A 2$\times$2 random switching model and its dual risk model

In this article a special case of an M/G/2-queue is considered, where the two servers are exposed to two types of jobs that are distributed among the servers via a random switch. In this model the asymptotic behaviour of the workload buffer exceedance probabilities for the two single servers/ both servers together/ one (unspecified) server is determined. Hereby one has to distinguish between jobs that are either heavy-tailed or light-tailed. The results are derived via the dual risk model of the studied M/G/2-queue for which the asymptotic behaviour of different ruin probabilities is determined.


Introduction
A general 2 × 2 switch is modelled by a two-server queueing system with two arrival streams. A well-studied special cases of such a switch is given by the 2 × 2 clocked buffered switch, where in a unit time interval each arrival stream can generate only one arrival and each server can serve only one customer; see e.g. [1,11,15] and others. This switch is commonly used to model a device used in data-processing networks for routing messages from one node to another.
In this paper we study a 2 × 2 switch that operates in continuous time, i.e. the arrivals are modelled by two independent compound Poisson processes. Every incoming job is of random size and it is then distributed to the two servers by a random procedure. This leads to a pair of coupled queues that form an an M/G/2 queue. In this model we study the equilibrium probabilities of the resulting workload processes. In particular we determine the asymptotic behaviour of the probabilities that the workloads exceed a prespecified buffer. Hereby we will distinguish between workload exceedance of a specific single server, both servers, or one unspecified server. As we will see, the behaviour of these workload exceedance probabilities strongly depends on whether jobs are heavy-tailed or light-tailed and we will therefore consider both cases separately.
A related model to the one we study has been introduced in [10] where a pair of coupled queues driven by independent spectrally-positive Lévy processes is introduced. The coupling procedure however is completely different to the switch we shall use. For this model, in [10], the joint transform of the stationary workload distribution in terms of Wiener-Hopf factors is determined. Two parallel queues are also considered e.g. in [19] for an M/M/2 queue where arriving customers simultaneously place two demands handled independently by two servers. We refer to [2] and [18] and references therein for more general information on Lévy-driven queueing systems.
As it is well known, there are several connections between queueing and risk models. In particular the workload (or waiting time) in an M/G/1 queue with compound Poisson input is related to the ruin probability in the prominent Cramér-Lundberg risk model, in which the arrival process of claims is defined to be just the same compound Poisson process; see e.g. [2] or [23]. To be more precise, let be a Cramér-Lundberg risk process with initial capital u > 0, premium rate c > 0, i.i.d. claims {X i , i ∈ N} with cdf F such that X 1 > 0 a.s. and E[X 1 ] = µ < ∞, and a claim number process (N (t)) t≥0 which is a Poisson process with rate λ > 0. Then it is well known that the ruin probability Ψ(u) = P(R(t) < 0 for some t ≥ 0) tends to 0 as u → ∞, as long as the net-profit condition λµ < c holds, while otherwise Ψ(u) ≡ 1. In particular, if the claims sizes are light-tailed in the sense that an adjustment coefficient κ > 0 exists, i.e. ∃κ > 0 : for some known constant C ≥ 0 depending on the chosen parameters of the model. On the contrary, for heavy-tailed claims whose tail-functions are regularly varying at infinity it is known that typically (cf. [3, Eq. I.(4.6)]) such that the ruin probability in this case decreases only polynomially. Via the mentioned duality these results can easily be translated into corresponding results on the workload exceedance probability of an M/G/1 queue.
In this paper we shall use an analogue duality between queueing and risk models in a multi-dimensional setting as it was introduced in [7]. This allows us to obtain results on the workload exceedance probabilities of the 2 × 2 switch by studying the corresponding ruin probabilities in the two-dimensional dual risk model.
Two-dimensional risk processes have e.g. been considered in [4,5,6,14,17,20,24]. In particular in [4] the asymptotic behaviour of ruin probabilities for light-tailed claims is studied under certain model assumptions. In general dimensions, multivariate ruin is studied e.g. in [9,12,13,25]. Note that in particular the model in [9], where a bipartite network induces the dependence between the risk processes, is in some sense similar to the dual risk model in this paper. Further, in [16], multivariate risk processes with heavytailed claims are treated and so-called ruin regions are studied, that is, sets in R d which are hit by the risk process with small probability. Heavy-tailed claims are also assumed e.g. in [21] where several business lines are considered that can balance out ruin, and some of these results will be applied on the dual risk model in this paper.
The paper is outlined as follows. In Section 2 we specify the random switch model that we are interested in and introduce the corresponding dual risk model. Section 3 is devoted to study both models under the assumption that jobs/claims are heavy-tailed. As we shall rely on results on the risk model studied in [21] we first concentrate on the risk model in Section 3.1 and then transfer our findings to the switch model in Section 3.2.
In Section 4 we assume all jobs/claims to be light-tailed and again first consider the risk model in Section 4.1 before converting the results to the switch context in Section 4.2.
Two particular examples of the switch will then be outlined in Section 5 where we also compare the behaviour of the exceedance probabilities for different specifications of the random switch via a short simulation study in Section 5.3. The final Section 6 collects the proofs of all our findings.
2 The switching model and its dual 2.1 The 2 × 2 random switching model Let S 1 , S 2 be servers with work speeds c 1 , c 2 > 0 and let J 1 , J 2 be two job generating objects. We assume that both objects generate jobs independently with Poisson rates λ 1 , λ 2 > 0, respectively, and that the workloads generated by one object are i.i.d. positive random variables. More specific, we identify the objects J j , j = 1, 2, with two independent compound Poisson processes The jobs shall be distributed to the two servers by a random switch that is modeled by a random (2 × 2)-matrix A = (A ij ) i,j=1,2 , independent of all other randomness and satisfying the following conditions: 1] for all i, j = 1, 2, meaning that a job can not be assigned more than totally or less than not at all to a certain server, A ij = 1 for all j = 1, 2, i.e. every job must be assigned entirely to the servers.
The switch matrix is triggered independently at every arrival of a job. We are interested in the M/G/2-queue defined by the resulting storage processes of the two servers, i.e.
In particular we aim to study the stationary distribution of the multivariate storage process W(t) = (W 1 (t), W 2 (t)) , that is the distributional limit of W(t) as t → ∞ whenever it exists. In this case we write for a generic random vector with this steady-state distribution. Note that here and in the following (·) denotes the transpose of a vector or matrix.
Let u > 0 be some fixed buffer barrier for the system and b = (b 1 , b 2 ) ∈ (0, 1) 2 with b 1 + b 2 = 1. Set u = bu, i.e. u i = b i u. Then we are in particular interested in the probabilities that the single servers exceed their barriers, the probability that at least one of the workloads exceeds the barrier u as 4) and the probability that both of the workloads exceed the barrier u as

