Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored $\delta$ -shock model, $\delta \ge 1$ , for which the system fails whenever no shock occurs within a $\delta$ -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob. 32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order $\delta$ (a geometric distribution of order $\delta$ under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored $\delta$ -shock model, for which the system fails when no shock occurs within a $\delta$ -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold $\gamma >0$ . Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.

We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.

In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.

In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.

In this paper, we establish some stochastic comparison results for largest claim amounts of two sets of independent and also for interdependent portfolios under the setup of the proportional odds model. We also establish stochastic comparison results for aggregate claim amounts of two sets of independent portfolios. Further, stochastic comparisons for largest claim amounts from two sets of independent multiple-outlier claims have also been studied. The results we obtained apply to the whole family of extended distributions, also known as the Marshall–Olkin family of distributions. We have given many numerical examples to illustrate the results obtained.

The Bruss–Robertson–Steele (BRS) inequality bounds the expected number of items of random size which can be packed into a given suitcase. Remarkably, no independence assumptions are needed on the random sizes, which points to a simple explanation; the inequality is the integrated form of an $\omega$ -by- $\omega$ inequality, as this note proves.

The principle of maximum entropy is a well-known approach to produce a model for data-generating distributions. In this approach, if partial knowledge about the distribution is available in terms of a set of information constraints, then the model that maximizes entropy under these constraints is used for the inference. In this paper, we propose a new three-parameter lifetime distribution using the maximum entropy principle under the constraints on the mean and a general index. We then present some statistical properties of the new distribution, including hazard rate function, quantile function, moments, characterization, and stochastic ordering. We use the maximum likelihood estimation technique to estimate the model parameters. A Monte Carlo study is carried out to evaluate the performance of the estimation method. In order to illustrate the usefulness of the proposed model, we fit the model to three real data sets and compare its relative performance with respect to the beta generalized Weibull family.

We consider the asset price as the weak solution to a stochastic differential equation driven by both a Brownian motion and the counting process martingale whose predictable compensator follows shot-noise and Hawkes processes. In this framework, we discuss the Esscher martingale measure where the conditions for its existence are detailed. This generalizes certain relationships not yet encountered in the literature.

We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

Negative dependence of sequences of random variables is often an interesting characteristic of their distribution, as well as a useful tool for studying various asymptotic results, including central limit theorems, Poisson approximations, the rate of increase of the maximum, and more. In the study of probability models of tournaments, negative dependence of participants’ outcomes arises naturally, with application to various asymptotic results. In particular, the property of negative orthant dependence was proved in several articles for different tournament models, with a special proof for each model. In this note we unify these results by proving a stronger property, negative association, a generalization leading to a very simple proof. We also present a natural example of a knockout tournament where the scores are negatively orthant dependent but not negatively associated. The proof requires a new result on a preservation property of negative orthant dependence that is of independent interest.

Consider the coupon collector problem where each box of a brand of cereal contains a coupon and there are n different types of coupons. Suppose that the probability of a box containing a coupon of a specific type is $1/n$ , and that we keep buying boxes until we collect at least m coupons of each type. For $k\geq m$ call a certain coupon a k-ton if we see it k times by the time we have seen m copies of all of the coupons. Here we determine the asymptotic distribution of the number of k-tons after we have collected m copies of each coupon for any k in a restricted range, given any fixed m. We also determine the asymptotic joint probability distribution over such values of k, and the total number of coupons collected.

We propose a stochastic model for the failure times of items subject to two external random shocks occurring as events in an underlying bivariate counting process. This is a special formulation of the competing risks model, which is of interest in reliability theory and survival analysis. Specifically, we assume that a system, or an item, fails when the sum of the two types of shock reaches a critical random threshold. In detail, the two kinds of shock occur according to a bivariate space-fractional Poisson process, which is a two-dimensional vector of independent homogeneous Poisson processes time-changed by an independent stable subordinator. Various results are given, such as analytic hazard rates, failure densities, the probability that the failure occurs due to a specific type of shock, and the survival function. Some special cases and ageing notions related to the NBU characterization are also considered. In this way we generalize certain results in the literature, which can be recovered when the underlying process reduces to the homogeneous Poisson process.

