Uniform Asymptotics for the Finite-Time Ruin Probability of a Time-Dependent Risk Model with Pairwise Quasiasymptotically Independent Claims

We consider a generalized time-dependent risk model with constant interest force, where the claim sizes are of pairwise quasiasymptotical independence structure, and the claim size and its interclaim time satisfy a dependence structure defined by a conditional tail probability of the claim size given the interclaim time before the claim occurs. As the claim-size distribution belongs to the dominated variation class, we establish some weakly asymptotic formulae for the tail probability of discounted aggregate claims and the finite-time ruin probability, which hold uniformly for all times in a relevant infinite interval.


Introduction
In the paper we will consider a generalized risk model of an insurance company, in which the claim sizes {X i , i ≥ 1} are nonnegative, identically distributed, but not necessarily independent random variables r.v.s with common distribution F and generic r.v.X, and their interarrival times {θ i , i ≥ 1} are other independent, identically distributed i.i.d., and nonnegative r.v.s with generic r.v.θ.To avoid triviality, X and θ are assumed not to be degenerate at 0. Denote the claim arrival times by τ 0 0, τ n n i 1 θ i , n ≥ 1, which constitute a renewal counting process as follows: with a finite-mean function λ t EN t ∞ i 1 P τ i ≤ t , t ≥ 0. Assume that, for every i ≥ 2, X i and τ i−1 are mutually independent.Meanwhile, as mentioned by Wang 1 , the total amount of premiums accumulated before t ≥ 0, denoted by C t with C 0 0 and C t < ∞

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almost surely for any fixed t > 0, is a nonnegative and nondecreasing stochastic process.Let r ≥ 0 be the constant interest force and x ≥ 0 be the insurer's initial reserve.Hence, the total reserve up to t ≥ 0 of the insurance company, denoted by U r t , satisfies as following: X i e −rτ i 1 {τ i ≤t} , 1.3 where S t N t i 1 X i is the aggregate claim amount before t ≥ 0 with S t 0 if N t 0. As usual, the ruin probability within a finite time t > 0 is defined by ψ r x, t P U r s < 0 for some 0 ≤ s ≤ t , 1.4 and the infinite-time ruin probability is ψ r x, ∞ P U r t < 0 for some 0 ≤ t < ∞ .

1.5
For the renewal risk model with i.i.d.claim sizes {X i , i ≥ 1} and i.i.d.interarrival times {θ i , i ≥ 1}, in which {X i , i ≥ 1} and {θ i , i ≥ 1} are mutually independent, there are many related works on ruin theory with a constant interest force r > 0, for example, see Kl üppelberg and Stadtim üller 2 , Kalashnikov and Konstantinides 3 , Konstantinides et al. 4 , Tang 5,6 , and Hao and Tang 7 , among others.However, the independence assumptions above are made mainly not for practical relevance but for theoretical interest.
In recent years, various extensions to the renewal risk model have been proposed to appropriately relax these independence assumptions.Generally, there are two directions to discuss the extensions.One is that a certain dependence structure is imposed on the claim sizes {X i , i ≥ 1} and/or their interarrival times {θ i , i ≥ 1}, but {X i , i ≥ 1} are assumed to be independent of {θ i , i ≥ 1}.See, for example, Chen and Ng 8 , Li et al. 9 , Yang and Wang 10 ,Wang et al. 11 ,and Liu et al. 12 , and references therein.The other is that the claim size X and its interarrival time θ follow a certain dependence structure, but { X i , θ i , i ≥ 1} are i.i.d.random pairs.In this direction, many researchers considered some ruin-related problems of a generalized risk model with a certain dependence between X and θ when the claim-sized distribution is light tailed, for example, Albrecher and Teugels 13 , Boudreault et al.The dependence structure between X and θ introduced by Asimit and Badescu 17 satisfies that the relation holds for some measurable function h • : 0, ∞ → 0, ∞ , where the symbol ∼ means that the quotient of both sides tends to 1 as x → ∞.When t is not a possible value of θ, the conditional probability in 1.6 is understood as an unconditional one, and then h t 1.If relation 1.6 holds uniformly for all t ∈ 0, ∞ , then, by conditioning on τ i−1 and θ i , i ≥ 1, it holds uniformly for all t ∈ 0, ∞ that where θ * is a r.v.independent of X and θ, with a proper distribution given by Remark 1.1.Note that the general dependence structure defined by 1.6 can cover both positive and negative dependence and is also easily verifiable for many common bivariate copulas, which can be found in Li et al. 18 .Furthermore, practitioners in insurance industry often meet the following situations: for autoinsurance or fire insurance, if the claim interarrival time is longer, then more measures can be taken to reduce the forthcoming losses of property, while if the deductible of the insured is raised, then the claim interarrival time will increase since some small claims can avoid.These situations tell us that the claim size and its interarrival time are interacted on each other, and the dependence structure defined by 1.6 is realistic in actuarial environments.
Based on the two study directions above to extend the renewal risk model, this paper will consider a more generalized risk model with the claim sizes following some dependence structure as well as the random pair X, θ satisfying relation 1.6 and establish the weakly asymptotic formulae for the tail probability of discounted aggregate claims and the finitetime ruin probability, which hold uniformly for all times in a relevant infinite interval.The method used in the paper is different from that in the above mentioned literatures, and the obtained results can extend and improve some existing results.
The rest of this paper is organized as follows: Section 2 will present the main results of this paper after preparing some preliminaries, and Section 3 will give some lemmas which are helpful to prove our main results in Section 4.

