Joint asymptotic distributions of smallest and largest insurance claims

Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalized sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.


Introduction
When dealing with heavy-tailed insurance claims, it is a classical problem to consider and quantify the influence of the largest among the claims on their total sum, see e.g. Ammeter (1964) for an early reference in actuarial literature. This topic is particularly relevant in non-proportional reinsurance applications when a significant proportion of the sum of claims is consumed by a small number of claims. The influence of the maximum of a sample on the sum has in particular attracted considerable attention over the last fifty years (see Ladoucette and Teugels [15] for a recent overview of existing literature on the subject). Different modes of convergence of the ratios sum over maximum or maximum over sum have been linked with conditions on additive domain of attractions of a stable law (see e.g. Darling [9], Bobrov [7], Chow and Teugels [8] and Bingham and Teugels [6]).
It is also of interest to study the joint distribution of normalized smallest and largest claims when the number of claims over time are described by a general counting process. This has an impact on the design of possible reinsurance strategies and risk management in general. In this paper we consider a homogeneous insurance portfolio, where the distribution of the individual claims has a regularly varying tail. The number of claims is generated by a near mixed Poisson process. For this rather general situation we derive a number of limiting results for the joint Laplace transforms of the smallest and largest claims, as the time t tends to infinity. These turn out to be quite explicit and crucially depend on the rule of what is considered to be a large claim as well as on the value of the tail index.
Let X 1 , X 2 , . . . be a sequence of independent positive random variables (representing claims) with common distribution function F . For n ≥ 1, denote by X * 1 ≤ X * 2 ≤ . . . ≤ X * n the corresponding order statistics. We assume that the claim size distribution satisfies the condition where α > 0 and ℓ is a slowly varying function at infinity. The tail index is defined as γ = 1/α and U (y) = F ← (1 − 1/y) is the tail quantile function of F . Under (1), U (y) = y 1/α ℓ 1 (y), where ℓ 1 is again a slowly varying function. For textbook treatments of regularly varying distributions and/or their applications in insurance modelling, see e.g. Bingham et al. [5], Embrechts et al. [11], Rolski et al. [18] and Asmussen and Albrecher [4]. Denote the number of claims up to time t by N (t) with p n (t) = P (N (t) = n). The probability generating function of N (t) is given by which is defined for |z| ≤ 1. Let r z N (t)−r be its derivative of order r with respect to z. In this paper we assume that N (t) is a near mixed Poisson (NMP) process, i.e. the claim counting process satisfies the condition for some random variable Θ, where D denotes convergence in distribution. This condition implies that Note also that, for β > 0 and r ∈ N, If the distribution of Θ is degenerate at a single point, then (N (t)) t≥0 has asymptotically the same behavior as a renewal process. One particular example of a renewal process is the homogeneous Poisson process, which is very popular in claims modelling and plays a crucial role in both actuarial literature and practice. The general class of NMP processes has found numerous applications in (re)insurance modelling because of its flexibility, its success in actuarial data fitting and its property of being more dispersed than the Poisson process (see Grandell [12]). The mixing may e.g. be interpreted as claims coming from a heterogeneity of groups of policyholders or of contract specifications. The aggregate claim up to time t is given by where it is assumed that (N (t)) t≥0 is independent of the claims (X i ) i≥1 . For s ∈ N and N (t) ≥ s + 2, we define the sum of the N (t) − s − 1 smallest and the sum of the s largest claims by Here Σ refers to small while Λ refers to large.
In this paper we study the limiting behavior of the triple (Λ s (t), X * N (t)−s , Σ s (t)) with appropriate normalisation coefficients depending on γ, the tail index, and on s, the number of terms in the sum of the largest claims. We will consider three asymptotic cases: s is fixed, s tends to infinity but slower than the expected number of claims, and s tends to infinity and is asymptotically equal to a proportion of the number of claims.
The paper is organized as follows. We first give the joint Laplace transform of the triple (Λ s (t), X * N (t)−s , Σ s (t)) for a fixed t in Section 2. Section 3 deals with asymptotic joint Laplace transforms in the case 0 < α < 1. We also discuss consequences for moments of ratios of the limiting quantities. The behavior for α = 1 depends on whether E[X i ] is finite or not. In the first case, the analysis for α > 1 applies, in the latter one has to adapt the analysis of Section 3 exploiting the slowly varying varying function x 0 y dF (y), but we refrain from treating this very special case in detail (see e.g. [2] for a similar adaptation in another context). Sections 4 and 5 treat the case α > 1 without and with centering, respectively. The proofs of the results in Sections 3-5 are given in Section 6. Section 7 concludes.

