The Subexponential Product Convolution of Two Weibull-type Distributions

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1 The Subexponential Product Convolution of Two Weibull-type Distributions Yan Liu School of Mathematics and Statistics Wuhan University Wuhan, Hubei 4372, P.R. China Qihe Tang Department of Statistics and Actuarial Science The University of Iowa 241 Schaeffer Hall, Iowa City, IA 52242, USA April 15, 21 Abstract Let X 1 and X 2 be two independent and nonnegative random variables with distributions F 1 and F 2, respectively. This paper proves that if both F 1 and F 2 are of Weibull type and fulfill certain easily verifiable conditions, then the distribution of the product X 1 X 2, called the product convolution of F 1 and F 2, belongs to the class S and, hence, is subexponential. Keywords: Asymptotics; class S ; product convolution; Weibull-type distribution. AMS 2 subject classifications: Primary 6E5; Secondary 62E2 1 Introduction Throughout this paper, all limit relations are for x unless otherwise stated and the symbol means that the quotient of both sides tends to 1. Let X 1 and X 2 be two independent and nonnegative random variables with distributions F 1 and F 2, respectively. The distribution of the sum X 1 + X 2, written as F 1 F 2, is called the sum convolution of F 1 and F 2 ; that is, F 1 F 2 x) = x Corresponding author. Tel.: ; Fax: F 1 x y)df 2 y), x. 1.1) 1

2 The distribution of the product X 1 X 2, written as F 1 F 2, is called the product convolution of F 1 and F 2 ; that is, F 1 F 2 x) = F 1 x/y) df 2 y), x. 1.2) Comparing 1.1) and 1.2), our experience tells that the product convolution is usually much more intractable than the sum convolution. A distribution F on [, ) is said to be subexponential, written as F S, if F x) = 1 F x) > for all x and F F x) 2F x). 1.3) A distribution F on [, ) is said to belong to the class S if F x) > for all x, µ = F y)dy <, and x F x y)f y)dy 2µF x). The class S forms an important subclass of the subexponential class S. Klüppelberg 1988) first introduced the class S and pointed out that the class S contains almost all cited subexponential distributions with finite means. In recent studies in applied probability, researchers have discovered that the class S enjoys a lot of nicer properties than the class S. In studies concerning asymptotic tail probabilities in many fields such as queueing theory and risk theory, it is often a standard assumption that underlying distributions belong to the class S. Recent in-depth studies revealing new properties and proposing important applications of the class S can be found in Asmussen 1998), Asmussen et al. 23), Foss and Zachary 23), Foss et al. 25), and Chen et al. 29), among others. All these works look at the asymptotic tail probabilities of sums and maxima of sums of random variables. The study of subexponentiality of product convolutions was initiated by Breiman 1965), reactivated by Embrechts and Goldie 198) and Cline and Samorodnitsky 1994), and further extended by Tang 26a, 26b), Denisov and Zwart 27), and Tang 28). This study is important because, like sums, products of random variables are a basic element of modelling in applied fields and because the study of the tail behavior of certain stochastic quantities of complicated structure can usually be reduced to the study of the tail behavior of sums and products. The reader is referred to Cline and Samorodnitsky 1994) for further discussions. However, the study of subexponentiality of products is often much more diffi cult than the study of subexponentiality of sums. This is not surprising since, as shown in 1.3), subexponentiality is defined in terms of sums and not products. In this paper we prove that the product convolution of two Weibull-type distributions fulfilling certain mild conditions belongs to the class S. Our work is motivated by an interesting 2

3 observation of Tang 28) that the product convolution of two exponential distributions is subexponential. The rest of this paper consists of three sections. After showing a main result and two related consequences in Section 2, we prepare two propositions in Section 3 and prove the main result in Section 4. 2 Main result A distribution F on [, ) is said to be of Weibull type if F x) = cx) exp { bx)x p, x, 2.1) where p > is a constant, called the shape parameter, c ) : [, ), ) and b ) : [, ), ) are two measurable functions such that the limits cx) c and bx) b exist and are positive. If only for the purpose of definition, the function c ) in 2.1) can actually be eliminated when replacing bx) by bx) = bx) x p log cx). Nevertheless, we still separate the two functions c ) and b ) as in 2.1) because in 2.5) we make an assumption on the differentiability of the function b ). Let X 1 and X 2 be two independent and nonnegative random variables with distributions F 1 and F 2, respectively, and let G be the distribution of their product Y = X 1 X ) Thus, G = F 1 F 2. Assume that both F 1 and F 2 are of Weibull type with tails F i x) = c i x) exp { b i x)x p i, x, i = 1, 2, 2.3) for some constants p i > and some measurable functions c i ) : [, ), ) and b i ) : [, ), ) satisfying c i x) c i > and b i x) b i >. Further assume that p p 1 2 > 1 2.4) and that, for i {1, 2 determined by p i = max {p 1, p 2, the function b i ) is eventually continuously differentiable with derivative b i ) satisfying b i p i < lim inf b ix)x lim sup b ix)x < b i p i p p ). 2.5) Theorem 2.1. Consider the product in 2.2). Under conditions 2.3) 2.5) we have G = F 1 F 2 S. 3