Preliminaries and Main Results
All limit relationships in the paper are for x → ∞ unless mentioned otherwise.For two positive functions a • and b • , we write a Following the first trend to extend the renewal risk model, in the paper we will discuss a risk model with the claim sizes satisfying the following dependence structure.Definition 2.1.Say that the r.v.s {ξ i , i ≥ 1} with their respective distributions V i , i ≥ 1 are pairwise quasiasymptotically independent if  Lehmann 28 , and even it can cover some sequences of positive dependent r.v.s.
For statement convenience, we denote by Λ the set of all t such that 0 Also we write Λ T 0, T ∩ Λ for any finite T ∈ Λ.Under the case that the claim sizes and/or interarrival times follow some dependence structure, Li et al. 9 obtained a weakly asymptotic formula for the finite-time ruin probability as follows.
Theorem 2.3.Consider the insurance risk model introduced in Section 1, in which the claim sizes are pairwise NQD with common distribution F ∈ D such that J − F > 0, the interarrival times {θ i , i ≥ 1} are NLOD, and the premium process {C t , t ≥ 0} is a deterministic linear function.If {X i , i ≥ 1} and {θ i , i ≥ 1} are mutually independent, then, for every fixed t ∈ Λ, In addition, if {C t , t ≥ 0} is a general stochastic process and {N t , t ≥ 0} is a delayed renewal counting process, Yang and Wang 10 also gave the formula 2.5 for ψ r x, t .Following the second extension direction, Li et al. 18 have investigated the tail behavior of discounted aggregate claims D r t for the renewal risk model with X and θ meeting the dependence structure defined by 1.6 .
Theorem 2.4.Consider the discounted aggregate claims defined in 1.3 with r > 0, in which { X i , θ i , i ≥ 1} are i.i.d.random pairs, and relation 1.6 holds uniformly for all t ∈ Λ.If F ∈ ERV −α, −β for some 0 < α ≤ β < ∞, and inf 0≤t≤t * h t > 0 for some t * ∈ Λ, then holds uniformly for t ∈ Λ, where Inspired by the results of Theorems 2.3 and 2.4, in this paper we will further discuss the following issues.
The following are the main results of this paper, among which the first two theorems discuss the tail behavior of the discounted aggregate claims D r t described by 1.3 .Theorem 2.5.Consider the discounted aggregate claims 1.3 described in Section 1 with r ≥ 0. If the claim sizes {X i , i ≥ 1} are pairwise quasiasymptotically independent r.v.s with common distribution F ∈ D such that J − F > 0, and relation 1.6 holds uniformly for all t ∈ Λ T , then it holds uniformly for all t ∈ Λ T that Additionally if F ∈ C, then relation 2.6 holds uniformly for all t ∈ Λ T .
The second theorem extends the set over which relations 2.6 and 2.8 hold uniformly to the whole set Λ.It is well known that if r 0, then D r t → ∞ almost surely as t → ∞, and hence it is not impossible to establish the uniformity of 2.6 and 2.8 for all t ∈ Λ.So we assume that r > 0 in the following result.In what follows, we will deal with the asymptotic behavior of the finite-time and infinite-time ruin probabilities, where we will discuss two cases: one is that the premium process {C t , t ≥ 0} is independent of {X i , i ≥ 1} and {N t , t ≥ 0}, and the other is that {C t , t ≥ 0} is not necessarily independent of {X i , i ≥ 1} or {N t , t ≥ 0}.For later use, we write the discounted value of premiums accumulated before time t as C t t 0 e −rs C ds . 2.9 Clearly, by the conditions on C t , it holds that 0 ≤ C t < ∞ almost surely for any fixed 0 < t < ∞.
where κ is any positive number.Further assume that F ∈ C, and then holds uniformly for all t ∈ Λ T .
Theorem 2.9.Consider the insurance risk model introduced with r > 0. Under the conditions of Theorem 2.6, relation 2.10 holds uniformly for t ∈ Λ, if either of the following conditions holds: 2 the total discounted amount of premiums satisfies the following:

