Asymptotic results for heavy-tailed distributions using defective renewal equations

Statistics and Probability Letters 79 (2009) 774–779

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Asymptotic results for heavy-tailed distributions using defective renewal equations Georgios Psarrakos Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, 83200 Samos, Greece

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Article history: Received 8 January 2008 Received in revised form 19 October 2008 Accepted 28 October 2008 Available online 8 November 2008

In this note, we investigate the asymptotic behavior of two special functions that satisfy defective renewal equations and have a key role in risk theory. Heavy-tailed distributions are important in our analysis. Furthermore, applying our results to the renewal and classical models of risk theory yields a generalisation of a well-known asymptotic formula of [Embrechts, P., Veraverbeke, N., 1982. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72]. © 2008 Elsevier B.V. All rights reserved.

1. Introduction and preliminaries Consider a distribution function (d.f.) F (x) supported on [0, ∞) with F (0) = 0. Let m(u) satisfies the defective renewal equation m(u) = φ

u

Z

m(u − t ) dF (t ) + υ(u),

u ≥ 0,

(1.1)

0

where 0 < φ < 1 and υ(u) is a continuous function supported on [0, ∞). Equations of the form (1.1) arise repeatedly in various areas of applications of probability theory such as risk theory, queueing theory and branching theory; for more details see Willmot and Lin (2001, Chapter 9), and the references therein. The solution of Eq. (1.1) is m(u) =

Z

1 1−φ

u

υ(u − t ) dH (t ),

(1.2)

0−

n ∗n n ∗n where H (u) = n=0 (1 − φ)φ F (u) is a compound geometric d.f. with tail H (u) = 1 − H (u) = n=1 (1 − φ) φ F (u). ∗n The function F (u) = 1 − F ∗n (u) is the n-fold convolution of the d.f. F . The tail H (u) satisfies the defective renewal equation (see Willmot and Lin (2001, p. 156))

P∞

H (u) = φ

P∞

u

Z

H (u − t ) dF (t ) + φ F (u),

(1.3)

0

whose solution is given by H (u) =

φ 1−φ

Z

u

F (u − t ) dH (t ).

(1.4)

0−

In Eqs. (1.2) and (1.4), the interval in the range of integration is considered closed. Note that H (u) has a mass of H (0) = 1 −φ at 0. Next we define an important class of heavy-tailed distributions; see Embrechts et al. (1997) for a review.

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G. Psarrakos / Statistics and Probability Letters 79 (2009) 774–779

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Definition 1.1. A d.f. B(x) with support on (0, ∞) is subexponential, and we denote B ∈ S , if lim

x→∞

B∗2 (x) B(x)

= 2.

(1.5)

The class of subexponential distributions was independently introduced by Chistyakov (1964) and Chover et al. (1973) in the context of branching processes. As a consequence of (1.5) it is well known that, if B ∈ S , then lim

x→∞

B∗n (x)

= n for all n ≥ 2,

B(x)

and B(x − y)

= 1 for every fixed y, (1.6) B(x) see Embrechts et al. (1979) and the references therein. Moreover, if the d.f. B satisfies the relation (1.6), then we say that B is long-tailed, and we denote B ∈ L. By Chistyakov (1964), it is known that S ⊂ L. Examples of subexponential distributions include Pareto distributions, lim

x→∞

B(x) =



a

b b+x

,

a, b > 0,

the Weibull distribution r B(x) = e−cx ,

c > 0, 0 < r < 1,

and the Benktander type I distribution 2 B(x) = (1 + 2(b/a) ln x) e−b(ln x) −(a+1) ln x ,

a, b > 0 .

Note that the result S 6= L is non-trivial; relevant examples are to be found in Embrechts and Goldie (1980) and Pitman (1980). For more details see Embrechts et al. (1997, p.51). The main purpose of this note is to obtain asymptotic formulas for functions satisfying defective renewal equations, see Section 2. As an application of our analysis, in Section 3, we present some new asymptotic results in the renewal and classical models of risk theory. In particular, in this section we extend a result of Embrechts and Veraverbeke (1982). 2. Asymptotic results We introduce the function m(u, y) that satisfies the defective renewal equation m(u, y) = φ

Z

u

m(u − t , y) dF (t ) + υ(u + y),

(2.1)

0

and is written in the form (solution) m(u, y) =

Z

1 1−φ

u

υ(u + y − t ) dH (t ).

(2.2)

0−

For more details about the solution of a defective renewal equation, see Asmussen (1987, Chapter VI). Theorem 2.1. If the function υ(u) is nonincreasing and H (u) ∈ L, then lim

u→∞

m(u, y) m(u + y)

= 1,

y ≥ 0.

Proof. By (1.2) and (2.2), we observe that m(u, y) = m(u + y) −

1 1−φ

u+y

Z

υ(u + y − t ) dH (t ).

