,ρ2 with auxiliary function A2 (t ) for ρ2 ≤ 0, β ∈ R, and c (t ) = eα t G(t ) ∈ RVγ , γ ∈ R. Then (i) if γ ≤ −1, for large t we have t 2

 7

F (t − u)dG(u) = F (t )M1 (β, γ , t )

(3.2)

0 8

9

with

   1   t 2 2   α tc ( t )  β− 1 −β   c (u)du α +  t β (1 − u) (log u)du + (log 2) 1 − 2   2  0 0  c ( u ) du  0      1 1 + tc ( t )    + o  t + |A2 (t )| , γ = −1 with mG (α) = ∞;  +t  2 2   c ( u ) du c ( u ) du  0 0      ∞  ∞     c (u)du + o c (u)du + |A2 (t )| , γ = −1 with mG (α) < ∞; mG (α) − α   t t   2 2   1  2   γ M1 (β, γ , t ) = 1 β  mG (α) + α (1 − u) − 1 u du + tc (t ) + o(tc (t ) + |A2 (t )|),   (1 + γ )21+γ  0     − 2 < γ < −1;     t    t   2 2  −1 −1   uc (u)du + o t uc (u)du + |A2 (t )| , mG (α) − αβ t    1 1    ∞    uc (u)du = ∞;  γ = −2 with   1    ∞  ∞   m (α) − β t −1 ueα u dG(u) + o(t −1 + |A2 (t )|), γ ≤ −2 with uc (u)du < ∞. G 0

10

11

1

(ii) if γ > −1, suppose c (t ) ∈ 2RVγ ,ρ3 with auxiliary function A3 (t ) and ρ3 ≤ 0, γ + ρ3 + 1 > 0, then (3.2) holds with 1 2

  M1 (β, γ , t ) = tc (t ) α

1 2

((1 − u)ρ2 − 1) (1 − u)β uγ duA2 (t ) ρ 2 0 0    1  1 2 2 β−1 γ β γ + β (1 − u) u (2u − 1)du − 2(1 + γ ) (1 − u) u du t −1

12

(1 − u)β uγ du + α



0

 +α

13

 +

14

15

16 17

18

19

(3.3)

0

β γ + ρ3 + 1 2

−ρ3

2−ρ3 − 1

ρ3

−

1 2



(1 − u)β−1 uγ +1

0

2−ρ3

γ + ρ3 + 1

1 2



ρ3

((2u) − 1) du ρ3  β γ

(1 − u) u du A3 (t ) + o t 

−1

  + |A2 (t )| + |A3 (t )|

(3.4)

0

for large t. Theorem 2. Let F , G ∈ Lα , α > 0. Assume that b(t ) = eα t F (t ) ∈ 2RVβ,ρ2 with auxiliary function A2 (t ) for ρ2 ≤ 0, β ∈ R, and c (t ) = eα t G(t ) ∈ 2RVγ ,ρ3 with auxiliary function A3 (t ) for ρ3 ≤ 0, γ ∈ R. Then F ∗ G(t ) = F (t )M1 (β, γ , t ) + G(t )M2 (β, γ , t ) for large t, where M1 (β, γ , t ) is given by Proposition 1, and M2 (β, γ , t ) is given as follows:

(3.5)

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

5

(i) if β ≤ −1, for large t we have

1

  t   1  2 2    α tb(t )  γ − 1 −γ  b(u)du α +  t γ (1 − u) (log u)du + (log 2) 1 − 2    2 0 0  b ( u ) du  0       1 1 + tb ( t )   +t + o  t + |A3 (t )| , β = −1 with mF (α) = ∞;    2 2  b ( u ) du b ( u ) du  0 0       ∞  ∞    mF (α) − α b(u)du + o b(u)du + |A3 (t )| , β = −1 with mF (α) < ∞;   t t   2 2   1  M2 (β, γ , t ) = 2 1 γ β  tb(t ) + o(tb(t ) + |A3 (t )|), ((1 − u) − 1) u du + mF (α) + α   (1 + β)21+β 0     − 2 < β < −1 ;     t    t   2 2   − 1 − 1 mF (α) − αγ t ub(u)du + o t ub(u)du + |A3 (t )| ,    1 1    ∞     β = − 2 with ub(u)du = ∞;    1  ∞  ∞    mF (α) − γ t −1 ueα u dF (u) + o(t −1 + |A3 (t )|), β ≤ −2 with ub(u)du < ∞. 0