2.16
Particularly, if F ∈ C, then relation 2.12 still holds uniformly for t ∈ Λ.
According to the uniformity of ψ r x, t for all t ∈ Λ in Theorem 2.9, we can immediately derive the corresponding result on the infinite-time ruin probability ψ r x, ∞ .Corollary 2.10.Under conditions of Theorem 2.9, one has 2.17

2.18
Remark 2.11.We would like to make some explanations of conditions F ∈ D with J − F > 0 in the main results.Wang and Yang 22 proposed the following assertions.

2.19
and D 1 , D 2 , and D 3 are pairwise disjoint sets.
ii Remark 2.12.By the expression of 2.7 , one can easily see that λ t is exactly the mean function of a delayed renewal process constituted by {θ * , θ i , i ≥ 2}.
Remark 2.13.The Lemma 3.3 below tells us that X i e −rτ i 1 {τ i ≤t} and X j e −rτ j 1 {τ j ≤t} are quasiasymptotically independent for all t ∈ Λ T and every fixed i / j ≥ 1 under the following assumptions: the claim sizes {X i , i ≥ 1} are pairwise quasiasymptotically independent with common distribution F ∈ D, and their interarrival times {θ i , i ≥ 1} are independent, identically distributed and such that 1.6 holds uniformly for all t ∈ Λ T .Hence, the dependence structures among claim sizes and that between claim size and its interarrival time in the paper are technically feasible.
Remark 2.14.In Theorems 2.7-2.9 and Corollaries 2.8-2.10, the independence between the premium process and the claim process in condition 1 has been extensively considered by Wang

Some Lemmas
In order to prove the main results, we need the following lemmas, among which the first lemma is a combination of Proposition 2. Lemma 3.1.For a distribution F on −∞, ∞ , the following assertions hold: 2 If F ∈ D, then for any p 1 < J − F and any p 2 > J F , there are positive numbers C i and D i , i 1, 2, such that Consider the insurance risk model introduced in Section 1 with r ≥ 0 and F ∈ D, where a generic pair X, θ is such that relation 1.6 holds uniformly for all t ∈ Λ T , and then 1 the distribution of the X i e −rτ i 1 {τ i ≤t} belongs to the class D for every fixed i ≥ 1 and all t ∈ Λ T ; moreover, if F ∈ C, then the distribution of the X i e −rτ i 1 {τ i ≤t} still belongs to the class C for every fixed i ≥ 1 and all t ∈ Λ T ; 2 for every fixed i ≥ 1 and all t ∈ Λ T , it holds that Proof. 1 If F ∈ D, then we have from 1.7 and Theorem 3.3 ii of Cline and Samorodnitsky 33 that, for all y > 0, lim sup holds for every fixed i ≥ 1 and all t ∈ Λ T , which implies that the distribution of the X i e which also implies that the distribution of the X i e −rτ i 1 {τ i ≤t} belongs to the class D for every fixed i ≥ 1 and all t ∈ Λ T .
2 By F ∈ D, 1.7 and Theorem 3.3 iv of Cline and Samorodnitsky 33 , we can prove that 3.3 holds.Lemma 3.3.Under the conditions of Theorem 2.5, if F ∈ D, then X i e −rτ i 1 {τ i ≤t} and X j e −rτ j 1 {τ j ≤t} are still quasiasymptotically independent for all t ∈ Λ T and every fixed i / j ≥ 1.
Proof.Without loss of generality, we assume that j > i ≥ 1.By conditioning on τ i−1 and θ i , i ≥ 1, and using 1.6 and 1.7 , it holds for all t ∈ Λ T that