Preliminaries
In this section, we state a versatile formula that will allow us later to derive almost all desired asymptotic properties of the joint distributions of the triple (Λ s (t), X * N (t)−s , Σ s (t)). We consider the joint Laplace transform of (Λ s (t), X * N (t)−s , Σ s (t)) to study their joint distribution in an easy fashion. For a fixed t, it is denoted by . Then the following representation holds: Proof: The proof is standard if we interpret X * r = 0 whenever r ≤ 0. Indeed, condition on the number of claims at the time epoch t and subdivide the requested expression into three parts.
The conditional expectation in the first term on the right simplifies easily to the form ( ∞ 0 e −ux dF (x)) n . For the conditional expectations in the second and third term, we condition additionally on the value y of the order statistic X * n−s ; the n − s − 1 order statistics X * 1 , X * 2 , . . . , X * n−s−1 are then distributed independently and identically on the interval [0, y] yielding the factor ( y 0 e −wx dF (x)) n−s−1 . A similar argument works for the s order statistics X * n−s+1 , X * n−s+2 , . . . , X * n . Combinations of the two terms yields Ω s (u, v, w; t) A straight-forward calculation finally shows Consequently, it is possible to easily derive the expectations of products (or ratios) of Λ s (t), X * N (t)−s , Σ s (t) and S(t) by differentiating (or integrating) the joint Laplace transform. We only write down their first moment for simplicity.
Proof: The individual Laplace transforms can be written in the following form: where Π s+1 (t) = s n=0 p n (t). By taking the first derivative, we arrive at the respective expectations.
3. Asymptotics for the joint Laplace transforms when 0 < α < 1 Before giving the asymptotic joint Laplace transform of the sum of the smallest and the sum of the largest claims, we first recall an important result about convergence in distribution of order statistics and derive a characterization of their asymptotic distribution. All proofs of this section are deferred to Section 6.
It is well-known that there exists a sequence E 1 , E 2 , ... of exponential random variables with unit mean such that (X * n , ..., X * 1 ) Lemma 1 in LePage et al. [16]). For 0 < α < 1, the series ( n k=1 Γ −1/α k ) n≥1 converges almost surely. Therefore, for a fixed s, we deduce that, as n → ∞, In particular, we derive by the Continuous Mapping Theorem that Note that the first moment of R (s) (but only the first moment) may be easily derived since We also recall that F belongs to the (additive) domain of attraction of a stable law with index α ∈ (0, 1) if and only if g. Theorem 1 in Ladoucette and Teugels [15]).
When (N (t)) t≥0 is a NMP process, we also have, as t → ∞, (see e.g. Lemma 2.5.6 in Embrechts et al. [11]). But note that, if the triple (Λ s (t), X * N (t)−s , Σ s (t)) is normalized by U (t) instead of U (N (t)) in (5), then the asymptotic distribution will differ due to the randomness brought in by the counting process (N (t)) t≥0 .
The following proposition gives the asymptotic Laplace transform when the triple (Λ s (t), X * N (t)−s , Σ s (t)) is normalised by U (t).
If Θ = 1 a.s., this expression simplifies to We observe that (N (t)) t≥0 modifies the asymptotic Laplace transform by introducing q s+1 into the integral (6). However, the moments of R (s) do not depend on the law of Θ: Note that this corollary only provides the moments of R (s) . In order to have moment convergence results for the ratios, it is necessary to assume uniform integrability of ( It is also possible to use the Laplace transform of the triple with a fixed t to characterize the moments of the ratios , and then to follow the same approach as proposed by Ladoucette [13] for the ratio of the random sum of squares to the square of the random sum under the condition that Remark 3.2. R (s) is the ratio of the sum Ξ s + Σ s over Ξ s . By taking the derivative of (6), it may be shown that, for 1 < γ < s + 1 and E {Θ γ } < ∞, Therefore the mean of Ξ s + Σ s will only be finite for sufficiently small γ. An alternative interpretation is that for given value of γ, the number s of removed maximal terms in the sum has to be sufficiently large to make the mean of the remaining sum finite. The normalisation of the sum by Ξ s , on the other hand, ensures the existence of the moments of the ratio R (s) for all values of s and γ > 1.