(3.6)

1

(ii) if β > −1 with β + ρ2 + 1 > 0, for large t we have 1 2

  M2 (β, γ , t ) = tb(t ) α

3 1 2

((1 − u)ρ3 − 1) (1 − u)γ uβ duA3 (t ) ρ3 0 0   1   1 2 2 γ −1 β γ β −β−γ + γ (1 − u) u (2u − 1)du − 2(1 + β) (1 − u) u du + 2 t −1 (1 − u)γ uβ du + α



0

 +α  +

4

5

0

2−ρ2 γ

β + ρ2 + 1 2−ρ2 − 1

ρ2

1 2



−

(1 − u)γ −1 uβ+1

0

2−ρ2

β + ρ2 + 1

1 2



((2u)ρ2 − 1) du ρ2  γ β

6

(1 − u) u du A2 (t ) + o t 

−1

+ |A2 (t )| + |A3 (t )|

 

.

(3.7)

−α t Example 2. Suppose X and Y are nonnegative random variables with distribution F and G, respectively.  Let F (t ) = b(t )e 

and G(t ) = c (t )e

α ζ −1 ζ −1 t 0 (ζ )

ζ −1 −1 t α

with b(t ) = t 1+ (t ) + o(A2 (t )) , |A2 (t )| ∈ RV−4 , c (t ) = 1+ + o( t ) , α > 0, ζ > 1. Here 0 (·) denotes the gamma function. From Lemma 3, it follows that b(t ) ∈ 2RV−3,−4 , c (t ) ∈ 2RVζ −1,−1 which imply that mF (α) < ∞, mG (α) = ∞. Then second-order expansion (3.5) leads to    ∞ F ∗ G(t ) = G(t ) mF (α) − (ζ − 1)t −1 ueα u dF (u) + o(t −1 ) , 0    ∞ −1 αu −1 F ∗ F (t ) = 2F (t ) mF (α) + 3t ue dF (u) + o(t ) −3

1 A 4 2



7

0

Remark 2. For F (t ) = b(t )e−α t , b(t ) ∈ RVβ , β ∈ R, it is clear that F ∈ Lα for all β ∈ R. Note that F ∈ Sα if and only if β ≤ −1 with mF (α) < ∞. So Theorem 2 also derives the second-order tail asymptotics of convolution of distributions from a subclass of Sα .

−α t

2



−1

8 9 10

11 12 13 14

15

16

0

and

17

  1     1 2 2 2 α ζ −1 ζ G ∗ G(t ) = t G(t ) α (1 − y)ζ −1 yζ −1 dy + (ζ − 1) (1 − y)ζ −2 yζ dy + 2−2(ζ −1)−1 t −1 + o(t −1 ) 0 (ζ ) 0 0 for large t.

18

19

6

1

2 3 4

5 6

7

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

4. Proofs The aim of this section is to prove our main results. Without loss of generality, we assume that the auxiliary functions of 2RV functions are positive eventually in the following proofs. In order to prove Theorem 1, we first give two lemmas as follows. Lemma 4. Let χ (t ) ∈ 2RVρ,ρ1 with auxiliary function A1 (t ) for 0 < ρ < 1, ρ1 ≤ 0. Assume that χ ′ (t ) is nonincreasing eventually. Then for large t, we have

 ′  1  t χ (t )  2    χ(t ) − ρ  ≤ cA1 (t ),

(4.1)

8

where c is a positive constant.