The dual risk model
In the one-dimensional case it is well known that there exists a duality between risk-and queueing models, see e.g. [2]. The multivariate analogue shown in [7] allows us to formulate the dual risk model to the above introduced random switching model as follows.
Let N (t) := N 1 (t) + N 2 (t) such that N (t) is a Poisson process with rate λ = λ 1 + λ 2 . Define the multivariate risk process where B k are i.i.d. random matrices, independent of all other randomness, such that Note that the components of (R(t)) t≥0 satisfy the net-profit condition, if We will therefore assume (2.7) throughout the paper. Note that as mentioned in [7], (2.7) implies existence of the stationary distribution of W(t), i.e. W in (2.2) is well-defined. For a proof of this fact in the univariate setting, see e.g. [23,Thm. 4.10].
For the buffer u > 0, in the risk model, we define the ruin probabilities of the single components the ruin probability for at least one component 9) and the ruin probability for all components where as before u = bu for b ∈ (0, 1) 2 with b 1 + b 2 = 1.
The following Lemma allows us to gather information about the bivariate storage process in the switching model by performing calculations on our dual risk model.
Note that in the ruin context it is common (see e.g. [4] or [9]) to consider the simultaneous ruin probability for all components (2.11) As we will see, results on Ψ ∧,sim can sometimes be shown in analogy to those on Ψ ∨ and we shall do so whenever it seems suitable. However, Ψ ∧,sim has no counterpart in the switching model.
It is clear from the above definitions that for all u = bu ∈ (0, ∞) 2 and likewise We will therefore focus in our study on Υ ∨ and Ψ ∨ and then derive the corresponding results for Υ ∧ and Ψ ∧ via (2.13) and (2.12).