Remark 3.3.
It is interesting to compare Formula (7) with the limiting moment of the statistic For instance, lim t→∞ E T N (t) = 1−α, lim t→∞ Var T N (t) = α (1 − α) /3 and the limit of the nth moment can be expressed as an nth-order polynomial in α, see Albrecher and Teugels [2], Ladoucette [13] and Albrecher et al. [1]. Motivated by this similarity, let us study the link in some more detail. By using once again Lemma 1 in LePage et al. [16], we deduce that Recall that R (0) is the weak limit of the ratio ( . Using (9) and E{R 2 (0) T ∞ } = 2/(2 − α) (which is a straight-forward consequence of the fact that X 2 i has regularly varying tail with tail index 2γ), one then obtains a simple formula for the covariance between R 2 (0) and T ∞ : Determining Var{R 2 (0) } by exploiting (7) for k = 4, we then arrive at the linear correlation coefficient .
The correlation coefficient allows to quantify the negative linear dependence between the two ratios (the dependence becomes weaker when α increases, as the maximum term will then typically be less dominant in the sum). Next, let us consider the case when the number of largest terms also increases as t → ∞, but slower than the expected number of claims. It is now necessary to change the normalisation coefficients of X * N (t)−s and Σ s (t).
Several messages may be derived from (10). First note that the asymptotic distribution of X * N (t)−s is degenerated for s = ⌊p(t)N (t)⌋, since X * N (t)−s /U (p −1 (t)) D → 1 as t → ∞. Second, the asymptotic distribution of the sum of the smallest claims is the distribution of Θ up to a scaling factor, since Σ s (t)/(tp(t)U (p −1 (t))) Finally, for a fixed proportion of maximum terms, it is also necessary to change the normalisation coefficients of X * N (t)−s and Σ s (t). We have As expected, X * If Θ = 1 a.s. and α = 1/2, then Λ p has an inverse Gamma distribution with shape parameter equal to 1/2.