4 Note that condition 2.4) cannot be removed from Theorem 2.1. A simple counterexample is that F 1 x) = F 2 x) = Φx ) for x, where Φ ) is the standard normal distribution. In this case, p 1 = p 2 = 2 and it is easy to verify that F 1 F 2 has certain finite exponential moments and, hence, it is even not heavy tailed. Professor Enkelejd Hashorva has kindly brought to our attention a closely related but different result obtained by Arendarczyk and Dębicki 29). According to this reference, a distribution F on [, ) is said to have a Weibullian tail if F x) Cx γ exp { βx α, α, β, C >, γ, ). Their Lemma 2.1 shows that if both F 1 and F 2 have Weibullian tails then their product convolution F 1 F 2 also has a Weibullian tail with explicitly given parameters. A distribution F on [, ) is said to belong to the class Lγ) for γ if the relation F x y) e γy F x) holds for all y. When γ > the distribution F Lγ) is usually said to have an exponential tail, while when γ = the class Lγ) reduces to the well-known class of long-tailed distributions. Studies on this and related distribution classes can be found in Cline 1986, 1987), Pakes 24, 27), Watanabe 28), and Albin 28), among others. Applying Karamata s representation theorem for regularly varying functions see Bingham et al. 1987) and Klüppelberg 1989)), we know that F Lγ) if and only if F ) can be expressed as { x F x) = cx) exp γy)dy, x, 2.6) where c ) : [, ), ) and γ ) : [, ), ) are measurable functions such that the limits cx) c > and γx) γ exist. As can be seen from the proof of Corollary 2.1 below, the class Lγ) for γ > forms a subclass of Weibull-type distributions with shape parameter 1. Corollary 2.1. If F 1 Lγ 1 ) for some γ 1 > and F 2 is of Weibull type with shape parameter < p 2 1, then G = F 1 F 2 S. In particular, if F i Lγ i ) for some γ i > for i = 1, 2, then G = F 1 F 2 S. Proof. According to 2.6), F 1 ) can be expressed as { x F 1 x) = c 1 x) exp γ 1 y)dy, x, for some measurable functions c 1 ) : [, ), ) and γ 1 ) : [, ), ) satisfying c 1 x) c 1 > and γ 1 x) γ 1. We can always construct a distribution F with { x F x) = exp γ y)dy, x, 2.7) 4

5 for some continuous function γ ) : [, ), ) such that γ 1 y) γ y) dy <. Clearly, there exists some positive constant c such that F 1 x) c F x). By Lemma A.5 of Tang and Tsitsiashvili 24), we obtain Rewrite 2.7) in the style of 2.3) so that Gx) c F F 2 x). F x) = exp { b x)x, x, with b x) = x 1 x γ y)dy. Note that b x)) x. Hence, by Theorem 2.1, F F 2 S. By Corollary 1.1C1) of Tang 28), G L). Therefore, by the closure of S under tail equivalence as shown in Theorem 2.1b) of Klüppelberg 1988), we know that G S. Let {Bt), t be a standard Brownian motion and let τ be a nonnegative random variable independent of {Bt), t. Denote by B τ) = sup Bt) t τ the maximum of the Brownian motion over a random time interval [, τ]. It is well known that if τ is exponentially distributed, so is B τ); see, for example, 1.1.2) of Borodin and Salminen 22). The following example catches a subexponential tail of B τ) when τ follows a heavy-tailed Weibull-type distribution. Example 2.1. If τ follows a Weibull-type distribution with shape parameter < p < 1, then the distribution of B τ) belongs to the class S. Proof. By conditioning on τ, Pr B τ) > x) = 2 Pr Bτ) > x), x. Thus, we only need to prove that the distribution of B + τ) = Bτ) belongs to the class S. Clearly, B + τ) D = X τ, 2.8) where X and τ are independent, X follows the distribution Φx ) for x with Φ ) the standard normal distribution, and D = means equality in distribution. Elementary calculation gives Pr X > x) 1 { exp log x ) x 2, 2π 2 x 2 which shows that the distribution of X is of Weibull type with shape parameter 2. By the assumption on τ, it is easy to see that τ follows a Weibull-type distribution too with shape parameter 2p. Therefore by Theorem 2.1, the distribution of X τ in 2.8) belongs to the class S. 5