Further notations
To keep notation short, we write R ≥0 , and R ≤0 for the positive/negative half line of the real numbers, respectively, and likewise use the notations R >0 , and R <0 such that in particular

The heavy-tailed case
In this section we will assume that the tail functions of the arriving jobs asymptotically show a power law behaviour. To specify what this means, let f : R → (0, ∞) be a measurable function and recall that f is regularly varying (at infinity) with index α > 0 if for all λ > 0 it holds that In this case we write f ∈ RV(α). A real-valued random variable X is called regularly varying with index α > 0, i.e. X ∈ RV(α), if its tail function F (·) = P(X > ·) is regularly varying with index −α.
Further we follow [21] and call a random vector Z on R q multivariate regularly varying if there exists a non-null measure µ on R q \{0} such that Here and ever after ∂M denotes the boundary of the set M and the norm · will typically be chosen to be the L 1 -norm in this article. If Z is multivariate regularly varying, necessarily there exists α > 0 satifsfying that for all M as in (3.1) and t > 0 it holds that Thus we write Z ∈ MRV(α, µ).
Note that in the one-dimensional case the above definitions coincide. We refer to [8] and [26] for references of the above and more detailed information on multivariate regular variation.

Results in the risk context
We will now present our first main result which we state in terms of the risk process defined in Section 2.2. The rather long and technical proof of Theorem 3.1 relies on results from [22] and will be given in Section 6.1.
Note that by conditioning on B we have Using the limiting-measure property of µ * it is further possible to explicitely compute the constants C ∨ , and C ∧,sim in Theorem 3.1 above. This then yields the following proposition whose proof is also postponed to Section 6.1.
such that clearly ζ ∈ (0, ∞) implies α 1 = α 2 . Then and We continue our study of the asymptotics of the risk model by determining the asymptotic behaviour of Ψ ∧ . It is clear from Equations (2.12) and (3.5) that in order to do this, we first have to determine the asymptotic behaviour of the ruin probabilities for single components (2.8), which will be given by the following lemma. Lemma 3.3. Assume X 1 ∈ RV(α 1 ), and X 2 ∈ RV(α 2 ) for α 1 , α 2 > 1. Then the ruin probability for a single component (2.8) fulfils With this the following proposition is straightforward. Again, the proof can be found in Section 6.1.
it holds that with C ∨ as defined in (3.6) and with the weighted integrated tail functions Otherwise, if (3.9) fails, then

Results in the switch context
With the help of Lemma 2.1 we may now directly summarize our findings from the last section to provide a rather explicit insight into the asymptotic behaviour of the workload barrier exceedance probabilities in the switching model defined in Section 2.1.
Example 3.6. In the setting of Corollary 3.5 assume that α 1 < α 2 . Then in all asymptotics given in Corollary 3.5 the terms including F 2 that are regularly varying with index −α 2 + 1 are dominated by the terms involving F 1 which are regularly varying with index −α 1 + 1. This yields that in this case With these observations at hand we may now conclude that (3.13) holds if and only if (3.14) Thus, given (3.14), we get Remark 3.7. The above example can be generalized in the sense that a regularly varying tail dominates any lighter tail, no matter whether this is regularly varying as well or not. Indeed, assuming that w.l.o.g. X 1 ∈ RV(α) for α > 1 and X 2 is such that one can prove in complete analogy to the results from the last subsection, that the workload exceedance probabilities (2.3), (2.4), and (2.5) fulfil and, assuming additionally that (3.13) holds,

The light-tailed case
In this section we will study the asymptotic behaviour of ruin/workload exceedance probabilities for claims/jobs that are typically small, i.e. we will assume throughout this section that the moment generating functions

Results in the risk context
As in the heavy-tailed setting we start by studying the dual risk model. Again, the ruin probabilities for the single components are particularly easy to treat. The following lemma is obtained by a direct application of Lundberg's well-known inequality and the Cramér-Lundberg approximation, see e.g. [3, Thms. IV.5.2 and IV.5.3]. In Section 6.2 a short proof is provided.
Lemma 4.1. Assume the claim size variables X 1 , X 2 fulfil (4.1) and assume there exist (unique) solutions κ 1 , κ 2 > 0 to Then the ruin probabilities of the single components fulfil we easily derive the following Lundberg-type bound for Ψ ∨ from the above Lemma.
Corollary 4.2. Assume the claim size variables X 1 , X 2 fulfil (4.1) and assume there exist (unique) solutions κ 1 , κ 2 > 0 to (4.2), then the ruin probability for at least one component fulfils Similarly to what has been done in [9, Thm. 6.1] it is also possible to derive a Lundberg bound for Ψ ∧,sim via classical martingale techniques. Indeed one can show that for any κ 1 , κ 2 > 0 such that As this has no implications for the considered queueing model we will not go into further details here.
To derive the asymptotics of Ψ ∧ , Ψ ∧,sim and Ψ ∨ we rely on results from [4], which lead to the following Theorem.

