Tail bounds for the joint distribution of the surplus prior to and at ruin

For the classical risk model with Poisson arrivals, we study the (bivariate) tail of the joint distribution of the surplus prior to and at ruin. We obtain some exact expressions and new bounds for this tail, and we suggest three numerical methods that may yield upper and lower bounds for it. As a by-product of the analysis, we obtain new upper and lower bounds for the probability and severity of ruin. Many of the bounds in the present paper improve and generalise corresponding bounds that have appeared earlier. For the numerical bounds, their performance is also compared against bounds available in the literature.     

Upper comonotonicity and convex upper bounds for sums of random variables

It is well-known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to convex order. However, replacing the (unknown) copula by the comonotonic copula will in most cases not reflect reality well. For instance, in an insurance context we may have partial information about the dependence structure of different risks in the lower tail. In this paper, we extend the aforementioned result, using the concept of upper comonotonicity, to the case where the dependence structure of a random vector in the lower tail is already known. Since upper comonotonic random vectors have comonotonic behavior in the upper tail, we are able to extend several well-known results of comonotonicity to upper comonotonicity. As an application, we construct different increasing convex upper bounds for sums of random variables and compare these bounds in terms of increasing convex order.     

Large deviations and the generalized processor sharing scheduling for a multiple-queue system

We establish asymptotic upper and lower bounds on the asymptotic decay rate of per-session queue length tail distributions for a multiple-queue system where a single constant rate server services the queues using the generalized processor sharing (GPS) scheduling discipline. In the special case where there are only two queues, the upper and lower bounds match, yielding the optimal bound proved in [15]. The dynamics of bandwidth sharing of a multiple-queue GPS system is captured using the notion of partial feasible sets, and the bounds are obtained using the sample-path large deviation principle. The results have implications in call admission control for high-speed communication networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

Comonotonic bounds on the survival probabilities in the Lee–Carter model for mortality projection

In the Lee–Carter framework, future survival probabilities are random variables with an intricate distribution function. In large homogeneous portfolios of life annuities, value-at-risk or conditional tail expectation of the total yearly payout of the company are approximately equal to the corresponding quantities involving random survival probabilities. This paper aims to derive some bounds in the increasing convex (or stop-loss) sense on these random survival probabilities. These bounds are obtained with the help of comonotonic upper and lower bounds on sums of correlated random variables.     

Analysis of transient queues with semidefinite optimization

We derive upper bounds on the tail distribution of the transient waiting time in the GI/GI/1 queue, given a truncated sequence of the moments of the service time and that of the interarrival time. Our upper bound is given as the objective value of the optimal solution to a semidefinite program (SDP) and can be calculated numerically by solving the SDP. We also derive the upper bounds in closed form for the case when only the first two moments of the service time and those of the interarrival time are given. The upper bounds in closed form are constructed by formulating the dual problem associated with the SDP. Specifically, we obtain the objective value of a feasible solution of the dual problem in closed from, which turns out to be the upper bound that we derive. In addition, we study bounds on the maximum waiting time in the first busy period.     

带有相依重尾冲击的Poisson噪音过程尾的一致渐近性质

本文考虑了带有某种相依重尾冲击的Poisson噪音过程尾的一致渐近性质.当冲击是二元上尾渐近独立的非负随机变量具有长尾和控制变化尾分布且噪音函数具有正的上下界时,得到了过程尾概率的一致渐近公式.进而,当冲击具有连续的一致变化尾分布时,去除了噪音函数具有正的下界的限制.对于噪音函数不一定具有正的上界的情形,当冲击具有两两负象限相依结构时,也得到了一致渐近性结果.     

Inequalities for the Moments and Distribution of the Ladder Height of a Random Walk

We obtain upper bounds for the tail distribution of the first nonnegative sum of a random walk and for the moments of the overshoot over an arbitrary nonnegative level if the expectation of jumps is positive and close to zero. In addition, we find an estimate for the expectation of the first ladder epoch.     

Tail bounds for occupancy and the satisfiability threshold conjecture

The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following well-known conjecture on threshold phenomenon for the satisfiability problem. Consider random 3-SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c* ≈? 4.2. We provide a strong upper bound on the value of c*, showing that for c > 4.758 a random 3-SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds.     

多参数情形下 MLE 的大偏差

在大样本点估计理论中,我们一般只考虑相合估计,对于任何一个相合估计 T_n 以及任何给定的ε>0,其尾概率     

Upper tail probabilities of integrated Brownian motions

We obtain new upper tail probabilities of m-times integrated Brownian motions under the uniform norm and the Lp norm. For the uniform norm, Talagrand’s approach is used, while for the Lp norm, Zolotare’s approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities(large ball probabilities) for general Gaussian random variables in Banach spaces. As applications,explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of m-times integrated Brownian motions are presented as well.     