3.7
Let H i and H j be the distributions of W i and W j , respectively, and H s, t be the joint distribution of W i and W j , then we see that

3.8
For A 1 x , by 2.2 , it follows that 3.9 Then, by Theorem 3.3 iv of Cline and Samorodnitsky 33 , we have

3.10
Similarly, we also have

3.11
Therefore, we attain 3.7 , and then we complete the proof.

3.12
Additionally, if F ∈ C, then it holds uniformly for all t ∈ Λ T that

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Proof.Clearly, for every fixed n ≥ 1 and all t ∈ Λ T , we have

3.14
Form Lemma 3.3, we see that lim sup 3.15 which, along with 3.14 , leads to that holds uniformly for all t ∈ Λ T .On the other hand, we have that, for any fixed 0 < v < 1, 3.17

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For A 4 , it follows from F ∈ D that for any fixed ε > 0, there exists a x 1 > 0 such that, for all x ≥ x 1 and all t ∈ Λ T , we have

3.18
Hence, we can derive by the arbitrariness of ε > 0 and 0 < v < 1 that holds uniformly for all t ∈ Λ T , where L −1 F is of sense from Lemma 3.1 1 .For A 5 , it holds uniformly for all t ∈ Λ T that where in the second last step we used Lemma 3.3, and in the last step we used F ∈ D and Lemma 3.2 1 .Hence, substituting 3.19 and 3.20 into 3.17 , it holds uniformly for all t ∈ Λ T that This, along with 3.16 , proves the uniformity of 3.12 for all t ∈ Λ T .Additionally, if F ∈ C, then L F 1, and thus we get the uniformity of 3.13 by 3.12 .
Lemma 3.5.Under the conditions of Theorem 2.6, relation 2.8 holds for every fixed t ∈ Λ.
Additionally, if F ∈ C, then relation 2.6 holds for every fixed t ∈ Λ.
Proof.Clearly, for every integer i ≥ 2 and every fixed t ∈ Λ, X i e −rτ i 1 {τ i ≤t} ≤ X i e −rτ i−1 , where X i and e −rτ i−1 are independent, note that, for any n ≥ 1 and all t ∈ Λ T ,

3.22
For A 6 , it follows by 3.1 that, for all x ≥ D 1 , Ee −rp 1 τ 1 i .

3.23
Then for any given ε > 0, there exists some positive integer n 0 such that for all n ≥ n 0 , A 6 ≤ εF x .

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For A 7 , by Markov's inequality, we obtain that, for some p 2 > J F ,

3.25
On the one hand when 0 < J F < 1, applying the inequality |a b| r ≤ |a| r |b| r for 0 < r < 1 and any number a, b, we have

3.28
Hence, combining 3.27 and 3.28 and letting 3.29 Substituting 3.29 into 3.26 and by 3.24 , we deduce that, for all n ≥ n 0 ,

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On the other hand when J F > 1, by Minkowski's inequality and along with the similar lines of the proof of the case when 0 < J F < 1, we also attain that, for some constant C > 0, where in the second last step we used 3.1 .Therefore, for all n ≥ n 0 , it still holds that A 7 ≤ εF x .

3.34
Let n 0 be fixed as above.Applying 3.12 in Lemma 3.4, 3.34 , and Lemma 3.2 2 in turn, we find that, for every fixed t ∈ Λ,

3.36
By contrast, for n 0 as above and any fixed 0 < v < 1, we obtain from 3.12 and 3.33 that, for every fixed t ∈ Λ,

3.37
For A 8 , arguing as the proof of 3.19 leads to

3.39
From 3.37 to 3.39 and by the arbitrariness of ε > 0, we obtain that, for every fixed t ∈ Λ,

Proofs of Main Results
Proof of Theorem 2.5.From the proof of Lemma 3.5, we can see that the relations 3.33 -3.40 still hold uniformly for all t ∈ Λ T , and then we get the uniformity of 2.8 for all t ∈ Λ T immediately.As F ∈ C, the uniformity of 2.6 over all t ∈ Λ T is clear.
Proof of Theorem 2.6.According to the proof of Lemma 4.2 of Hao and Tang 7 or see the proof of 4.3 of Tang 6 , we know that, for an arbitrarily fixed ε > 0, there exists some Combining with Theorem 2.5, it suffices to prove that relation 2.8 holds uniformly for all t ∈ T 0 , ∞ .On the one hand, by Lemma 3.5 with t T 0 and 4.1 , it holds uniformly for all t ∈ T 0 , ∞ that