Results in the switch context
Again, using Lemma 2.1 we summarize our findings from the last section to obtain the following corollary on the asymptotic behaviour of the workload barrier exceedance probabilities in the switching model defined in Section 2.1.
Further, with C i , i = 1, 2, as in (4.3), it holds while the probability that both workloads exceed their barrier fulfils Remark 4.6. Note that the light-tail assumption (4.1) does not necessarily imply existence of κ 1 , κ 2 > 0 solving (4.2). Assuming for j = 1, 2 the slightly stronger condition Either however is sufficient for existence of κ 1 , κ 2 > 0.
In case that the above condition fails, i.e. for some j ∈ {1, 2} there exists x * j such that ϕ X j (x j ) < ∞ for all x j ≤ x * j and ϕ X j (x j ) = ∞ for all x j > x * j , then existence of κ 1 , κ 2 depends on the chosen parameters of the model; see e.g. [3, Chapter IV.6a] for a more thorough discussion of this.
Remark 4.7. If κ 1 b 1 = κ 2 b 2 then the summand of lower order on the right hand side of (4.4) can be omitted in the asymptotic equivalence. Thus, in contrary to the heavy-tailed case, the vector b here is crucial for the exact asymptotic behaviour and contributes more than just inside the constant. On the other hand we immediately see that, given two job distributions and hence given κ 1 , κ 2 > 0, we can choose b 1 , b 2 in order to minimize the joint exceedance probabilities. The optimal b then solves which leads to

Examples and simulation study
In this section we consider two special choices of the random switch for which we will evaluate the above results and compare to simulated data. The first part is dedicated to the special case of the Bernoulli switch, where the queueing processes become independent of each other. In the second part we discuss the special case of a non-random switch, where every job is shared between the servers with some predefined deterministic proportions. We finish in Section 5.3 with a short comparison to study the influence of the chosen type of randomness on the exceedance probabilities.

The Bernoulli switch
The Bernoulli switch does not split any jobs, but assigns the arriving jobs randomly to one of the two servers. More precisely we set independent of each other with p, q ∈ [0, 1]. This yields independence of the components of the process (R(t)) t≥0 which can now be represented as where X j,k and X j,k are independent copies of X j,k , j = 1, 2, k ∈ N, and the counting processes (N 2 (t)) t≥0 , and (N 2 (t)) t≥0 are independent Poisson processes with rates λ 1 p λ 1 +λ 2 , λ 1 (1−p) λ 1 +λ 2 , λ 2 q λ 1 +λ 2 , and λ 2 (1−q) λ 1 +λ 2 , respectively. In particular from (2.5) and (2.13) we obtain in the Bernoulli switch and hence Υ ∧ (u) and Υ ∨ (u) can be expressed in terms of Υ 1 (b 1 u), and Υ 2 (b 2 u). For these we obtain by direct application of Corollary 3.5 that for X 1 ∈ RV(α 1 ), X 2 ∈ RV(α 2 ) , .

(5.2)
In the light-tailed case an application of Corollary 4.5 yields as long as there exist κ 1 , κ 2 > 0 such that (4.2) holds, which in the Bernoulli switch simplifies to The asymptotic behaviour of Υ ∨ and Υ ∧ can now be described via (5.1).
In Figures 2 and 3 we compare the asymptotics in the Bernoulli switch obtained in this way with data that has been simulated using standard Monte-Carlo techniques. As one can see in all cases the obtained asymptotics fit the data very well for u large enough. Here job sizes are Pareto distributed with F 1 (x) = x −3/2 , x ≥ 1, and F 2 (x) = 4x −2 , x ≥ 2. Further λ 1 = λ 2 = 1, c 1 = 5, c 2 = 8, and b 1 = 0.8 = 1 − b 2 . The Bernoulli switch is characterized by p = 0.4 and q = 0.7. For these parameters from (5.2) we derive Υ 1 (u 1 ) Note that a direct evaluation of the asymptotics of Υ ∨ as given in Corollary 3.5 yields the same result. Note that in the latter case we keep both summands, since the exponents are close together.