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.     

On upper bounds for the tail distribution of geometric sums of subexponential random variables

The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.     

The deficit at ruin in the Sparre Andersen model with interest

In this paper, we consider the Sparre Andersen risk model modified by the inclusion of interest on the surplus. By using the techniques of Cai and Dickson [Ins.: Math. Econ. 32(2003)], we give the functional and also the exponential type upper bounds for the tail probability of the deficit at ruin. Some special cases are also discussed.     

Sojourn time asymptotics in Processor Sharing queues with varying service rate

This paper addresses the sojourn time asymptotics for a GI/GI/? queue operating under the Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the job-size distribution has a light tail. Whereas upper bounds on the decay rate can be derived under fairly general conditions, the establishment of the corresponding lower bounds requires that the service process satisfies a sample-path large-deviation principle. We show that the class of allowed service processes includes the case where the service rate is modulated by a Markov process. Finally, we extend our results to a similar system operation under the Discriminatory Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations techniques.     

上尾渐近独立随机变量和的大偏差的渐近下界(英文)

本文研究了多元风险模型中服从长尾分布的带上尾渐近独立的随机变量和的大偏差渐近下界.利用大偏差的经典求法,得到了随机变量的非随机和和随机和的大偏差表达式,推广了独立同分布情形下的相关结论.     

Tail asymptotic for discounted aggregate claims with one-sided linear dependence and general investment return

In this study, we investigate the tail probability of the discounted aggregate claim sizes in a dependent risk model. In this model, the claim sizes are observed to follow a one-sided linear process with independent and identically distributed innovations. Investment return is described as a general stochastic process with c`adl`ag paths. In the case of heavy-tailed innovation distributions, we are able to derive some asymptotic estimates for tail probability and to provide some asymptotic upper bounds to improve the applicability of our study.     

Bounded size bias coupling: a Gamma function bound,and universal Dickman-function behavior

Under the assumption that the distribution of a nonnegative random variable \(X\) admits a bounded coupling with its size biased version, we prove simple and strong concentration bounds. In particular the upper tail probability is shown to decay at least as fast as the reciprocal of a Gamma function, guaranteeing a moment generating function that converges everywhere. The class of infinitely divisible distributions with finite mean, whose Lévy measure is supported on an interval contained in \([0,c]\) for some \(c < \infty \), forms a special case in which this upper bound is logarithmically sharp. In particular the asymptotic estimate for the Dickman function, that \(\rho (u) \approx u^{-u}\) for large \(u\), is shown to be universal for this class. A special case of our bounds arises when \(X\) is a sum of independent random variables, each admitting a 1-bounded size bias coupling. In this case, our bounds are comparable to Chernoff–Hoeffding bounds; however, ours are broader in scope, sharper for the upper tail, and equal for the lower tail. We discuss bounded and monotone couplings, give a sandwich principle, and show how this gives an easy conceptual proof that any finite positive mean sum of independent Bernoulli random variables admits a 1-bounded coupling with the same conditioned to be nonzero.     

Tail bounds for the distribution of the deficit in the renewal risk model

We obtain upper and lower bounds for the tail of the deficit at ruin in the renewal risk model, which are (i) applicable generally; and (ii) based on reliability classifications. We also derive two-side bounds, in the general case where a function satisfies a defective renewal equation, and we apply them to the renewal model, using the function Λ_u introduced by [Psarrakos, G., Politis, K., 2007. A generalisation of the Lundberg condition in the Sparre Andersen model and some applications (submitted for publication)]. Finally, we construct an upper bound for the integrated function and an asymptotic result when the adjustment coefficient exists.     

New upper solution bounds of the discrete algebraic Riccati matrix equation

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.     

Tails of waiting times and their bounds

Tails of distributions having the form of the geometric convolution are considered. In the case of light-tailed summands, a simple proof of the famous Cramér asymptotic formula is given via the change of probability measure. Some related results are obtained, namely, bounds of the tails of geometric convolutions, expressions for the distribution of the 1st failure time and failure rate in regenerative systems, and others. In the case of heavy-tailed summands, two-sided bounds of the tail of the geometric convolution are given in the cases where the summands have either Pareto or Weibull distributions. The results obtained have the property that the corresponding lower and upper bounds are tailed-equivalent. This revised version was published online in June 2006 with corrections to the Cover Date.