9

Proof. First we consider the case of ρ1 < 0. Note that

ρ(ρ − 1)x2

< (1 + x)ρ < 1 + ρ x,

10

1 + ρx +

11

1 − ρ x + 1 − c2−1

12

1 + ρ1 x < (1 + x)ρ1 < 1 + ρ1 x +

13

1 − ρ1 x < (1 − x)ρ1 < 1 − ρ1 x + 1 − c2−1

2



ρ−2 ρ(ρ − 1)x2 2

(4.2)

< (1 − x)ρ < 1 − ρ x,

ρ1 (ρ1 − 1)x2 

ρ1 −2 ρ1 (ρ1 − 1)x2

χ (t )t ≤ χ (t )

21

22 23

24 25

1

2



2

ρ+ρ1 −δ

for large t. Similarly,

 ρ    ρ+ρ1 +δ 1 1 1 ρ(ρ − 1) 12 ρ1 − 1 12 χ ′ (t )t 1+ ≥ρ+ A1 (t ) + A1 (t ) 1 + A12 (t ) A1 (t ) − ε A12 (t ) 1 + A12 (t ) χ (t ) 2 2 for large t, which implies

 ′  1  t χ (t )  2    χ (t ) − ρ  ≤ cA1 (t ) with c > 0. The arguments for the case of ρ = 0 are similar.



Lemma 5. Let F (t ) = e−α t +χ(t ) such that χ ′ (t ) is nonincreasing eventually. Suppose χ (t ) ∈ 2RVρ,ρ1 with auxiliary function A1 (t ) and 0 < ρ < 1, ρ + ρ1 < 0. Then 1

t

f (2t )

F (2t − u)dF (u) 2

tF (t )

27

28

1 2

+ εA1 (t ) 1 − A1 (t )

χ 2 (t ) 26

χ ′ (y)dy

A12 (t )χ (t )

1 2

17

20





 ρ   1  ρ−2 ρ(ρ − 1) 12  ρ1 −2 ρ1 − 1 12 ≤ ρ − 1 − c2−1 A1 (t ) + A1 (t ) 1 − A12 (t ) 1 − 1 − c2−1 A1 (t )

16

19

1

t 1−A12 (t )

′

18

(4.5)

2

1 for 0 < ρ < 1, ρ1 < 0, 0 < x < c2−1 , 1 < c2 < 1/ 1 − (2 − 2ρ ) ρ . By using (2.5), (4.2)–(4.5), we can get

 t 15

(4.4)

2



14

(4.3)

for large t, where f (t ) = 1/χ ′ (t ).

=

√ π α π (ρ − 2)(ρ − 3) 1 − 3 2 ρ(1 − ρ) χ (t ) 32(ρ(1 − ρ)) 2    1 1 1 ρπ χ (t ) χ 2 (t ) 1 χ 2 (t ) − − + o A1 (t )χ (t ) + + 2 1−ρ t 2t χ (t ) t α



(4.6)

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

7

Proof. From Lemma 1, we have limt →∞ t χ ′ (t )/χ (t ) = ρ , which implies that χ (t ) is increasing eventually. For sufficiently large t and A satisfied f (2t ) < A < t, we can get



1 t −f (2t ) 2 χ (t ) t



 α−χ

1 t −A 2 t χ (t )

′

t−



ty

exp χ

1

χ 2 (t ) 1

 = o A1 (t )χ (t ) + χ

−1

(t ) +

 

χ 2 (t )

t+



ty 1

χ 2 (t )

 +χ t −



ty

2

 − 2χ (t ) dy

1

χ 2 (t )

1

3

 (4.7)

t

and

4

5



t −A t χ 1 c2

χ

1 2 (t )



 α − χ′ t −

1 2 (t )



ty

exp χ

1 2

χ (t ) 1

 = o A1 (t )χ (t ) + χ

−1

(t ) + 1



 

χ 2 (t )

t+

ty 1 2

χ (t )



 +χ t −

ty 1 2

χ (t )



 − 2χ (t ) dy

6

 ,

t

(4.8)



where 1 < c2 < 1/ 1 − (2 − 2ρ ) ρ .