6 3 Preliminaries In the rest of this paper, for two positive functions a ) and b ), we write ax) bx) if < lim inf ax)/bx) lim sup ax)/bx) < and write ax) bx) or bx) ax) if lim sup ax)/bx) 1. Let C denote an absolute positive constant whose value may vary from line to line. Proposition 3.1. Let F 1 and F 2 be as given in 2.3). Assume that b 1 = b 2 = 1 and < p 2 p 1 <, and write a = p 1 /p 2 ) 1/p 1+p 2 ). If b 1 ) is differentiable and its derivative satisfies then, for all ε >, Gx) p 1 < lim inf 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) b 1x)x lim sup b 1x)x <, 3.1) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 b 2 y)y p 2 y p 2 1 dy. 3.2) Proof. For arbitrarily fixed small ε >, we choose δ > such that 2 + ε)1 δ) > 21 + δ). Since then Gx) Pr X 1 > x p 2/p 1 +p 2 ) ) Pr X 2 > x p 1/p 1 +p 2 ) ) exp { b 1 x p 2 /p 1 +p 2 ) ) x p 1p 2 /p 1 +p 2 ) b 2 x p 1 /p 1 +p 2 ) ) x p 1p 2 /p 1 +p 2 ) exp { 21 + δ)x p 1p 2 /p 1 +p 2 ), F ε) 1/p 2 ax p 2/p 1 +p 2 ) ) exp { b ε) 1/p 2 ax p 2/p 1 +p 2 ) ) 2 + ε) p 1/p 2 a p 1 x p 1p 2 /p 1 +p 2 ) exp { 2 + ε)1 δ)x p 1p 2 /p 1 +p 2 ) = ogx)). 3.3) Likewise, F ε) 1/p 2 ax p 1/p 1 +p 2 ) ) = ogx)). 3.4) Therefore, Gx) = F 1 x/y) df 2 y) 2+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 2+ε) 1/p 2 a 1 x p 1 /p 1 +p 2 ) F 1 x/y) df 2 y). 3.5) Using integration by parts and substituting 3.3) and 3.4), we obtain that Gx) 2+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 2+ε) 1/p 2 a 1 x p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 df 2 y) 2+ε) 1/p 2 ax p 1 /p 1 +p 2 ) = ogx)) + F 2 y)d exp { b 1 x/y) x/y) p 1. 2+ε) 1/p 2 a 1 x p 1 /p 1 +p 2 ) 6

7 By 3.1), for all large x, the exponential function behind the differential operator is strictly increasing in y in the indicated interval. It follows from 2.3) and 3.1) that Gx) 2+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 2+ε) 1/p 2 a 1 x p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 b 2 y)y p 2 y p 2 1 dy. Split the integral on the right-hand side above into three parts as I 1 x) + I 2 x) + I 3 x) = To obtain 3.2), it suffi ces to prove that 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 2+ε) 1/p 2 ax p 1 /p 1 +p 2 ) ε) 1/p 2 a 1 x p 1 /p 1 +p 2 ) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) I 1 x) + I 3 x) = oi 2 x)). 3.6) Note that x/y) p 1 +y p 2, as a function of y, decreases when < y ax p 1/p 1 +p 2 ) and increases when y ax p 1/p 1 +p 2 ). On the one hand, we have and 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) I 1 x) exp { 1 ε 3) x/y) p 1 + y p 2 ) y p2 1 dy 2+ε) 1/p 2 a 1 x p 1 /p 1 +p 2 { ) Cx p 1p 2 /p 1 +p 2 ) exp 1 ε 3) ) 1 + ε) p 1/p ε) 1 a p 2 x p 1p 2 /p 1 +p 2 ) I 3 x) 2+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) exp { 1 ε 3) x/y) p 1 + y p 2 ) y p 2 1 dy Cx p 1p 2 /p 1 +p 2 ) exp { 1 ε 3) 1 + ε) p 1/p ε) a p 2 ) x p 1p 2 /p 1 +p 2 ). On the other hand, it is easy to see that I 2 x) ax p 1 /p 1 +p 2 ) 1+ε/2) 1/p 2 ax p 1 /p 1 +p 2 ) + 1+ε/2) 1/p 2 ax p 1 /p 1 +p 2 ) ax p 1 /p 1 +p 2 ) ) exp { 1 + ε 3) x/y) p 1 + y p 2 ) y p2 1 dy { Cx p 1p 2 /p 1 +p 2 ) exp 1 + ε 3) ) 1 + ε/2) p 1/p ε/2) 1 a p 2 { +Cx p 1p 2 /p 1 +p 2 ) exp 1 + ε 3) 1 + ε/2) p 1/p ε/2) a p 2 To prove 3.6), use Taylor s expansion to expand both 1 + ε/2) p 1/p ε/2) 1 a p 2 x p 1p 2 /p 1 +p 2 ) ) x p 1p 2 /p 1 +p 2 ). and 1 + ε) p 1/p ε) 1 a p 2 7