Asymptotics for the joint Laplace transforms when α > 1
In this section, we assume that α > 1 and hence the expectation of the claim distribution is finite. We let µ = E {X 1 }. The normalisation coefficient of the sum of the smallest claims, Σ s (t), will therefore be t −1 as it is the case for S(t) for the Law of Large Numbers. In Section 5, we will then consider the sum of the smallest centered claims with another normalisation coefficient.
Again, consider fixed s ∈ N first. The normalisation coefficients of Λ s (t) and X * N (t)−s are the same as for the case 0 < α < 1, but the normalisation coefficient of Σ s is now t −1 .
If Θ = 1 a.s., We first note that and therefore Σ s (t)/t D → µΘ as t → ∞ for any fixed s ∈ N. The influence of the largest claims on the sum becomes less and less important as t is large and is asymptotically negligible. This is very different from the case 0 < α < 1. In Theorem 1 in Downey and Wright [10], it is moreover shown that, as n → ∞, (1)) .
This result is no more true in our framework when Θ is not degenerate at 1. Assume that E {Θ γ } < ∞.
Using (2) and under a uniform integrability condition, one has Next, we consider the case with varying number of maximum terms. The normalisation coefficients of Λ s (t) and X * N (t)−s now differ.
As for the case 0 < α < 1, X * N (t)−s /U (p −1 (t)) P → 1 as t → ∞. Moreover the asymptotic distribution of the sum of the largest claim is the distribution of Θ up to a scaling factor since Λ s (t)/(tp(t)U (p −1 (t))) D → Θ/(1−γ) as t → ∞. Finally note that Σ s (t)/t D → µΘ as t → ∞ as for the case when s was fixed.
Finally we fix p. Only the normalisation coefficient of Λ s (t) and its asymptotic distribution differ from the case 0 < α < 1. We note that the normalisation of Λ s (t) is the same as for Σ s (t) and that Λ s (t)/t D → ΘE {X|X > x p } as t → ∞.

Asymptotics for the joint Laplace transform for α > 1 with centered claims when s is fixed
In this section, we consider the sum of the smallest centered claims: instead of the sum of the smallest claims Σ s (t). Like for the Central Limit Theorem, we have to consider two subcases: 1 < α < 2 and α > 2.
Corollary 5.1. We have This result is to compare with the one obtained by Bingham and Teugels [6] for s = 0 (see also Ladoucette and Teugels [15]).

Proofs
Proof of Proposition 3.1: In formula (6), we first use the substitution F (y) = z/t, i.e. y = U (t/z): Next, the substitution F (x) = ρz/t, i.e. x = U (t/(zρ)) leads to Note that the integral is well defined since γ > 1. Moreover e −vU(t/z)/U(t) → e −vz −γ and Proof of Corollary 3.1: From Proposition 3.1 we have This gives indeed, using (2), which extends (4) to the case of NMP processes. Next, we focus on (7) for general k. We first consider the case s = 0. We have By Proposition 3.1 and clearly Note that x j j! mj . Therefore ∂ n ∂w n q 1 (θ(z, w))

By de Faa di Bruno's formula
Subsequently, with definition (8). This gives cf. (2), and the result follows. For the case s > 0, we proceed in an analogous way. Equation (11) becomes Then Σ 0 /Ξ 0 is replaced by Σ s /Ξ s , q 1 (z) by q s+1 (z), q (k) 1 by q (k) s+1 and, by following the same path as for s = 0, we get

Proof of Proposition 3.2:
The proof is similar to the previous one, so we just highlight the differences here: Conditioning on N (t) = n we have × y 0 e −wx/(tp(t)U(p −1 (t))) dF (x) n(1−p(t))−1 dF (y). We first replace F by the substitution F (y) = p(t)z, i.e. y = U (1/(p(t)z)) : The factor involving v converges to The factor containing w behaves as and hence for the power Finally, for the factor containing u, replace F by the substitution F (x) = ρz/t, i.e. x = U (t/(zρ)): For the factor with the factorials, we have by Stirling's formula n 1/2 e (np(t)+1/2) ln(p(t)) e (n(1−p(t))+1/2) ln (1−p(t) .
Proof of Proposition 4.1: We first replace F by the substitution F (y) = z/t, i.e. y = U (t/z) Then we have (1)) .
which this influence is asymptotically negligible. We further related the dominance of the maximum term in such a random sum to another quantity that exhibits the effect of the tail index on the aggregate claim rather explicitly, namely the ratio of sum of squares of the claims over the sum of the claims squared. The results allow to further quantify the effect of large claims on the total claim amount in an insurance portfolio, and could hence be helpful in the design of appropriate reinsurance programs when facing heavy-tailed claims with regularly varying tail. Particular emphasis is given to the case when the tail index exceeds 1, which corresponds to infinite-mean claims, a situation that is particularly relevant for catastrophe modelling.