Precise Large Deviations Research Articles

Precise large deviations of the net loss process in a non-standard two-dimensional risk model

Abstract This paper investigates the precise large deviations of the net loss process in a two-dimensional risk model with consistently varying tails and dependence structures, and gives some asymptotic formulas which hold uniformly for all x varying in t-intervals. The study is among the initial efforts to analyze potential risk via large deviation results for the net loss process of the two-dimensional risk model, and can provide a novel insight to assess the operation risk in a long run by fully considering the premium income factors of the insurance company.

Precise Large Deviations for the Total Population of Heavy-Tailed Subcritical Branching Processes with Immigration

Precise Large Deviations for the Total Population of Heavy-Tailed Subcritical Branching Processes with Immigration

Tail asymptotics and precise large deviations for some Poisson cluster processes

Abstract We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

Precise large deviations for sums of dependent random variables with subexponential distribution

. In this article, we study the precise large deviations for sums of non identically distributed random variables { X i , i ≥ 1 } with distribution functions { F i , i ≥ 1 } that are widely upper orthant dependent (WUOD, in short) and ϕ-mixing. Moreover, the random variables are dependent, according to some copulas. We assume that the average of the right tails of distribution functions F i is equivalent to the subexponential distribution F.

Precise large deviations for non-centralized sums of partial sums and random sums of heavy-tailed END random variables

This paper obtains some asymptotic upper and lower bounds of precise large deviations for non-centralized sums of partial sums and random sums, where the summands are extended negatively dependent random variables with dominatedly-varying-tailed distributions supported on (−∞,+∞).

Precise large deviations for a multidimensional risk model with regression dependence structure

AbstractIn this paper, we consider a nonstandard multidimensional risk model, in which the claim sizes $\{\vec{X}_k, k\ge 1\}$ form an independent and identically distributed random vector sequence with dependent components. By assuming that there exists the regression dependence structure between inter-arrival time and the claim-size vectors, we extend the regression dependence to a more practical multidimensional risk model. For the univariate marginal distributions of claim vectors with consistently varying tails, we obtain the precise large deviation formulas for the multidimensional risk model with the regression size-dependent structure.

Precise large deviations of aggregate claims in bidimensional risk model with dependence structures

In this article, we consider a bidimensional risk model with strong asymptotic independence between marginal claims having consistent varying tails and study the asymptotics of precise large deviations of aggregate claims by allowing claim-size vectors and their inter-arrival times to be arbitrarily dependent. This study can provide novel insights into the various dependence structures imposed on the involved modeling components.

Precise large deviations in a non stationary risk model with arbitrary dependence between subexponential claim sizes and waiting times

Consider a risk model in which the claims follow a non stationary arrival process that satisfies a large deviation principle. Supposing that the claim sizes form a sequence of i.i.d. random variables with subexponential tail, precise large deviations for the aggregate claims are obtained, by allowing the claim-sizes and claim inter-arrival (waiting) times to be arbitrarily dependent.

Precise large deviations for aggregate claims in a multidimensional risk model with arbitrarily dependent claims and accident-arriving times

Precise large deviations for aggregate claims in a multidimensional risk model with arbitrarily dependent claims and accident-arriving times

Precise large deviations for aggregate claims in a two-dimensional compound dependent risk model

<abstract><p>This paper considers a two-dimensional compound risk model. We mainly investigate the claim sizes and inter-arrival times are size-dependent. When the claim sizes have consistently varying tails, we obtain the precise large deviations for aggregate amount of claims in the above dependent compound risk model.</p></abstract>

Precise large deviations for aggregate claims of a compound renewal risk model with arbitrary dependence between claim sizes and waiting times*

Precise large deviations for aggregate claims of a compound renewal risk model with arbitrary dependence between claim sizes and waiting times*

Precise large deviation estimates for branching process in random environment

Nous considérons un processus de branchement dans un environnement aléatoire {Zn}n≥0, qui est le processus de croissance d’une population où les individus se reproduisent indépendamment les uns des autres avec la loi de reproduction choisie au hasard à chaque génération. Nous décrivons des asymptotiques précises des grandes déviations supérieures, c’est-à-dire de P[Zn>eρn]. De plus dans le cas sous-critique, sous la condition de Cramér pour la moyenne de la loi de reproduction, nous étudions les estimations des grandes déviations pour le premier temps de passage de ce processus de branchement et de la taille totale de sa population.