8

In order to derive the desired result, we need some more precise inequalities as follows: 1 + ρx +

ρ(ρ − 1) ρ

< (1 + x)

2

x2 +

7

9

ρ(ρ − 1)(ρ − 2) 3 ρ(ρ − 1)(ρ − 2)(ρ − 3) 4 x + x 3! 4!

10 11

ρ(ρ − 1)

ρ(ρ − 1)(ρ − 2) 3 ρ(ρ − 1)(ρ − 2)(ρ − 3) 4 < 1 + ρx + x + x + x 2 3! 4! ρ(ρ − 1)(ρ − 2)(ρ − 3)(ρ − 4) 5 + x 5! 2

12

(4.9)

and

13

14

ρ(ρ − 1)

ρ(ρ − 1)(ρ − 2) 3 ρ(ρ − 1)(ρ − 2)(ρ − 3) 4 1 − ρx + x2 − x + x 2 3! 4! ρ(ρ − 1)(ρ − 2)(ρ − 3)(ρ − 4) 5 − (1 − c2−1 )ρ−5 x 5! < (1 − x)ρ ρ(ρ − 1) 2 ρ(ρ − 1)(ρ − 2) 3 ρ(ρ − 1)(ρ − 2)(ρ − 3) 4 < 1 − ρx + x − x + x 2 3! 4!

15

16 17

(4.10)

for 0 < x < c2−1 , 0 < ρ < 1. Combining with (2.5), (4.2)–(4.5), we have

19 20

ρ(ρ − 1)(ρ − 2)(ρ − 3) y4 ρ(ρ − 1)(ρ − 2)(ρ − 3)(ρ − 4) y5 − (1 − c2−1 )ρ−5 3 12 χ (t ) 5! χ 2 (t )       1 + (ρ + ρ1 )(ρ + ρ1 − 1) 1 + (1 − c2−1 )ρ+ρ1 −2 − ρ(ρ − 1) 1 + (1 − c2−1 )ρ−2 y2 A1 (t ) 2ρ1  ρ+ρ1 +ε  ρ+ρ1 −ε 

ρ(ρ − 1)y2 +

− ε A1 (t )χ (t )  ≤χ t+

yt

1+



y

+ 1−

1 2

χ (t ) 

21

22

y

23

1 2

χ (t )



− 2χ(t ) 1 χ 2 (t ) ρ(ρ − 1)(ρ − 2)(ρ − 3) y4 ρ(ρ − 1)(ρ − 2)(ρ − 3)(ρ − 4) y5 ≤ ρ(ρ − 1)y2 + + 3 12 χ(t ) 5! χ 2 (t )  ρ+ρ1 +ε  ρ+ρ1 −ε  1

χ 2 (t )

+ ε A1 (t )χ (t )

+χ t −

yt

1+

y

1 2

χ (t )

1 for large t and 0 < y < c2−1 χ 2 (t ).

18

+ 1−

y

1 2

χ (t )

24

25

(4.11)

26

27

8

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx



1

2

3



Note that ρ(ρ − 1)(ρ − 2)(ρ − 3)y4 /(12χ (t )) + ρ(ρ − 1)(ρ − 2)(ρ − 3)(ρ − 4)y5 / 5!χ 2 (t ) 1 2

x2 2

0 < y < c2 χ (t ). From (4.11) and the inequality 1 + x < ex < 1 + x + −1

 3

1 c2

1

χ 2 (t )

  exp χ



yt





yt

< 0 for 0 < ρ < 1,

, x < 0, it follows that



+χ t − 1 − 2χ (t ) dy 1 χ 2 (t ) χ 2 (t ) √    1 π π (ρ − 2)(ρ − 3) 1 = − + o χ −1 (t ) + A1 (t )χ (t ) 3 2 ρ(1 − ρ) χ ( t ) 32 (ρ(1 − ρ)) 2 t+