8 in ε up to the ε 2 term. Then we find that the coeffi cients of the constant terms and the ε terms are equal, but the coeffi cient of the ε 2 term of the first is smaller than the corresponding coeffi cient of the second. Therefore, for all small ε >, 1 + ε 3 ) 1 + ε/2) p 1/p ε/2) 1 a p 2) < 1 ε 3) 1 + ε) p 1/p ε) 1 a p 2 ), 3.7) which implies that I 1 x) = oi 2 x)). Similarly as above, for all small ε >, ) 1 + ε ε/2) p 1/p ε/2) a 2) p < 1 ε 3) ) 1 + ε) p 1/p ε)a p 2, 3.8) which implies that I 3 x) = oi 2 x)). Hence, relation 3.6) holds. This proves that relation 3.2) holds for all small ε > hence, for all ε > ). Proposition 3.2. Let F 1 and F 2 be as given in 2.3) with < p 2 p 1 < satisfying 2.4). Further assume that b 1 ) is differentiable with b 1x) = O x 1 ). Then G = F 1 F 2 L). Proof. It suffi ces to prove that Gx + 1) Gx). For arbitrarily fixed small ε >, write ε, x) = 2 + ε) 1/p 2 a 1 x p 1/p 1 +p 2 ), 2 + ε) 1/p 2 ax p 1/p 1 +p 2 ) ], where a = p 1 /p 2 ) 1/p 1+p 2 ) as before. By 3.3) 3.5), F 1 x + 1)/y) Gx + 1) F 1 x/y) df 2 y) Lx)Gx), F 1 x/y) where Lx) = F 1 x + 1)/y) inf y ε,x) F 1 x/y) y ε,x) = inf y ε,x) We need to prove that, uniformly for all y ε, x), c 1 x + 1)/y) exp { b 1 x + 1)/y) x + 1)/y) p 1 c 1 x/y) exp { b 1 x/y) x/y) p 1. Ix, y) = b 1 x + 1)/y) x + 1)/y) p 1 b 1 x/y) x/y) p ) Since b 1 x) is differentiable with b 1x) = O x 1 ), we have, uniformly for all y ε, x), Ix, y) = b 1 x + 1)/y) b 1 x/y)) x + 1)/y) p 1 + b 1 x/y) x + 1)p1 x p 1 y p 1 x+1)/y x + 1)/y) p 1 b 1z) dz + b 1 x/y) x + 1)p 1 x p 1 Cx p 1p 2 /p 1 +p 2 ) Cx p 1p 2 /p 1 +p 2 ) 1. x/y x+1)/y x/y Hence, by 2.4), relation 3.9) holds. z 1 dz + Cx p 1p 2 /p 1 +p 2 ) 1 y p 1 8

