Precise Large Deviations Research Articles

Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of Nagaev (Theory Probab Appl 14:51–64, 193–208, 1969) and Nagaev (Ann Probab 7:745–789, 1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (Stoch Proc Appl 44:291–327, 1993; 68:1–20, 1997) in the context of central limit theorems with infinite variance stable limits. We illustrate the principle for stochastic volatility models, real valued functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations.

Precise large deviations of aggregate claims in a size-dependent renewal risk model

Consider a renewal risk model in which claim sizes and inter-arrival times correspondingly form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure described via the conditional distribution of the inter-arrival time given the subsequent claim size being large. We study large deviations of the aggregate amount of claims. For a heavy-tailed case, we obtain a precise large-deviation formula, which agrees with existing ones in the literature.

Precise large deviations for compound random sums in the presence of dependence structures

In this paper, we deal with the compound random sums of dependent real-valued random variables with heavy-tailed distributions. We establish a precise large deviation result for a nonstandard renewal risk model in which innovations are extended negatively dependent real-valued random variables with a common dominatedly varying distribution function, their interarrival times are extended negatively dependent nonnegative random variables, and the numbers of innovations caused by individual events are also extended negatively dependent positive random variables. As an illustration of the obtained result, we give two applications related to some insurance risk models.

Lower bounds of large deviation for sums of long-tailed claims in a multi-risk model

In view of the actual condition of the insurance company, a multi-risk model is proposed. The lower bound for the sums of long-tailed claims in this model is given. The proof method is based on the results of precise large deviation for long-tailed distributions.

Stochastic Risk Networks: Modeling, Analysis and Efficient Monte Carlo

We propose a dynamic insurance network model that allows to deal with reinsurance default contagion risks with a particular aim of capturing cascading effects at the time of defaults. We capture these effects by finding an equilibrium allocation of settlements which can be found as the unique optimal solution of an optimization problem. This equilibrium allocation recognizes the correlation among the risk factors, the contractual obligations, which are assumed to follow popular contracts in the insurance industry (such as stop-loss), and the interconnections of the insurance-reinsurance network. We are able to obtain an asymptotic description of the most likely ways in which the default of a specific group of insurers can occur, by means of solving a multidimensional Knapsack integer programming problem. Finally, we propose a class of strongly efficient estimators (in a precise large deviations sense) for computing the expected loss of the network conditioning on the failure of a specific set of companies.

Extended Precise Large Deviations of Random Sums in the Presence of END Structure and Consistent Variation

The study of precise large deviations of random sums is an important topic in insurance and finance. In this paper, extended precise large deviations of random sums in the presence of END structure and consistent variation are investigated. The obtained results extend those of Chen and Zhang (2007) and Chen et al. (2011). As an application, precise large deviations of the prospective‐ loss process of a quasirenewal model are considered.

Precise Large Deviations for Sums of Negatively Dependent Random Variables with Common Long-Tailed Distributions

We investigate the precise large deviations for negatively dependent random variables. We prove general asymptotic relations for both the partial sums S n for the long tailed distributions and the random sums S t for the subexponential distributions, where the N t is an integer counting process. It is found out that the precise large deviations for negatively dependent random variables are insensitive to this kind of dependence. Finally, we present applications on the classical counting processes, Poisson, and renewal.

Precise Large Deviations for Long-Tailed Distributions

In this paper, we investigate the precise large deviations for sums of independent identically distributed random variables with heavy-tailed distributions. We prove asymptotic relations for non-random sums and for random sums of random variables with long-tailed distributions. We apply the results on two useful counting processes, namely, renewal and compound-renewal processes.

Precise large deviations for a customer-based individual risk model

In this paper, we propose a customer-based individual risk model, in which potential claims by customers are described as i.i.d. heavy-tailed random variables, but different insurance policy holders are allowed to have different probabilities to make actual claims. Some precise large deviation results for the prospective-loss process are derived under certain mild assumptions, with emphasis on the case of heavy-tailed distribution function class ERV (extended regular variation). Lundberg type limiting results on the finite time ruin probabilities are also investigated.