(2.3)

u

By hypothesis, the function υ(u + y − t ) is nondecreasing in t ∈ [u, u + y]. Thus, by (2.3), we have L(u, y) ≤ m(u, y) ≤ U (u, y),

(2.4)

where L(u, y) := m(u + y) −

1 1−φ

υ(y) [H (u) − H (u + y)]

and U (u, y) := m(u + y) −

1 1−φ

[H (u) − H (u + y)].

776

G. Psarrakos / Statistics and Probability Letters 79 (2009) 774–779

Dividing the function L(u, y) by H (u + y) yields L ( u, y ) H (u + y)

"

m(u + y)

=

− υ(y)

H (u + y)

#

H (u) H (u + y)

−1 .

By the fact that H (u) ∈ L, letting above u → ∞, it follows lim

u→∞

L(u, y)

m(u + y)

= lim

H (u + y)

H (u + y)

u→∞

.

Similarly we obtain lim

u→∞

U (u, y)

m(u + y)

= lim

H (u + y)

H (u + y)

u→∞

.

As a consequence, by dividing (2.4) by H (u + y) and taking u → ∞, we complete the proof.



In the following proposition, we derive an asymptotic result for m(u, y) when y tends to infinity. Proposition 2.1. If the function υ(u) is nonincreasing and limu→∞ υ(u + y)/υ(u) = 1 for any y ≥ 0, then lim

y→∞

m(u, y) 1 = [1 − H (u)], υ(u + y) 1−φ

u ≥ 0.

Proof. By (2.2) and the fact that υ(u + y − t ) is nondecreasing in t ∈ [0, u], we obtain 1 1−φ

1

υ(u + y) [1 − H (u)] ≤ m(u, y) ≤

1−φ

υ(y) [1 − H (u)].

Dividing by υ(u + y) and letting y → ∞, the result follows.



Next we consider the function m(u, x, y) satisfying the defective renewal equation u

Z

m(u, x, y) = φ

m(u − t , x, y) dF (t ) + υ(max{u + y, x + y}),

(2.5)

0

whose solution is m(u, x, y) =

u

Z

1 1−φ

υ(max{u + y − t , x + y}) dH (t ).

(2.6)

0−

Theorem 2.2. If the function υ(u) is nonincreasing and H (u) ∈ L, then lim

u→∞

m(u, x, y) m(u + y)

= 1,

x, y ≥ 0.

Proof. When u > x, by (2.2) and (2.6), we have u −x

Z

1

m(u, x, y) =

1

υ(u + y − t ) dH (t ) + υ(x + y) [H (u − x) − H (u)] 1−φ Z u 1 1 = m(u, y) − υ(u + y − t ) dH (t ) + υ(x + y) [H (u − x) − H (u)]. 1 − φ u −x 1−φ 1−φ

0−

Dividing each side of the above equation with H (u + y) yields m(u, x, y) H (u + y)

m(u, y)

=

H (u + y)

−

1

Z

1

1 − φ H (u + y)

u

υ(u + y − t ) dH (t ) + u −x

1 1−φ

" υ(x + y)

H (u − x) H (u + y)

−

H ( u) H (u + y)

# .

Since the function υ(u + y − t ) is nondecreasing in t ∈ [u − x, u], it is straightforward to see that m(u, y) H (u + y)

−

"

1 1−φ

[υ(y) − υ(x + y)]

H ( u − x) H (u + y)

By Theorem 2.1 and the fact that H (u) ∈ L, we derive lim

u→∞

m(u, y) H (u + y)

= lim

u→∞

m(u + y) H (u + y)

−

H ( u) H (u + y)

# ≤

m(u, x, y) H (u + y)

≤

m(u, y) H (u + y)

.

(2.7)

G. Psarrakos / Statistics and Probability Letters 79 (2009) 774–779

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and lim

u→∞

H ( u − x) H (u + y)

= lim

H ( u)

u→∞

H (u + y)

= 1.

Thus, by (2.7), and letting u → ∞, the result follows.



Finally, we present asymptotic formulas for m(u, x, y) when x and y tend to infinity, respectively. Proposition 2.2. If the function υ(u) is nonincreasing and limu→∞ υ(u + y)/υ(u) = 1 for any y ≥ 0, then lim

x→∞

m(u, x, y)

υ(u + x + y)

=

1 1−φ

[1 − H (u)],

u, y ≥ 0.

Proof. When u ≤ x, by dividing (2.6) with υ(u + x + y), m(u, x, y)

=

υ(u + x + y)

υ(x + y) [1 − H (u)]. 1 − φ υ(u + x + y) 1

Taking x → ∞, the proof is complete.



