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.

Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and then study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, $\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.

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.

In this paper, we study the asymptotic behavior of the generalized Zagreb indices of the classical Erdős–Rényi (ER) random graph G(n, p), as $n\to\infty$. For any integer $k\ge1$, we first give an expression for the kth-order generalized Zagreb index in terms of the number of star graphs of various sizes in any simple graph. The explicit formulas for the first two moments of the generalized Zagreb indices of an ER random graph are then obtained from this expression. Based on the asymptotic normality of the numbers of star graphs of various sizes, several joint limit laws are established for a finite number of generalized Zagreb indices with a phase transition for p in different regimes. Finally, we provide a necessary and sufficient condition for any single generalized Zagreb index of G(n, p) to be asymptotic normal.