9 4 Proof of Theorem 2.1 If we have proven Theorem 2.1 for b 1 = b 2 = 1, then using the identity 1 ) ) Y = b 1/p 1 b 1/p 1 1 b 1/p 2 1 X 1 b 1/p 2 2 X 2, 2 the extension to the general case is straightforward because b 1/p 1 1 X 1 and b 1/p 2 2 X 2 still have Weibull-type distributions and, by the definition of the class S, the) distribution ) of Y belonging to S is equivalent to the distribution of the product b 1/p 1 1 X 1 b 1/p 2 2 X 2 belonging to S. Therefore, we may assume without loss of generality that b 1 = b 2 = 1. We may also assume without loss of generality that < p 2 p 1 <. Hence, relation 2.5) holds with i = 1. Recall that a = p 1 /p 2 ) 1/p 1+p 2 ). In the proof below ε denotes a positive constant which can be arbitrarily small. Denote by w ) the function on the right-hand side of 3.2); that is, wx) = 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 b 2 y)y p 2 y p 2 1 dy, x. We need to verify that wx) is eventually non-increasing. Direct computation shows that where J 1 x) = p 1 p 1 + p ε)a p 2 x p 1p 2 /p 1 +p 2 ) 1 w x) = J 1 x) J 2 x) J 3 x), 4.1) exp { b ε) 1/p 2 a 1 x p 2/p 1 +p 2 ) ) 1 + ε) p 1/p 2 x p 1p 2 /p 1 +p 2 ) exp { b ε) 1/p 2 ax p 1/p 1 +p 2 ) ) 1 + ε)a p 2 x p 1p 2 /p 1 +p 2 ), and J 2 x) = p 1 p 1 + p ε) 1 a p 2 x p 1p 2 /p 1 +p 2 ) 1 exp { b ε) 1/p 2 a 1 x p 2/p 1 +p 2 ) ) 1 + ε) p 1/p 2 x p 1p 2 /p 1 +p 2 ) exp { b ε) 1/p 2 ax p 1/p 1 +p 2 ) ) 1 + ε) 1 a p 2 x p 1p 2 /p 1 +p 2 ), J 3 x) = It is easy to see that 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 b 2 y)y p 2 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) b 1 x/y) x/y) p 1y ) 1 + b x p x/y) p 1 y p2 1 dy. J 1 x) Cx p 1p 2 /p 1 +p 2 ) 1 exp { 1 ε 3) 1 + ε) p 1/p ε)a p 2 ) x p 1p 2 /p 1 +p 2 ). 9 y p 1

10 By 2.5) and the monotonicity of the function x/y) p 1 + y p 2 in y, J 3 x) C 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 b 2 y)y p 2 xp1 1 y p2 1 dy 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) y p 1 Cx 2p 1p 2 /p 1 +p 2 ) 1 1+ε/2) 1/p 2 ax p 1 /p 1 +p 2 ) ax p 1 /p 1 +p 2 ) exp { b 1 x/y) x/y) p 1 b 2 y)y p 2 y 1 dy Cx 2p 1p 2 /p 1 +p 2 ) 1 exp { 1 + ε 3) 1 + ε/2) p 1/p ε/2)a p 2 ) x p 1p 2 /p 1 +p 2 ). Therefore for all small ε > such that 3.7) 3.8) hold, J 1 x) J 3 x) C exp { 1 ε 3 ) ) 1 + ε) p 1/p ε)a p 2 x p 1 p 2 /p 1 +p 2 ) x p 1p 2 /p 1 +p 2 ) exp { 1 + ε 3 ) 1 + ε/2) p 1/p ε/2)a p 2) x p 1 p 2 /p 1 +p 2), which implies that J 1 x)/ J 3 x). Likewise, J 2 x)/ J 3 x). It follows from 4.1) that w x) J 3 x). 4.2) Hence, w x) is negative for all large x. This proves the eventual monotonicity of wx). Introduce a distribution W on [, ) with tail { 1, x x, W x) = wx), x > x, where x > is some large number such that wx) is non-increasing for x x and wx ) 1. By Propositions above and Theorem 2.1b) of Klüppelberg 1988), it suffi ces to verify that W S, which amounts to x lim W x y)w y) dy = 2 W x) W y)dy. 4.3) By the definition of W, following the proof of Proposition 3.2, we can obtain that W L). Write Rx) = log W x) and rx) = R x). Recall that x/y) p 1 + y p 2, as a function of y, attains its minimum at y = ax p 1/p 1 +p 2 ), W x) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) 1+ε) 1/p 2 ax p 1 /p 1 +p 2 ) exp { 1 ε) x/y) p 1 + y p 2 ) y p 2 1 dy Cx p 1p 2 /p 1 +p 2 ) exp { 1 ε) + a p 2 ) x p 1p 2 /p 1 +p 2 ), which implies that Rx) 1 ε) + a p 2 ) x p 1p 2 /p 1 +p 2 ). 4.4) Write l 1 = lim sup b 1x)x. It follows from 4.2) and 2.5) that, for all x > x, rx) = W x) ) W x) w x) wx) J 3x) wx) 1 + ε)p 1/p 2 l 1 + p 1 ) x p 1p 2 /p 1 +p 2 ) ) 1

