We consider a stochastic model, called the replicator coalescent, describing a system of blocks of k different types that undergo pairwise mergers at rates depending on the block types: with rate $C_{ij}\geq 0$ blocks of type i and j merge, resulting in a single block of type i. The replicator coalescent can be seen as a generalisation of Kingman’s coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called replicator equations from evolutionary game theory. By dilating time arbitrarily close to zero, we see that initially, on coming down from infinity, the replicator coalescent behaves like the solution to a certain replicator equation. Thereafter, stochastic effects are felt and the process evolves more in the spirit of a multi-type death chain.

Competing and complementary risk (CCR) problems are often modelled using a class of distributions of the maximum, or minimum, of a random number of independent and identically distributed random variables, called the CCR class of distributions. While CCR distributions generally do not have an easy-to-calculate density or probability mass function, two special cases, namely the Poisson–exponential and exponential–geometric distributions, can easily be calculated. Hence, it is of interest to approximate CCR distributions with these simpler distributions. In this paper, we develop Stein’s method for the CCR class of distributions to provide a general comparison method for bounding the distance between two CCR distributions, and we contrast this approach with bounds obtained using a Lindeberg argument. We detail the comparisons for Poisson–exponential, and exponential–geometric distributions.

We show that $\alpha $-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for $0<\alpha <1$ and every $\alpha $-stable Lévy motion ${\mathbb {W}}$, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1\leq \alpha <2$ and every symmetric $\alpha $-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1< \alpha <2$ and every $-1\leq \beta \leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_\alpha (\ln (2), \beta ,0)$ random variable.

Motivated by the investigation of probability distributions with finite variance but heavy tails, we study infinitely divisible laws whose Lévy measure is characterized by a radial component of geometric (tempered) stable type. We closely investigate the univariate case: characteristic exponents and cumulants are calculated, as well as spectral densities; absolute continuity relations are shown, and short- and long-time scaling limits of the associated Lévy processes analyzed. Finally, we derive some properties of the involved probability density functions.

We introduce a comprehensive method for establishing stochastic orders among order statistics in the independent and identically distributed case. This approach relies on the assumption that the underlying distribution is linked to a reference distribution through a transform order. Notably, this method exhibits broad applicability, particularly since several well-known nonparametric distribution families can be defined using relevant transform orders, including the convex and the star transform orders. Moreover, for convex-ordered families, we show that an application of Jensen’s inequality gives bounds for the probability that a random variable exceeds the expected value of its corresponding order statistic.

Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics, vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165(1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.

For a continuous-time phase-type (PH) distribution, starting with its Laplace–Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

We propose Rényi information generating function (RIGF) and discuss its properties. A connection between the RIGF and the diversity index is proposed for discrete-type random variables. The relation between the RIGF and Shannon entropy of order q > 0 is established and several bounds are obtained. The RIGF of escort distribution is derived. Furthermore, we introduce the Rényi divergence information generating function (RDIGF) and discuss its effect under monotone transformations. We present nonparametric and parametric estimators of the RIGF. A simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. A comparison study between the nonparametric and parametric estimators is made in terms of the standard deviation, absolute bias, and mean square error. We have observed superior performance for the newly proposed estimators. Some applications of the proposed RIGF and RDIGF are provided. For three coherent systems, we calculate the values of the RIGF and other well-established uncertainty measures, and similar behavior of the RIGF is observed. Further, a study regarding the usefulness of the RDIGF and RIGF as model selection criteria is conducted. Finally, three chaotic maps are considered and then used to establish a validation of the proposed information generating function.

For coherent systems with components and active redundancies having heterogeneous and dependent lifetimes, we prove that the lifetime of system with redundancy at component level is stochastically larger than that with redundancy at system level. In particular, in the setting of homogeneous components and redundancy lifetimes linked by an Archimedean survival copula, we develop sufficient conditions for the reversed hazard rate order, the hazard rate order and the likelihood ratio order between two system lifetimes, respectively. The present results substantially generalize some related results in the literature. Several numerical examples are presented to illustrate the findings as well.

In this paper, we define weighted failure rate and their means from the stand point of an application. We begin by emphasizing that the formation of n independent component series system having weighted failure rates with sum of weight functions being unity is same as a mixture of n distributions. We derive some parametric and non-parametric characterization results. We discuss on the form invariance property of baseline failure rate for a specific choice of weight function. Some bounds on means of aging functions are obtained. Here, we establish that weighted increasing failure rate average (IFRA) class is not closed under formation of coherent systems unlike the IFRA class. An interesting application of the present work is credited to the fact that the quantile version of means of failure rate is obtained as a special case of weighted means of failure rate.