Precise large deviation for sums of sub-exponential claims with the [formula omitted]-dependent semi-Markov type structure

Consider a renewal risk model in which the current inter-arrival time depends on a fixed number of previous claims but is independent of all other claims. We are interested in the precise large deviations of the aggregate amount of claims for the case of sub-exponential claims. In addition, an asymptotic estimate for the expectation of the aggregate amount of claims is also obtained. The asymptotic formulas are fully consistent with existing works in this study.

Precise large deviations in a bidimensional risk model with arbitrary dependence between claim-size vectors and waiting times

Consider a bidimensional risk model in which an insurer simultaneously confronts two types of claims sharing a common non-stationary arrival process, and the claim-sizes {X→k;k≥1} form a sequence of i.i.d. random vectors with nonnegative components being dependent on each other. Supposing that the univariate marginal distributions of the claim-size vectors have dominatedly varying tails, precise large deviations for the aggregate amount of claims are obtained, by allowing that the claim-size vectors and claim inter-arrival (waiting) times are arbitrarily dependent.

Precise large deviations of aggregate claims in a nonstandard risk model with arbitrary dependence between claim sizes and waiting times

<abstract><p>Recently, Chen et al.<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup> investigated the precise large deviations of aggregate claims in a renewal risk model with arbitrary dependence between claim sizes and their waiting times. In this paper, we extend their results to a nonstandard risk model in which various dependence structures are imposed on the modeling components, and obtain the asymptotic lower and upper bounds of precise large deviations for aggregate claims, which hold uniformly for all $ x $ in a $ t $-interval.</p></abstract>

Precise Large Deviations of Aggregate Claims in a Compound Risk Model with Arbitrary Dependence between Claim Sizes and Waiting Times

Precise Large Deviations of Aggregate Claims in a Compound Risk Model with Arbitrary Dependence between Claim Sizes and Waiting Times

Estimations for aggregate amount of claims in a risk model with arbitrary dependence between claim sizes and inter-arrival times

<abstract><p>This paper considers a compound risk model, in which the individual claim sizes and their inter-arrival times can be arbitrarily dependent. We mainly investigate the claim sizes are extended negatively dependent. When the claim sizes have consistently-varying-tailed distributions, we obtain precise large deviations of the aggregate amount of claims in the above dependent compound risk model.</p></abstract>

Critical fluctuations in renewal models of statistical mechanics

We investigate the sharp asymptotic behavior at criticality of the large fluctuations of extensive observables in renewal models of statistical mechanics, such as the Poland–Scheraga model of DNA denaturation, the Fisher–Felderhof model of fluids, the Wako–Saitô–Muñoz–Eaton model of protein folding, and the Tokar–Dreyssé model of strained epitaxy. These models amount to Gibbs changes of measure of a classical renewal process and can be identified with a constrained pinning model of polymers. The extensive observables that enter the thermodynamic description turn out to be cumulative rewards corresponding to deterministic rewards that are uniquely determined by the waiting time and grow no faster than it. The probability decay with the system size of their fluctuations switches from exponential to subexponential at criticality, which is a regime corresponding to a discontinuous pinning–depinning phase transition. We describe such decay by proposing a precise large deviation principle under the assumption that the subexponential correction term to the waiting time distribution is regularly varying. This principle is, in particular, used to characterize the fluctuations of the number of renewals, which measures the DNA-bound monomers in the Poland–Scheraga model, the particles in the Fisher–Felderhof model and the Tokar–Dreyssé model, and the native peptide bonds in the Wako–Saitô–Muñoz–Eaton model.

Precise Large Deviations for Sums of Claim-size Vectors in a Two-dimensional Size-dependent Renewal Risk Model

Precise Large Deviations for Sums of Claim-size Vectors in a Two-dimensional Size-dependent Renewal Risk Model

Point process convergence for the off-diagonal entries of sample covariance matrices

We study point process convergence for sequences of i.i.d. random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a (2+δ)th moment. We make heavy use of precise large deviation results for sums of i.i.d. random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.