The deterministic switch
The deterministic switch is characterized by setting for some predefined constants d 1 , d 2 ∈ [0, 1]. Note that for d 1 = d 2 the corresponding dual risk model coincides with the so-called "two-dimensional degenerate risk model" considered in [4]. Clearly, for any choice of d 1 , d 2 in the deterministic switch one can easily evaluate the asymptotics of the exceedance probabilities as given in Corollaries 3.5 and 4.5 since all appearing expectations disappear.
In Figures 4 and 5 we compare the asymptotics and bounds in the deterministic switch obtained in this way with data that has been simulated using standard Monte-Carlo techniques. Again simulations and theoretical asymptotics fit well in all cases.

A comparison of different switches
In this section we aim to compare the two above special cases of the Bernoulli switch and the deterministic switch with a non-trivial random switch, which we chose to be a Beta switch characterized by setting for some constants β 1 , β 2 , γ 1 , γ 2 > 0, where Beta(β, γ) is the Beta distribution with density 11 ] and E[A 12 ] such that the scenarios only differ in the behaviour of the switch and the job sizes. Figure 6 shows the approximate exceedance probabilities obtained by Monte Carlo simulation for the Bernoulli switch, the deterministic switch und two different Beta switches.
As we can see, in the presence of heavy tails the probability that at least one of the workloads exceeds the barrier Υ ∨ tends to zero with the same index of regular variation  for all choices of the random switch. In case of the probability that both components exceed their barrier Υ ∧ , the Bernoulli switch yields a faster decay due to the independence of the two workload processes in this model. Further, the figure indicates the intuitive behaviour: The more correlated the co-ordinates of the workload process are, the closer together are Υ ∨ and Υ ∧ . This leads to a tradeoff between the two probabilities: Changing the switch towards reducing one probability raises the other and the Beta switches may serve here as a compromise to control both probabilities.
In the light-tailed case the trade-off between Υ ∨ and Υ ∧ can not be observed. Quite the contrary, the more correlated the co-ordinates of the workload process are, the lower tend to be the exceedance probabilities. Hence in this case the Bernoulli switch yields the highest exceedance probabilities, while the deterministic switch obtains the best results.
Thus, for keeping Υ ∨ small, in general the simple deterministic switch yields good results. On the contrary, if one is interested to keep Υ ∧ small, the tail-behaviour of the appearing jobs is crucial for the choice of the optimal switch. Here again Beta switches or other nontrivial random switches may serve as a compromise in situations where the tail-behaviour of the appearing jobs is unknown.