The pth ($p\geq 1$) moment exponential stability, almost surely exponential stability and stability in distribution for stochastic McKean–Vlasov equation are derived based on some distribution-dependent Lyapunov function techniques.

A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

Random bridges have gained significant attention in recent years due to their potential applications in various areas, particularly in information-based asset pricing models. This paper aims to explore the potential influence of the pinning point’s distribution on the memorylessness and stochastic dynamics of the bridge process. We introduce Lévy bridges with random length and random pinning points, and analyze their Markov property. Our study demonstrates that the Markov property of Lévy bridges depends on the nature of the distribution of their pinning points. The law of any random variables can be decomposed into singular continuous, discrete, and absolutely continuous parts with respect to the Lebesgue measure (Lebesgue’s decomposition theorem). We show that the Markov property holds when the pinning points’ law does not have an absolutely continuous part. Conversely, the Lévy bridge fails to exhibit Markovian behavior when the pinning point has an absolutely continuous part.

We propose a monotone approximation scheme for a class of fully nonlinear degenerate partial integro-differential equations which characterize nonlinear $\alpha$-stable Lévy processes under a sublinear expectation space with $\alpha\in(1,2)$. We further establish the error bounds for the monotone approximation scheme. This in turn yields an explicit Berry–Esseen bound and convergence rate for the $\alpha$-stable central limit theorem under sublinear expectation.

In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV’s asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV.

We introduce a bivariate tempered space-fractional Poisson process (BTSFPP) by time-changing the bivariate Poisson process with an independent tempered $\alpha$-stable subordinator. We study its distributional properties and its connection to differential equations. The Lévy measure for the BTSFPP is also derived. A bivariate competing risks and shock model based on the BTSFPP for predicting the failure times of items that undergo two random shocks is also explored. The system is supposed to break when the sum of two types of shock reaches a certain random threshold. Various results related to reliability, such as reliability function, hazard rates, failure density, and the probability that failure occurs due to a certain type of shock, are studied. We show that for a general Lévy subordinator, the failure time of the system is exponentially distributed with mean depending on the Laplace exponent of the Lévy subordinator when the threshold has a geometric distribution. Some special cases and several typical examples are also demonstrated.

In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$, where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$, for $i=1,\ldots, n$, which are coupled by Archimedean copulas having different generators. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$, in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of [5] (Barmalzan, G., Ayat, S.M., Balakrishnan, N., & Roozegar, R. (2020). Stochastic comparisons of series and parallel systems with dependent heterogeneous extended exponential components under Archimedean copula. Journal of Computational and Applied Mathematics 380: Article No. 112965).

Qu, Dassios, and Zhao (2021) suggested an exact simulation method for tempered stable Ornstein–Uhlenbeck processes, but their algorithms contain some errors. This short note aims to correct their algorithms and conduct some numerical experiments.

In this paper, we study the functional given by the integral of the mean curvature of a convex set with Gaussian weight with Gaussian volume constraint. It was conjectured that the ball centred at the origin is the only minimizer of such a functional for certain values of the mass. We prove that this is the case in dimension 2 while in higher dimension the situation is different. In fact, for small values of mass, the ball centred at the origin is a local minimizer, while for larger values the ball is a maximizer among convex sets with a uniform bound on the curvature.

Let $\mathcal{C}$ denote the family of all coherent distributions on the unit square $[0,1]^2$, i.e. all those probability measures $\mu$ for which there exists a random vector $(X,Y)\sim \mu$, a pair $(\mathcal{G},\mathcal{H})$ of $\sigma$-fields, and an event E such that $X=\mathbb{P}(E\mid\mathcal{G})$, $Y=\mathbb{P}(E\mid\mathcal{H})$ almost surely. We examine the set $\mathrm{ext}(\mathcal{C})$ of extreme points of $\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of $\mathrm{ext}(\mathcal{C})$. We apply these results to obtain the asymptotic sharp bound $\lim_{\alpha \to \infty}\alpha\cdot(\sup_{(X,Y)\in \mathcal{C}}\mathbb{E}|X-Y|^{\alpha}) = {2}/{\mathrm{e}}$.