Large deviations and moderate deviations for sums of negatively dependent random variables

In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (−∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure.

Precise large deviations for consistently varying-tailed distributions in the compound renewal risk model

We investigate precise large deviations for heavy-tailed random sums. We prove a general asymptotic relation in the compound renewal risk model for consistently varying-tailed distributions. This model was introduced in [Q. Tang, C. Su, T. Jiang, and J.S. Zang, Large deviation for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52:91–100, 2001] as a more practical risk model. The proof is based on the inequality found in [D. Fuk and S.V. Nagaev, Probability for sums of independent random variables, Theory Probab. Appl., 16:600–675, 1971].

Precise Large Deviations of Random Sums in Presence of Negative Dependence and Consistent Variation

The study of precise large deviations for random sums is an important topic in insurance and finance. In this paper, we extend recent results of Tang (Electron J Probab 11(4):107–120, 2006) and Liu (Stat Probab Lett 79(9):1290–1298, 2009) to random sums in various situations. In particular, we establish a precise large deviation result for a nonstandard renewal risk model in which innovations, modelled as real-valued random variables, are negatively dependent with common consistently-varying-tailed distribution, and their inter-arrival times are also negatively dependent.

Local precise large deviations for sums of random variables with [formula omitted]-regularly varying densities

In this paper we establish a local precise large deviation result for sums S n , n = 1 , 2 , … of independent and identically distributed random variables X 1 , X 2 , … with O -regularly varying densities f . The asymptotic behavior of the probability P ( x < S n − E S n ≤ x + T ) is comparable, for fixed T , with quantities n T f ( x + E X 1 ) or n ( F ( x + T ) − F ( x ) ) .

Total progeny in killed branching random walk

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite (Gantert and Muller in Markov Process. Relat. Fields 12:805–814, 2006; Hu and Shi in Ann. Probab. 37(2):742–789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388–412, 1998; and Power laws and killed branching random walks) that E [Z] < ∞ while \({{\mathbf E}\left[Z\,{\rm log}\, Z\right]=\infty}\) . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.

Precise Large Deviations for the Actual Aggregate Loss Process

In this article, we propose a customer-arrival-based insurance risk model, in which customer's actual claim sizes are described as i.i.d. heavy-tailed r.v.'s multiplied with a generalized shot function, and the model can be treated as a generalized Poisson shot noise process. We obtain some precise large deviation results for the actual loss process and extend the results to the multi-risk model.

Precise large deviations for dependent random variables with heavy tails

By extending the negatively dependent (ND) structure, the paper puts forth the concept of extended negative dependence (END). The results show that the END structure has no effect on the asymptotic behavior of precise large deviations of partial sums and random sums for non-identically distributed random variables on ( − ∞ , + ∞ ) .

Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails

Let {Xk,k≥1} be a sequence of negatively dependent random variables with common distribution F and mean 0. Suppose that F̄(x)=1−F(x) is positive for all x and consistently varying as x→∞. Let {θk,k≥1} be another sequence of random variables independent of {Xk,k≥1} satisfying P(a≤θk≤b)=1 for some 0<a≤b<∞, k≥1. The paper investigates large deviations for the randomly weighted sums.

Precise large deviation results for the total claim amount under subexponential claim sizes

The paper deals with the renewal risk model. A precise large deviation result in the case of subexponential claim sizes is proved. As a special case, the example of Pareto distributed claim sizes and inter-occurrence times is investigated.

Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {X ij , j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n 1,…,n k ) = ∑ i=1 k ∑ j=1 n i X ij and random sums S(k; t) = ∑ i=1 k ∑ j=1 N i (t) X ij , where N i (t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.

Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {Xij, j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n1,…,nk) = ∑i=1k ∑j=1niXij and random sums S(k; t) = ∑i=1k ∑j=1Ni (t)Xij, where Ni(t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.