11 Furthermore, by 2.4), uniformly for all y x/2, x x y v p 1p 2 /p 1 +p 2 ) 1 dv yx y) p 1p 2 /p 1 +p 2 ) 1 y p 1p 2 /p 1 +p 2 ). 4.6) Using 4.4) 4.6) we obtain that, for all y x/2 and all large x >, W x y)w y) W x) { x = exp rv)dv Ry) x y { exp 1 + 2ε) p 1/p 2 l 1 + p 1 ) x x y v p 1p 2 /p 1 +p 2 ) 1 dv 1 2ε) + a ) p 2 y p 1p 2 /p 1 +p 2 ) exp { 1 + 2ε) p 1/p 2 l 1 + p 1 ) 1 2ε) + a p 2 )) y p 1p 2 /p 1 +p 2 ). 4.7) By 2.5) and the definition of a, we obtain Consequently, for all small ε >, l 1 + p 1 ) < p 1 + p 2 p 2 = + a p ε) p 1/p 2 l 1 + p 1 ) < 1 2ε) + a p 2 ). Therefore, the right-hand side of 4.7) as a function of y is integrable over [, ). Applying the dominated convergence theorem, we obtain 4.3), as x lim W x y)w y) W x) x/2 dy = 2 lim W x y)w y) dy = 2 W x) W y)dy. Acknowledgments. The authors would like to thank Enkelejd Hashorva, Dimitrios Konstantinides, and an anonymous referee for their useful comments on an earlier version of this paper. The majority of this work was completed during a research visit of the first author to the University of Iowa. She would like to thank the Department of Statistics and Actuarial Science for its excellent hospitality. This work was partially supported by the National Natural Science Foundation of China No and No ), the Ministry of Education of China No ), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars. References [1] Albin, J. M. P. A note on the closure of convolution power mixtures random sums) of exponential distributions. J. Aust. Math. Soc ), no. 1,

12 [2] Arendarczyk, M.; Dębicki, K. Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli 16 21), to appear. [3] Asmussen, S. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab ), no. 2, [4] Asmussen, S.; Foss, S.; Korshunov, D. Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Probab ), no. 2, [5] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular Variation. Cambridge University Press, Cambridge, [6] Borodin, A. N.; Salminen, P. Handbook of Brownian Motion Facts and Formulae. Second edition. Birkhäuser Verlag, Basel, 22. [7] Breiman, L. On some limit theorems similar to the arc-sin law. Theory Prob. Appl ), no. 2, [8] Chen, G.; Wang, Y.; Cheng, F. The uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. Stoch. Models 25 29), no. 3, [9] Cline, D. B. H. Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields ), no. 4, [1] Cline, D. B. H. Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A ), no. 3, [11] Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent random variables. Stochastic Process. Appl ), no. 1, [12] Denisov, D.; Zwart, B. On a theorem of Breiman and a class of random difference equations. J. Appl. Probab ), no. 4, [13] Embrechts, P.; Goldie, C. M. On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A ), no. 2, [14] Foss, S.; Palmowski, Z.; Zachary, S. The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk. Ann. Appl. Probab ), no. 3, [15] Foss, S.; Zachary, S. The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Probab ), no. 1, [16] Klüppelberg, C. Subexponential distributions and integrated tails. J. Appl. Probab ), no. 1, [17] Klüppelberg, C. Subexponential distributions and characterizations of related classes. Probab. Theory Related Fields ), no. 2, [18] Pakes, A. G. Convolution equivalence and infinite divisibility. J. Appl. Probab ), no. 2,

13 [19] Pakes, A. G. Convolution equivalence and infinite divisibility: corrections and corollaries. J. Appl. Probab ), no. 2, [2] Tang, Q. On convolution equivalence with applications. Bernoulli 12 26a), no. 3, [21] Tang, Q. The subexponentiality of products revisited. Extremes 9 26b), no. 3-4, [22] Tang, Q. From light tails to heavy tails through multiplier. Extremes 11 28), no. 4, [23] Tang, Q.; Tsitsiashvili, G. Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. in Appl. Probab ), no. 4, [24] Watanabe, T. Convolution equivalence and distributions of random sums. Probab. Theory Related Fields ), no. 3-4,