Layer reinsurance treaty is a common form obtained in the problem of optimal reinsurance design. In this paper, we study allocations of policy limits in layer reinsurance treaties with dependent risks. We investigate the effects of orderings and heterogeneity among policy limits on the expected utility functions of the terminal wealth from the viewpoint of risk-averse insurers faced with right tail weakly stochastic arrangement increasing losses. Orderings on optimal allocations are presented for normal layer reinsurance contracts under certain conditions. Parallel studies are also conducted for randomized layer reinsurance contracts. As a special case, the worst allocations of policy limits are also identified when the exact dependence structure among the losses is unknown. Numerical examples are presented to shed light on the theoretical findings.

We study large-deviation probabilities of Telecom processes appearing as limits in a critical regime of the infinite-source Poisson model elaborated by I. Kaj and M. Taqqu. We examine three different regimes of large deviations (LD) depending on the deviation level. A Telecom process $(Y_t)_{t \ge 0}$ scales as $t^{1/\gamma}$ , where t denotes time and $\gamma\in(1,2)$ is the key parameter of Y. We must distinguish moderate LD ${\mathbb P}(Y_t\ge y_t)$ with $t^{1/\gamma} \ll y_t \ll t$ , intermediate LD with $ y_t \approx t$ , and ultralarge LD with $ y_t \gg t$ . The results we obtain essentially depend on another parameter of Y, namely the resource distribution. We solve completely the cases of moderate and intermediate LD (the latter being the most technical one), whereas the ultralarge deviation asymptotics is found for the case of regularly varying distribution tails. In all the cases considered, the large-deviation level is essentially reached by the minimal necessary number of ‘service processes’.

In this short note we introduce two notions of dispersion-type variability orders, namely expected shortfall-dispersive (ES-dispersive) order and expectile-dispersive (ex-dispersive) order, which are defined by two classes of popular risk measures, the expected shortfall and the expectiles. These new orders can be used to compare the variability of two risk random variables. It is shown that either the ES-dispersive order or the ex-dispersive order is the same as the dilation order. This gives us some insight into parametric measures of variability induced by risk measures in the literature.

Let f be the density function associated to a matrix-exponential distribution of parameters $(\boldsymbol{\alpha}, T,\boldsymbol{{s}})$ . By exponentially tilting f, we find a probabilistic interpretation which generalizes the one associated to phase-type distributions. More specifically, we show that for any sufficiently large $\lambda\ge 0$ , the function $x\mapsto \left(\int_0^\infty e^{-\lambda s}f(s)\textrm{d} s\right)^{-1}e^{-\lambda x}f(x)$ can be described in terms of a finite-state Markov jump process whose generator is tied to T. Finally, we show how to revert the exponential tilting in order to assign a probabilistic interpretation to f itself.

The paper proves transportation inequalities for probability measures on spheres for the Wasserstein metrics with respect to cost functions that are powers of the geodesic distance. Let $\mu$ be a probability measure on the sphere ${\bf S}^n$ of the form $d\mu =e^{-U(x)}{\rm d}x$ where ${\rm d}x$ is the rotation invariant probability measure, and $(n-1)I+{\hbox {Hess}}\,U\geq {\kappa _U}I$, where $\kappa _U>0$. Then any probability measure $\nu$ of finite relative entropy with respect to $\mu$ satisfies ${\hbox {Ent}}(\nu \mid \mu ) \geq (\kappa _U/2)W_2(\nu,\, \mu )^2$. The proof uses an explicit formula for the relative entropy which is also valid on connected and compact $C^\infty$ smooth Riemannian manifolds without boundary. A variation of this entropy formula gives the Lichnérowicz integral.

Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.

Almost stochastic dominance has been receiving a great amount of attention in the financial and economic literatures. In this paper, we characterize the properties of almost first-order stochastic dominance (AFSD) via distorted expectations and investigate the conditions under which AFSD is preserved under a distortion transform. The main results are also applied to establish stochastic comparisons of order statistics and receiver operating characteristic curves via AFSD.

We show that load-sharing models (a very special class of multivariate probability models for nonnegative random variables) can be used to obtain basic results about a multivariate extension of stochastic precedence and related paradoxes. Such results can be applied in several different fields. In particular, applications of them can be developed in the context of paradoxes which arise in voting theory. Also, an application to the notion of probability signature may be of interest, in the field of systems reliability.