4.2
On the other hand, by Lemma 3.5 and 4.1 , it holds uniformly for all t ∈ T 0 , ∞ that In the following, we establish the uniform asymptotic lower bound of ψ r x, t for all t ∈ Λ T .Under the condition 1 of Theorem 2.7, we deduce from 4.7 , Theorem 2.5, and F ∈ D that, for any fixed 0 < w < 1 and any given ε > 0, there exists some x 2 > 0 such that for all x ≥ x 2 and uniformly for all t ∈ Λ T , ψ r x, t ≥ P D r t > x C T

4.11
For I 1 , using Theorem 2.5 and F ∈ D, there exists some x 3 > 0 such that for all x ≥ x 3 and uniformly for all t ∈ Λ T ,

4.12
For I 2 , by the condition 2 of Theorem 2.7 and F ∈ D, we have lim sup When t ∈ Λ, by 4.13 and 3.2 in Lemma 3.1, there exists some x 4 > 0 such that for all x ≥ max{D 2 , x 4 } and uniformly for all t ∈ Λ T , where C C 2 e rp 2 t / λ t .When t / ∈ Λ, choose some 0 < σ < 1 such that t σ ≤ t, again by 4.13 , 3.2 , and arguing as 4.14 , there exists some x 4 > 0 such that for all x ≥ max{D 2 , x 4 } and uniformly for all t ∈ Λ T , where C C 2 e rp 2 t σ / λ t σ .Hence, from 4.11 to 4.15 and by the the arbitrariness of ε > 0 and 0 < δ < 1, we still obtain the uniformity of 4.10 for all t ∈ Λ T .This ends the proof for the uniformity of 2.10 over all t ∈ Λ T .
If F ∈ C, then L F 1. Therefore, we conclude from 2.10 that relation 2.12 uniformly holds for all t ∈ Λ T .
Proof of Corollary 2.8.Clearly, by Theorem 2.7, we know that relation 2.13 holds uniformly for all t ∈ Λ T .Note that for all finite t ∈ Λ, λ t F x κ λ t ≤ κ −1 x κ λ t x F y dy ≤ λ t F x .

4.16
By F ∈ D, it follows that, for any given ε > 0 and any fixed 0 < α < 1, there exists some x 5 > 0 such that for all x ≥ x 5 and uniformly for all t ∈ Λ T , λ t F x κ λ t ≥ λ t F x κ λ T 4.17 which, along with the arbitrariness of ε > 0 and 0 < α < 1, yields that Proof of Theorem 2.9.Applying 4.6 and Theorem 2.6, relation 4.8 still holds uniformly for all t ∈ Λ.On the other hand, by Theorem 2.7, we also attain that relation 4.10 holds uniformly for all t ∈ Λ T under conditions 1 and 2 of Theorem 2.9.Hence, we only need to show the uniformity of 4.10 for all t ∈ T, ∞ .Under the condition 1 of Theorem 2.9, similarly to the proof of 4.9 , we prove that, uniformly for all t ∈ T, ∞ , where the last step is due to 4.1 .Because ε > 0 and 0 < w < 1 are arbitrary, the relation 4.10 holds uniformly for all t ∈ T, ∞ .
Under the condition 2 of Theorem 2.9, we derive from 4.7 that, for the fixed 0 < δ < 1 as above and all t ∈ T, ∞ ,

4.21
For I 3 , by Theorem 2.6 with t T and the similar proof of 4.12 , it holds uniformly for all t ∈ T, ∞ that Consequently, using 4.21 -4.24 and by the arbitrariness of ε > 0 and 0 < δ < 1, we can obtain the uniformity of 4.10 for all t ∈ T, ∞ , and then we prove the uniformity of 2.10 over all t ∈ Λ.
From F ∈ C and 2.10 , it is easy that relation 2.12 holds uniformly for all t ∈ Λ T .
aggregate claims up to t ≥ 0 are expressed as

Theorem 2 . 6 .
Consider the discounted aggregate claims 1.3 described in Section 1 with r > 0. If the claim sizes {X i , i ≥ 1} are pairwise quasiasymptotically independent r.v.s with common distribution F ∈ D such that J − F > 0, and relation 1.6 holds uniformly for all t ∈ Λ, then relation 2.8 still holds uniformly for all t ∈ Λ.Further assume that F ∈ C, then relation 2.6 still holds uniformly for all t ∈ Λ.

