Applied Probability Problems Research Articles

Introduction to the Applied Probability Society’s “Open Problems in Applied Probability” Session at the INFORMS Annual Meeting, Phoenix, Arizona, November 4–7, 2018

This Special Section of Stochastic Systems includes a number of open problems that were presented at a session organized by the Applied Probability Society during the INFORMS Annual Meeting held in Phoenix, Arizona, November 4–7, 2018, on topics of current interest in applied probability.

Relational exchangeability

AbstractA relationally exchangeable structure is a random combinatorial structure whose law is invariant with respect to relabeling its relations, as opposed to its elements. Historically, exchangeable random set partitions have been the best known examples of relationally exchangeable structures, but the concept now arises more broadly when modeling interaction data in modern network analysis. Aside from exchangeable random partitions, instances of relational exchangeability include edge exchangeable random graphs and hypergraphs, path exchangeable processes, and a range of other network-like structures. We motivate the general theory of relational exchangeability, with special emphasis on the alternative perspective it provides and its benefits in certain applied probability problems. We then prove a de Finetti-type structure theorem for the general class of relationally exchangeable structures.

Tail asymptotic of Weibull-type risks

With motivation from Arendarczyk and De¸bicki [Asymptotics of supremum distribution of a Gaussian process over a Weibullian time. Bernoulli. 2011;17:194–210], in this paper we derive the tail asymptotics of the product of two dependent Weibull-type risks, which is of interest in various statistical and applied probability problems. Our results extend some recent findings of Schlueter and Fischer [The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution. Extremes. 2012;15:159–174] and Bose et al. [Product of exponentials and spectral radius of random k circulants. Annales de l'Institut Henri Poincare Probabilites et Statistiques. 2012;48:424–443].

Implicit renewal theorem for trees with general weights

Consider distributional fixed point equations of the form R=Df(Q,Ci,Ri,1≤i≤N), where f(⋅) is a possibly random real-valued function, N∈{0,1,2,3,…}∪{∞}, {Ci}i∈N are real-valued random weights and {Ri}i∈N are iid copies of R, independent of (Q,N,C1,C2,…); =D represents equality in distribution. Fixed point equations of this type are important for solving many applied probability problems, ranging from the average case analysis of algorithms to statistical physics. We develop an Implicit Renewal Theorem that enables the characterization of the power tail behavior of the solutions R to many equations of multiplicative nature that fall into this category. This result extends the prior work in Jelenković and Olvera-Cravioto (2012) [16], which assumed nonnegative weights {Ci}, to general real-valued weights. We illustrate the developed theorem by deriving the power tail asymptotics of the solution R to the linear equation R=D∑i=1NCiRi+Q.

Reward distributions associated with some block tridiagonal transition matrices with applications to identity by descent

Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.

Reward distributions associated with some block tridiagonal transition matrices with applications to identity by descent

Markov and semi-Markov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships.

A Set of Algorithms for the Incomplete Gamma Functions

This paper gives fast and reliable algorithms for the numerical evaluation of the incomplete gamma functions and for auxiliary functions, such as functions related with the gamma function and error function. All these functions are of basic importance in applied probability problems.

Disruption of the representativeness heuristic: Can we be perturbed into using correct probabilistic reasoning?

Some misconceptions in learning and understanding statistical or probabilistic concepts stem from the use of what is called the “representativeness heuristic” for solving such problems. Incorrect solutions to problems is the result. This study provided graduate students enrolled in beginning inferential statistics with a series of experiences which seemed to disrupt such faulty logic by having them experience logical inconsistencies. Such direct experimental involvement with applied probability problems was recommended as a means to enhance the use of more appropriate probabilistic reasoning in subsequent statistical or probability problems.

Generalized inverses and their application to applied probability problems

The main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a Markov chain is present either directly or indirectly through some form of imbedding. By characterizing all generalized inverses of I— P, where P is the transition matrix of a finite irreducible discrete time Markov chain, we are able to obtain general procedures for finding stationary distributions, moments of the first passage time distributions, and asymptotic forms for the moments of the occupation-time random variables. It is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework. More generally, in the context of a Markov renewal process setting the aforementioned problems are also examined using generalized inverses of I— P. As a special case, Markov chains in continuous time are considered, and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process, instead of to I— P, where P is the transition matrix of the discrete time jump Markov chain.