In a family, parameterized by θ, of non-negative random variables with finite, positive second moment, Taylor's law (TL) asserts that the population variance is proportional to a power of the population mean as θ varies: σ2 (θ) = a[μ(θ)]b, a > 0. TL, sometimes called fluctuation scaling, holds widely in science, probability theory, and stochastic processes. Here we report diverse examples of TL with b = 2 (equivalent to a constant coefficient of variation) arising from a difference of random variables in normed vector spaces of dimension 1 and larger. In these examples, we compute a exactly using, in some cases, a simple, new technique. These examples may prove useful in future models that involve differences of random variables, including models of the spatial distribution and migration of human populations.

For a Markov chain on a finite partially ordered state space, we show that its Siegmund dual exists if and only if the chain is Möbius monotone. This is an extension of Siegmund's result for totally ordered state spaces, in which case the existence of the dual is equivalent to the usual stochastic monotonicity. Exploiting the relation between the stationary distribution of an ergodic chain and the absorption probabilities of its Siegmund dual, we present three applications: calculating the absorption probabilities of a chain with two absorbing states knowing the stationary distribution of the other chain; calculating the stationary distribution of an ergodic chain knowing the absorption probabilities of the other chain; and providing a stable simulation scheme for the stationary distribution of a chain provided we can simulate its Siegmund dual. These are accompanied by concrete examples: the gambler's ruin problem with arbitrary winning/losing probabilities; a non-symmetric game; an extension of a birth and death chain; a chain corresponding to the Fisher–Wright model; a non-standard tandem network of two servers, and the Ising model on a circle. We also show that one can construct a strong stationary dual chain by taking the appropriate Doob transform of the Siegmund dual of the time-reversed chain.

The coefficient of tail dependence is a quantity that measures how extreme events in one component of a bivariate distribution depend on extreme events in the other component. It is well known that the Gaussian copula has zero tail dependence, a shortcoming for its application in credit risk modeling and quantitative risk management in general. We show that this property is shared by the joint distributions of hitting times of bivariate (uniformly elliptic) diffusion processes.

We consider a single-item, two-echelon spare parts inventory model for repairable parts for capital goods with high downtime costs. The inventory system consists of multiple local warehouses, a central warehouse, and a central repair facility. When a part at a customer fails, if possible his request for a ready-for-use part is fulfilled by his local warehouse. Also, the failed part is sent to the central repair facility for repair. If the local warehouse is out of stock, then, via an emergency shipment, a ready-for-use part is sent from the central warehouse if it has a part in stock. Otherwise, it is sent via a lateral transshipment from another local warehouse, or via an emergency shipment from the external supplier. We assume Poisson demand processes, generally distributed leadtimes for replenishments, repairs, and emergency shipments, and a basestock policy for the inventory control.

Our inventory system is too complex to solve for a steady-state distribution in closed form. We approximate it by a network of Erlang loss queues with hierarchical jump-over blocking. We show that this network has a product-form steady-state distribution. This enables an efficient heuristic for the optimization of basestock levels, resulting in good approximations of the optimal costs.

Cossette, Marceau, and Perreault derived formulas for aggregation and capital allocation based on risks following two bivariate exponential distributions. Here, we derive formulas for aggregation and capital allocation for 18 mostly commonly known families of bivariate distributions. This collection of formulas could be a useful reference for financial risk management.

Log-concave random variables and their various properties play an increasingly important role in probability, statistics, and other fields. For a distribution F, denote by 𝒟F the set of distributions G such that the convolution of F and G has a log-concave probability mass function or probability density function. In this paper, we investigate sufficient and necessary conditions under which 𝒟F ⊆ 𝒟G, where F and G belong to a parametric family of distributions. Both discrete and continuous settings are considered.

We consider polling models in the sense of Takagi [19]. In our case, the feature of the server is that it may be forced to wait idly for new messages at an empty queue instead of switching to the next station. We propose four different wait-and-see strategies that govern these waiting periods. We assume Poisson arrivals for new messages and allow general service and switchover time distributions. The results are formulas for the mean average queueing delay and characterizations of the cases where the wait-and-see strategies yield a lower delay compared with the exhaustive strategy.

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m-dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m-dependent sequence.

Bayesian networks are convenient graphical expressions for high-dimensional probability distributions which represent complex relationships between a large number of random variables. They have been employed extensively in areas such as bioinformatics, artificial intelligence, diagnosis, and risk management. The recovery of the structure of a network from data is of prime importance for the purposes of modeling, analysis, and prediction. There has been a great deal of interest in recent years in the NP-hard problem of learning the structure of a Bayesian network from observed data. Typically, one assigns a score to various structures and the search becomes an optimization problem that can be approached with either deterministic or stochastic methods. In this paper, we introduce a new search strategy where one walks through the space of graphs by modeling the appearance and disappearance of edges as a birth and death process. We compare our novel approach with the popular Metropolis–Hastings search strategy and give empirical evidence that the birth and death process has superior mixing properties.

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.