A Lundberg-type inequality for an inhomogeneous renewal risk model

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.


Introduction
The classical risk model and the renewal risk model are two models that are traditionally used to describe the nonlife insurance business. The classical risk model was introduced by Lundberg and Cramér about a century ago (see [8,14,15] for the source papers and [18] for the historical environment). In this risk model, it is assumed that interarrival times are identically distributed, exponential, and independent random variables. In 1957, the Danish mathematician E. Sparre Andersen proposed the renewal risk model to describe the surplus process of the insurance company. In the renewal risk model, the claim sizes and the interarrival times are independent, identically distributed, nonnegative random variables (see [2] for the source paper and [22] for additional details). In this paper, we assume that interoccurrence times and claim sizes are nonnegative random variables (r.v.s) that are not necessarily identically distributed. We call such a model the inhomogeneous model and present its exact definition. It is evident that the inhomogeneous renewal risk model reflects better the real insurance activities in comparison with the classical risk model or with the renewal (homogeneous) risk model.
A typical path of the surplus process of an insurance company is shown in Fig. 1. If all claim sizes {Z 1 , Z 2 , . . .} and all interarrival times {θ 1 , θ 2 , . . .} are identically distributed, then the inhomogeneous renewal risk model becomes the homogeneous renewal risk model.
The time of ruin and the ruin probability are the main critical characteristics of any risk model. Let B denote the event of ruin. We suppose that that is, that ruin occurs if at some time t 0 the surplus of the insurance company becomes negative or, in other words, the insurer becomes unable to pay all the claims. The first time τ when the surplus drops to a level less than zero is called the time of ruin, that is, τ is the extended r.v. for which The ruin probability ψ is defined by the equality Usually, we suppose that the main parameter of the ruin probability is the initial reserve x, though actually the ruin probability, together with time of ruin, depends on all components of the renewal risk model. All trajectories of the process U (t) are increasing functions between times T n and T n+1 for all n = 0, 1, 2, . . . . Therefore, the random variables U (θ 1 + θ 2 + · · · + θ n ), n 1, are the local minimums of the trajectories. Consequently, we can express the ruin probability in the following way (for details, see [9] or [16]): Further, in this paper, we restrict our study to the so-called Lundberg-type inequality. An exponential bound for the ruin probability is usually called a Lundberg-type inequality. We further give the well-known exponential bound for ψ(x) in homogeneous renewal risk model (see, for instance, Chapters "Lundberg Inequality for Ruin Probability", "Collective Risk Theory", "Adjustment Coefficient," or "Cramer-Lundberg Asymptotics" in [21]). Theorem 1. Let the net profit condition EZ 1 − cEθ 1 < 0 hold, and let Ee hZ1 < ∞ for some h > 0 in the homogeneous renewal risk model. Then, there is a number for all x 0. If Ee R(Z1−cθ1) = 1 for some positive R, then we can take H = R in (1).
There exist a lot of different proofs of this theorem. The main ways to prove inequality (1) are described in Chapter "Lundberg Inequality for Ruin Probability" of [21]. Details of some existing proofs were given, for instance, by Asmussen and Albrecher [3], Embrechts, Klüppelberg, and Mikosch [9], Embrechts and Veraverbeke [10], Gerber [11], and Mikosch [16]. We note only that the bound (1) can be proved using the exponential tail bound of Sgibnev [19] and the inequality ψ(0) < 1.
The following theorem is the main statement of the paper.
Then, there are constants c 1 > 0 and The inhomogeneous renewal risk model differs from the homogeneous one because the independence and/or homogeneous distribution of sequences of random variables {Z 1 , Z 2 , . . . } and/or {θ 1 , θ 2 , . . . } are no longer required. The changes depend on how the inhomogeneity in a particular model is understood. In Definition 1, we have chosen one of two possible directions used in numerous articles that deal with inhomogeneous renewal risk models. This is due to the fact that an inhomogeneity can be considered as the possibility to have either differently distributed or dependent r.v.s in the sequences.
The possibility to have differently distributed r.v.s was considered, for instance, in [5,6,12,17]. In the first three works, the discrete-time inhomogeneous risk model was considered. In such a model, the interarrival times are fixed, and the claims {Z 1 , Z 2 , . . . } are independent, not necessary identically distributed, integer valued r.v.s. In [17], the authors consider the model where the interarrival times are identically distributed and have a particular distribution, whereas the claims are differently distributed with distributions belonging to a particular class. Bernackaitė and Šiaulys [4] deal with an inhomogeneous renewal risk model where the r.v.s {θ 1 , θ 2 , . . . } are not necessarily identically distributed, but the claim sizes {Z 1 , Z 2 , . . . } have a common distribution function. In this article, we consider a more general renewal risk model. In the main theorem, we assume that not only r.v.s {θ 1 , θ 2 , . . . } are not necessarily identically distributed, but also the same holds for the sequence of claim sizes There is another approach to inhomogeneous renewal risk models, which implies the possibility to have dependence in sequences and mainly found in works by Chinese researchers. In this kind of models, the sequences {Z 1 , Z 2 , . . . } and {θ 1 , θ 2 , . . . } consist of identically distributed r.v.s, but there may be some kind of dependence between them. Results for such models can be found, for instance, in [7] and [23]. Another interpretation of dependence is also possible, where r.v.s in both sequences {Z 1 , Z 2 , . . . } and {θ 1 , θ 2 , . . . } still remain independent. Instead of that, the mutual independence of these two sequences is no longer required. The idea of this kind of dependence belongs to Albrecher and Teugels [1], and this encouraged Li, Tang, and Wu [13] to study renewal risk models having this dependence structure.
The rest of the paper consists of two sections. In Section 2, we formulate and prove an auxiliary lemma. The proof of the main theorem is presented in Section 3.
Then, there are constants c 3 > 0 and c 4 > 0 such that for all x 0.

Proof of Theorem 2
In this section, we prove Theorem 2.
Next, it follows from (C1) that So, the requirement (C1 * ) holds too. It remains to prove that To establish this, we use the inequality Taking the limit as u → ∞ in the first summand of the right side of inequality (11), we get Since x e γx /γ, x 0, condition (C1) implies that In addition, by condition (C2). Therefore, relations (12), (13), and (14) imply that Now taking the limit as u → ∞ in the second summand of the right side of inequality (11), by condition (C2) we have We now see that the desired equality (10) follows from (11), (15), and (16). This means that all requirements of Lemma 1 hold for r.v.s Z i − cθ i , i ∈ N.