0

4

5

for large t, if ρ + ρ1 < 0. 1

6





1

Note that A12 (t ) = o χ − 2 (t ) as t → ∞, if ρ + ρ1 < 0. Due to (2.4), (2.5), (4.1), (4.5) and (4.12), we obtain

       1 χ 12 (t ) t 1 − 1y χ′ t −   c2 2 (t ) χ       yt  0 χ t− 1 

  yt χ t − 1 1  χ 2 (t ) χ 2 (t )  − ρ  y 1− 1 χ (t ) χ 2 (t ) χ 2 (t )          yt yt × exp χ t + 1 +χ t − 1 − 2χ (t ) dy  χ 2 (t ) χ 2 (t )  1       ρ−1  1

7

8

c2

≤c

9

χ 2 (t )

1 2

t−

A1

0

  × exp χ t +

10

13

14

15

1 c2

1

χ 2 (t )

=

1



2

1

t



yt



  t+

ρπ χ (t ) χ 2 (t ) + +o 1−ρ t 2t

t

′

t−

1

χ 2 (t ) 1



yt 1

χ 2 (t )  1  χ 2 (t )

 +χ t −



yt 1

χ 2 (t )

 − 2χ (t ) dy (4.13)

F (2t − u)dF (u) 2

tF (t ) 1 t −f (2t ) 2 χ (t ) t

 =



 α−χ

′

t−

0

23

− 2χ (t ) dy

1

χ 2 (t )

exp χ

χ

f (2t )

20

22

1

χ 2 (t )



for large t. Combining (4.7)–(4.8) and (4.12)–(4.13), we have

χ 2 (t )

21

χ (t )

yt



1−

as t → ∞, which implies that

0

19



+χ t −

1 2

1+ε

1

χ 2 (t )

−ε 

y

               1 χ 12 (t ) χ t − 1yt  c2   (ρ − 1)y  yt χ 2 (t )   exp χ t + yt   −1+ +χ t − 1 − 2χ (t ) dy 1 1    y  0  χ 2 (t ) χ 2 (t ) χ 2 (t ) 1− 1 χ (t )   χ 2 (t )  1  = o χ − 2 (t )

16

18

y

1−

1

χ 2 (t ) 



and



17

yt

yt



yt

 1  = o χ − 2 (t )

11

12

(4.12)

=

α 2



yt 1

χ 2 (t )



  exp χ

t+

yt 1

χ 2 (t )



 +χ t −

yt 1



χ 2 (t ) 

 − 2χ (t ) dy

 √  1 1 π α π (ρ − 2)(ρ − 3) 1 1 ρπ χ (t ) χ 2 (t ) 1 χ 2 (t ) − − − + o A1 (t )χ (t ) + + 3 ρ(1 − ρ) χ (t ) 2 1 − ρ t 2t χ (t ) t 32(ρ(1 − ρ)) 2

for large t, which completes the proof.



Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

9

By arguments similar to Lemma 5, we can get the following result, and omit its proofs here.

1

Lemma 6. Under the conditions of Lemma 5, we have F



t 2

0

ρ

2 t  2

=

22− 2

√

F (t − u)dF (u)

2

1

ρ(1 − ρ)χ 2 (t ) (1 + o(1)) √ α πt

(4.14)

as t → ∞.