D
−α, −β, A, B .2.21 In the particular case where A B 1, the class ERV ⊂ D 3 and this inclusion are proper.For example, the Peter and Paul distribution see Example 1.4.2 in Embrechts et al. 20 belongs to D 3 , but it does not belong to L; thus, it does not belong to the class ERV.

− ε t 0 F
with 1.7 and 2.7 , proves that for every fixed t ∈ Λ, P D r t > x 1 xe rs d λ t .

∞ 0 Pt 0 F∞ 0 t 0 F 1 ≥ 1 − ε F * 1 w t 0 F xe rs d λ s , 4 . 9 which
D r t > x y P C T ∈ dy ∞ 0 x y e rs d λ s P C T ∈ dy ≥ w xe rs d λ s P C T ∈ dy , along with the arbitrariness of ε > 0 and 0 < w <

0 T 0 F 0 T 0 F 1 0 F 0 F
ψ r x, t ≥ P D r T > x C ∞ 0 P D r T > x y P C ∈ dy ∞ x y e rs d λ s P C ∈ dy ≥ ∞ w xe rs d λ s P C ∈ dy ≥ 1 − ε F * 1 w T xe rs d λ s ≥ 1 − 2ε F * 1 w t xe rs d λ s , 4.20 ψ r x, t ≥ P D r T > x C ≥ P D r T > 1 δ x − P C > δx I 3 − I 4 .

F 1 0 F 0 FI 4 ≤ ε t 0 F
δ xe rs d λ s ≥ 1 − ε F * 1 δ T xe rs d λ s ≥ 1 − 2ε F * 1 δ t xe rs d λ s , 4.22where the last step is also due to 4.1 .For I 4 , by condition 2 of Theorem 2.9 and F ∈ D, we get that, for all t ∈ T, ∞ , , for all t ∈ T, ∞ and all large x, xe rs d λ s .4.24 14 , Cossette et al. 15 , and Badescu et al. 16 .Besides, Asimit and Badescu 17 introduced a general dependence structure for X, θ and presented the tail behavior of discounted aggregate claims D r t for the compound Poisson model with constant interest force and heavy-tailed claims.Under the dependence structure of Asimit and Badescu 17 , Li et al. 18 extended the tail behavior of D r t to the renewal risk model.
20r more details of heavy-tailed distributions and their applications to insurance and finance, the readers are referred to Bingham et al.19and Embrechts et al.20.Additionally, some reviews on the class D and its application can be found in Shneer 21 , Wang and Yang 22 , and others. 2 v.s see Wang et al. 11 , extended negatively dependent r.v.s see Liu 26 , negatively upper orthant dependent/negatively lower orthant dependent r.v.s NUOD/NLOD, see Block et al. 27 , pairwise negatively quadrant dependent r.v.s NQD, see 1 , Yang andWang 10 , Wang et al. 11 , Liu et al. 12, and many others, while condition 2, which does not require the independence between the premium process and the claim process, allows for a more realistic case that the premium rate varies as a deterministic or stochastic function of the insurer's current reserve, as that considered by Petersen 29 , Michaud 30 , Jasiulewicz 31 , andTang 5 .
34RN Probability and Statisticsthen by 1.7 and Lemma 2.5 of Wang et al.34, we derive that, for every fixed i ≥ 1 and all t ∈ Λ T , −rτ i 1 {τ i ≤t} belongs to the class D for every fixed i ≥ 1 and all t ∈ Λ T .Moreover, if F ∈ C, i 1 {τ i ≤t} > xy xe rs d λ s .By relations 4.2 , 4.3 , and the arbitrariness of ε > 0, we get the uniformity of relation 2.8 for all t ∈ T 0 , ∞ .As F ∈ C, we have L F 1, and hence relation 2.6 holds uniformly for all t ∈ Λ.
t F x κ λ t L F λ t F x 4.18 holds uniformly for all t ∈ Λ T .Thus, we obtain from 4.16 and 4.18 that, uniformly for all t ∈ Λ T , So, combining 2.13 and 4.19 leads to the uniformity of 2.14 for all t ∈ Λ T .If F ∈ C, then L F 1, and hence the uniformity of 2.15 for all t ∈ Λ T is proved immediately.