Proofs for Section 3
We start to prove the first statement of Theorem 3.1 which we restate below as Lemma 6.2.
Proposition 6.1. Let Z ∈ MRV(α, µ) be a random vector in R d and let M be a random where B c 1 := {x ∈ R q : x > 1} denotes the complement of the unit sphere in R q .
Proof. First note that our definition of regular variation corresponds to Definition 2.16 (Theorem 2.1.4 (i)) in [8], setting E = B c 1 , which implies P(Z ∈ tE) = P( Z > t). Now double application of [8, Proposition 2.1.18] implies the statement, since for M ⊆ R 2 measurable and bounded away from 0 Lemma 6.2. Consider the notation of Section 2. If X 1 and X 2 are regularly varying in the univariate sense with indices α 1 , α 2 , then there exists a measure µ * as in Proposition 6.1 such that ABX ∈ MRV(min{α 1 , α 2 }, µ * ).
Proof. Obviously X = (X 1 , X 2 ) ∈ MRV(α, µ) for some non-null measure µ concentrated on the axes, and α = min(α 1 , α 2 ) since the random variables X 1 , X 2 are independent and both regularly varying with indices α 1 , α 2 . To prove the Lemma it is thus enough to check the prerequisites of Proposition 6.1. Clearly, using the properties of A and B we compute E[ AB γ ] = 1 < ∞ for any γ. Further for M ⊆ R 2 measurable and bounded away from 0 Thus for M = B c 1 and recalling property (ii) of the matrix A we obtaiñ where we have used that, due to positivity of X, µ is zero on R 2 \R 2 >0 . This finishes the proof.
To prove the remainder of Theorem 3.1 we will use a result from [22]. To do so, first recall the bivariate compound Poisson process R from our dual risk model from Section 2.2. Let (T k ) k∈N be the independent identically Exp(λ)-distributed interarrival times of the Poisson process N (t), i.e.
We define the random walk and directly observe that (I n ) n∈N is compensated, i.e. for all n ∈ N The following Lemma explains the relationship between the risk process (R(t)) t≥0 and the random walk (I n ) n∈N . Lemma 6.3. Let F ⊆ R 2 be a ruin set, i.e. assume that Then . Thus by assumption (i) R(t) may enter F only by a jump and since N (t) t ∞ −→ ∞ a.s. we get which yields the claim.
We proceed with a Lemma that specifies the ruin sets that we are interested in.
(iv) The set b + F is p-increasing for all p ∈ R 2 >0 , i.e., for all v ≥ 0 it holds that Proof. That ABX ∈ RV(min(α 1 , α 2 ), µ * ) has been shown in Lemma 6.2. Recalling the definitions of I n and Ψ F (u) we may write All the other prerequisites ensure that we may apply [22,Thm. 3.1 and Rem. 3.2] to obtain the desired asymptotics.
The following Lemma justifies the usage of Proposition 6.5 for our problem.
Lemma 6.6. The sets F ∨ and F ∧,sim from Lemma 6.4 satisfy conditions (i)-(iv) of Lemma 6.3 and Proposition 6.5.
Proof. Properties (i), (ii) and (iv) are obvious. Consider (iii). Fix an arbitrary a = (a 1 , a 2 ) ∈ R 2 >0 . It holds that ∂(a + F ) = a + ∂(F ) and we have Thus for t 1 = t 2 we have t 1 M 1 (a) ∩ t 2 M 1 (a) = ∅. Further the set t∈(1,∞)∩Q tM 1 (a) is obviously bounded away from zero, since (a 1 , a 2 ) > 0. We thus obtain Since the last sum is infinite, µ * (M 1 (a)) must be zero. The same argument applied on M 2 (a) thus yields the result for F ∨ . The proof for F ∧,sim is analogue.
Proof of Theorem 3.1. The first statement has been shown in Lemma 6.2. The asymptotics for Ψ ∨ and Ψ ∧,sim are direct consequences of Lemma 6.6 and Proposition 6.5.
For the proof of Proposition 3.2 we will use the following lemma.
Proof. Obviously it holds that Proof of Proposition 3.2. We concentrate first on the ∨-case and start by determining the constant C ∨ . Using the limiting-measure property of µ * , (3.4) and the properties of A and B we obtain Hence A similar computation for A 12 X 2 A 22 X 2 thus leads to where P A ( · ) denotes the probability measure induced by A and A denotes the set of all possible realisation of A. Hereby the second equality has been obtained by conditioning on A = a while the last equality follows from Lebesgue's theorem of dominated convergence. Note that Lebesgue's theorem is applicable since and thus there exists t 0 > 0 independent of the realisation a such that for all t > t 0 the integrand is smaller than which, as a constant (with respect to A), is clearly P A -integrable. By Tonelli's theorem we thus obtain Applying Lemma 6.7 now yields (3.5). The proof of (3.7) can be carried out in complete analogy.
Proof of Lemma 3.3. Note that by definition where the random variables {Y i,k , k ∈ N} are i.i.d. copies of two generic random variables Y i , i = 1, 2. Fix i ∈ {1, 2} and assume that P(A i1 + A i2 = 0) < 1. Otherwise R i (t) is constant, Ψ i (u) = 0 and the statement is proven. Further assume for the moment, that neither A i1 = 0 a.s., nor A i2 = 0 a.s. Then, using Proposition 6.1 and the same argumentation as in the proof of Lemma 6.2 we obtain that Y i ∈ RV(min{α 1 , α 2 }). Thus the corresponding integrated tail functions are regularly varying as well, with index − min{α 1 , α 2 } + 1, and from [3, Thm. X.2.1] we obtain lim u→∞ Ψ i (u) By direct computation as a special case of the above. Clearly, the same argument also works for A i2 = 0 a.s. Finally note that by Tonelli's theorem for all i, j ∈ {1, 2} which yields (3.8).
Proof of Proposition 3.4. Assume (3.9) holds true. From Lemma 3.3 and its proof we obtain directly as u = u 1 + u 2 → ∞

Proofs for Section 4
Proof of Lemma 4.1. We take up the notation used in the proof of Lemma 3.3 and denote the jumps of the resulting one-dimensional risk processes by {Y i,k , k ∈ N}, i = 1, 2. Then the given bound for Ψ i (u) follows from [3, Thm. IV.5.2] with κ i > 0 such that c i κ i = λ(ϕ Y i (κ i ) − 1). (Note that in [3] the constants c and λ are combined as β = λ/c.) But since by conditioning with E[Y i ] as given in (6.3) and where, again by conditioning, which yields the given asymptotics.