3

4

Proof of Theorem 1. For large t and A satisfying 0 < A < f (t ) and χ (A) < α , we have ′

5

   t  f (t ) αu ′ χ (f (t )) e dF ( u ) + α − χ ( A ) f ( t ) − A e eχ (t )−2χ ( 2 ) ( ) F (t − u)dF (u) 0 0 ≤  2t  2t F (t −u) f (t ) F (t − u)dF (u) f (t ) 2 t dF (u) F (2)   1 χ(t ) = o A1 (t )χ (t ) + + χ(t ) t  A

6

7

by using Lemma 5, where f (t ) = χ ′1(t ) . Combining Lemmas 5 and 6, we can get 1

χ2

t  2

tF

2

F ∗ F (t ) 9

t  2

1

=

=

χ2

t  

t 2 F 2

t 

α

8



2

2

2

t 2

f (t )



 f (t )

F (t − u)dF (u) 1 + 0 t

2

f (t )

F (t − u)dF (u) F (t − u)dF (u)



F

 1 + 2



t 2

0

2



t  2

F (t − u)dF (u)

10



√    π 2ρ−5 α π (ρ − 2)(ρ − 3) 1 1 ρπ χ (t ) 1 χ (t ) − − + o A ( t )χ ( t ) + + 1 3 ρ(1 − ρ) χ(t ) 2ρ 1 − ρ t χ (t ) t (ρ(1 − ρ)) 2

for large t, which deduces the desired result.

11



12

Now we turn to prove Theorem 2. To get the desired results, the following three lemmas are needed. The following lemma shows that the product of two 2RV functions is still a 2RV function. Lemma 7. Assume that b(t ) ∈ 2RVβ,ρ2 with auxiliary function A2 (t ) and ρ2 ≤ 0, β ∈ R, and c (t ) ∈ 2RVγ ,ρ3 with auxiliary function A3 (t ) and ρ3 ≤ 0, γ ∈ R. Then b(t )c (t ) ∈ 2RVβ+γ ,max(ρ2 ,ρ3 ) . Proof. By using Lemma 3 for ρ2 < 0 and ρ3 < 0, it follows that b(tx)c (tx) b(t )c (t )

= xβ+γ



13 14

15 16 17

 xρ2 − 1 xρ 3 − 1 1 + A2 (t ) + A3 (t ) + o (A2 (t ) + A3 (t )) ρ2 ρ3

(4.15)

for large t and fixed x > 0. For the cases of ρ2 = 0 or ρ3 = 0, (4.15) also holds due to Definition 2. The proof is complete. 

18

19 20

To prove Theorem 2, we need the following two lemmas the proofs of which are given in Peng and Liao (2015). For c (t ) ∈ RVγ , Lemma 8 deals with the case of γ ≤ −1, while Lemma 9 investigates the case of γ > −1 by assuming that c (t ) is 2RV function further.

22

Lemma 8. Let F , G ∈ Lα , α > 0. Assume that b(t ) = eα t F (t ) ∈ 2RVβ,ρ2 with auxiliary function A2 (t ) for ρ2 ≤ 0, β ∈ R, and

24

αt

c (t ) = e G(t ) ∈ RVγ for γ ≤ −1. For large t, we have



t 2

0

F (t − u)dG(u) = F (t )M1 (β, γ , t ) with M1 (β, γ , t ) given by (3.3).

Lemma 9. Let F , G ∈ Lα , α > 0. Assume that b(t ) = eα t F (t ) ∈ 2RVβ,ρ2 with auxiliary function A2 (t ) and ρ2 ≤ 0, β ∈ R, and c (t ) = eα t G(t ) ∈ 2RVγ ,ρ3 with auxiliary function A3 (t ) and ρ3 ≤ 0, γ > −1, γ + ρ3 + 1 > 0. Then for large t, we have



t 2

0

F (t − u)dG(u) = F (t )M1 (β, γ , t ) holds with M1 (β, γ , t ) given by (3.4).

Proof of Proposition 1. Combining Lemmas 8 and 9, we can derive the desired results.