Proposition 2.3. If the function υ(u) is nonincreasing and limu→∞ υ(u + y)/υ(u) = 1 for any y ≥ 0, then lim

y→∞

m(u, x, y) 1 = [1 − H (u)], υ(u + x + y) 1−φ

u, x ≥ 0.

Proof. We have two cases. If u ≤ x, then we work as in Proposition 2.2, and if u > x, then we work as in Proposition 2.1. Both cases conclude the desired result.  Remark 2.1. If F (u) ∈ S , then by Theorem A3.20 of Embrechts et al. (1997), H (u) ∈ S . Thus, by the fact that S ⊂ L, it follows that the assumption H (u) ∈ S in Theorems 2.1 and 2.2 can be substituted by the assumption F (u) ∈ S . 3. Applications in risk theory Consider the renewal risk model, often referred to as the Sparre Andersen risk model. In this model, the insurer’s surplus at time t, denoted by U (t ), is given by U (t ) = u + ct −

Nt X

Yk ,

k =1

where u ≥ 0 is the initial surplus, c is the rate of premium income per unit time and Nt is the number of claims in the time interval (0, t ]. The individual claim amounts Y1 , Y2 , . . . are positive, independent and identically distributed (i.i.d.) random variables with a continuous common d.f. P (t ) = Pr (Y1 ≤ t ), tail P (t ) = 1 − P (t ) = Pr (Y1 > t ) and mean E (Y1 ) = µ < ∞. These claim amounts are also independent of Nt , and the corresponding interclaim times T1 , T2 , . . . are generally distributed with common mean E (T1 ). We assume that c = (1 + θ ) µ/E (T1 ), where θ > 0 is known as the relative safety loading. The probability of (ultimate) ruin is defined by

  ψ(u) = Pr inf U (t ) < 0|U (0) = u , t >0

u ≥ 0.

The time of ruin (if it occurs) is defined to be T = inf{t > 0|U (t ) < 0}. Thus, the probability of ruin can be written as

ψ(u) = Pr (T < ∞|U (0) = u), and since we assume that µ < c E (T1 ), ruin is not certain to occur. A well-known expression for thePruin probability in the ∞ n ∗n Sparre Andersen model is the Pollaczeck–Khinchine formula, namely, ψ(u) = H (u) = 1 − H (u) = n=1 (1 − φ) φ F (u), where F (u) is the ladder height d.f. (see Asmussen (2000)), φ = ψ(0) and H (u) is the probability of non-ruin, see for instance Rolski et al. (1999). The probability of ruin satisfies also the defective renewal equation

ψ(u) = φ

u

Z

ψ(u − t ) dF (t ) + φ F (u),

(3.1)

0

see, e.g., Willmot and Lin (2001). It is of interest to consider the size of the deficit at ruin, and thus, we define H (u, y) = Pr (|U (T )| ≤ y, T < ∞|U (0) = u). This d.f. was introduced by Gerber et al. (1987) and represents the probability that, starting with a surplus u, ruin occurs and the deficit |U (T )| at the time of ruin T does not exceed y ≥ 0. It is a defective d.f. with tail H (u, y) = ψ(u) − H (u, y) = Pr (|U (T )| > y,

T < ∞|U (0) = u),

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G. Psarrakos / Statistics and Probability Letters 79 (2009) 774–779

and satisfies limy→∞ H (u, y) = H (u, 0) = ψ(u) < 1. By Willmot (2002), the function H (u, y) satisfies the defective renewal equation H (u, y) = φ

u

Z

H (u − t , y) dF (t ) + φ F (u + y).

(3.2)

0

By relations (1.1), (2.1), (3.1) and (3.2), we observe that if υ(u) = φ F (u), then m(u) = ψ(u) and m(u, y) = H (u, y). Thus, in the renewal risk model by Theorem 2.1 and Proposition 2.1, we have the following results. Corollary 3.1. If probability of non-ruin H (u) ∈ L, then lim

u→∞

H (u, y) = 1, ψ(u + y)

y ≥ 0.

Corollary 3.2. If the ladder height d.f. F (u) ∈ L, then lim

y→∞

H (u, y)

=

F (u + y)

φ

[1 − ψ(u)],

1−φ

u ≥ 0.

Next we generalise the well-known asymptotic result of Embrechts and Veraverbeke (1982, Theorem 4.6) for the ratio Rx P¯ (t )dt, defined for x ≥ 0 with tail 0

ψ(u)/Pe (u). At this point, we consider the equilibrium d.f. of P, Pe (x) = µ−1 R∞ Pe (x) = µ−1 x P¯ (t )dt.

Proposition 3.1 (Embrechts and Veraverbeke, 1982, Theorem 4.6). If the equilibrium d.f. of the claim amount d.f. is subexponential, i.e. Pe (u) ∈ S , then lim

u→∞

ψ(u) Pe (u)

=

φ 1−φ

.