F

t

2

G

t

2

= 2−β−γ



1 + A2 (t )

−ρ2

2

23

25

26 27 28



Proof of Theorem 2. From (4.15), it follows that

   

21

29 30

−ρ3

 −1 + o (A2 (t ) + A3 (t )) G(t )b(t )

−1 2 + A3 (t ) ρ2 ρ3

31

10

1

Z. Peng, X. Liao / Statistics and Probability Letters xx (xxxx) xxx–xxx

for large t. By arguments similar to Proposition 1, we have t 2

 2

G(t − u)dF (u) + F

0 3

4

    t

2

G

t

2

= G(t )M2 (β, γ , t )

for large t, where M2 (β, γ , t ) is given by (3.6) and (3.7). With the decomposition of convolution tail, for large t we can get F ∗ G(t ) =

t 2

 0

F (t − u)dG(u) +

t 2



G(t − u)dF (u) + F

   

0

6

with M1 (β, γ , t ) given by (3.3) and (3.4). The proof is complete.

7

Acknowledgments

9

10

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

2

G

t

2

= F (t )M1 (β, γ , t ) + G(t )M2 (β, γ , t )

5

8

t



This work was supported by the National Natural Science Foundation of China Grant no. 11171275, the Natural Science Foundation Project of CQ no. cstc2012jjA00029. References Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. University Press, Cambridge. Chistyakov, V.P., 1964. A theorem on sums of independent positive random variables and its applications to branching process. Theory Probab. Appl. 9, 640–648. Chover, J., Ney, P., Wainger, S., 1973a. Functions of probability measures. J. Anal. Math. 26 (1), 255–302. Chover, J., Ney, P., Wainger, S., 1973b. Degeneracy properties of subcritical branching process. Ann. Probab. 1 (4), 663–673. Cline, D.B.H., 1986. Convolution tails, product tails and domains of attraction. Probab. Theory Related Fields 72 (4), 529–557. Cline, D.B.H., 1987. Convolutions of distributions with exponential and subexponential tails. J. Aust. Math. Soc. Ser. A 43, 347–365. Degen, M., Lambrigger, D.D., Segers, J., 2010. Risk concentration and diversification: second-order properties. Insurance Math. Econom. 46 (3), 541–546. de Haan, L., Ferreira, A., 2006. Extreme Value Theory. In: Springer Serries in Operations Research and Financial Engineering, Springer Verlag, New York. Drees, H., 1998. On smooth statistical tail functionals. Scand. J. Statist. 25, 187–210. Embrechts, P., Goldie, C.M., 1980. On closure and factorization properties of subexponential tails. J. Aust. Math. Soc. Ser. A 29, 243–256. Foss, S., Korshunov, D., Zachary, S., 2009. Convolutions of long-tailed and subexponential distributions. J. Appl. Probab. 46 (3), 756–767. Geluk, J.L., de Haan, L., Resnick, S., Stărică, C., 1997. Second-order regular variation, convolution and the central limit theorem. Stochastic Process. Appl. 69, 139–159. Hashorva, E., Ling, C., Peng, Z., 2014. Tail asymptotic expansions for L-statistics. Sci. China Math. 10, 1993–2012. Hua, L., Joe, H., 2011. Second order regular variation and conditional tail expectation of multiple risks. Insurance Math. Econom. 49, 537–546. Lin, J., 2012. Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims. Insurance Math. Econom. 51 (2), 422–429. Lin, J., 2014. Second order tail behaviour for heavy-tailed sums and their maxima with applications to ruin theory. Extremes 17 (2), 247–262. Mao, T., Hu, T., 2013. Second-order properties of risk concentrations without the condition of asymptotic smoothness. Extremes 16 (4), 383–405. Mao, T., Ng, K.W., 2015. Second-order properties of tail probabilities of sums and randomly weighted sums. Extremes http://dx.doi.org/10.1007/s10687015-0218-0. Peng, Z., Liao, X., 2015. Second-order asymptotics for convolution of distributions with light tails. arXiv:1505.03768. Resnick, S.I., 1987. Extreme Values, Regular Variation and Point Processes. Springer Verlag, New York. Yang, Y., Wang, Y., 2013. Tail behavior of the product of two dependent random variables with applications to risk theory. Extremes 16, 55–74.