Theorem 3.1. If Pe (u) ∈ S , then lim

u→∞

H (u, y) Pe (u + y)

φ

=

1−φ

,

y ≥ 0.

Proof. A well-known result is that the d.f. Pe (u) ∈ S if and only if H (u) ∈ S (see for example Goldie and Klüppelberg (1998)). Thus, by Corollary 3.1 and Proposition 3.1, lim

u→∞

H (u, y) Pe (u + y)

= lim

u→∞

H (u, y) ψ(u + y) φ .  = ψ(u + y) Pe (u + y) 1−φ

In the following results, weP consider the classical risk model, that is, Nt is an ordinary Poisson process. Here, it holds that ∞ n ∗n n=1 (1 − φ) φ Pe (u) (see Bowers et al. (1986, Chapter 12)). The probability of ruin satisfies the defective renewal equation in (3.1), where we set Pe instead of F . Consider the tail of the joint distribution of the surplus prior to and at ruin

ψ(u) = H (u) = 1 − H (u) =

H (u, x, y) = P (U (T −) > x, |U (T )| > y,

T < ∞|U (0) = u).

By Schmidli (1999); see also Dickson (1992) and Psarrakos and Politis (2008), the function H (u, x, y) satisfies the defective renewal equation H (u, x, y) = φ

u

Z

H (u − t , x, y) dPe (t ) + φ Pe (max{u + y, x + y}).

(3.3)

0

By (2.5) and (3.3), we have that if υ(u) = φ Pe (u), then m(u, x, y) = H (u, x, y). Thus, Theorem 2.2 and Propositions 2.2 and 2.3 imply the following results. Corollary 3.3. If H (u) ∈ L, then lim

u→∞

H (u, x, y) = 1, ψ(u − x + y)

x, y ≥ 0 .

G. Psarrakos / Statistics and Probability Letters 79 (2009) 774–779

779

Corollary 3.4. If the equilibrium d.f. Pe (u) ∈ L, then lim

x→∞

H (u, x, y) Pe (u + x + y)

= lim

y→∞

H (u, x, y) Pe (u + x + y)

=

φ 1−φ

[1 − ψ(u)].

The following result is an extension of Theorem 3.1 in the classical model. Its proof is similar to the proof of Theorem 3.1, and it is omitted. Theorem 3.2. If Pe (u) ∈ S , then lim

u→∞

H (u, x, y) Pe (u − x + y)

=

φ , 1−φ

x, y ≥ 0.

Acknowledgements The author is grateful to K. Politis and an anonymous referee for the helpful suggestions. References Asmussen, S., 1987. Applied Probability and Queues. Wiley, New York. Asmussen, S., 2000. Ruin Probabilities. World Scientific, Singapore. Bowers, N., Gerber, H., Hickman, J., Jones, D., Nesbitt, C., 1986. Actuarial Mathematics. Society of Actuaries, Ithaca, IL. Chistyakov, V.P., 1964. A theorem on sums of positive random variables and its applications to branching random processes. Theory Probab. Appl. 9, 640–648. Chover, J., Ney, P., Wainger, S., 1973. Degeneracy properties of subcritical branching processes. Ann. Probability 1, 663–673. Dickson, D.C.M., 1992. On the distribution of the surplus prior to ruin. Insurance Math. Econom. 11, 191–207. Embrechts, P., Goldie, C.M., 1980. On closure and factorization theorems for subexponential and related distributions. J. Aust. Math. Soc. Ser. A 29, 243–256. Embrechts, P., Veraverbeke, N., 1982. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72. Embrechts, P., Goldie, C.M., Veraverbeke, N., 1979. Subexponentiality and infinity divisibility. Z. Wahrscheinlichkeitsth 49, 335–347. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin. Gerber, H.U., Goovaerts, M.J., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bull. 17, 151–163. Goldie, C.M., Klüppelberg, C., 1998. Subexponential distributions. In: Alder, R., Feldman, R., Taqqu, M.S. (Eds.), A Practical Guide to Heavy-Tails: Statistical Techniques for Analysing Heavy-Tailed Distributions. Birkhäuser, Boston, pp. 435–459. Pitman, E.J.G., 1980. Subexponential distribution functions. J. Aust. Math. Soc. Ser. A 29, 337–347. Psarrakos, G., Politis, K., 2008. Tail bounds for the joint distribution of the surplus prior to and at ruin. Insurance Math. Econom. 42, 163–176. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes for Insurance and Finance. Wiley, New York. Schmidli, H., 1999. On the distribution of the surplus prior to and at ruin. ASTIN Bull. 29, 227–244. Willmot, G.E., 2002. Compound geometric residual lifetime distributions and the deficit at ruin. Insurance Math. Econom. 30, 421–438. Willmot, G.E., Lin, X.S., 2001. Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York.