## Grad Student Seminar: Michael Conroy & Adam Waterbury

**Michael Conroy-**** UNC-Chapel Hill**

**A direct Approach to Renewal Theory on Trees**

In a variety of applications ranging from computer science to statistical physics, a class of recursive stochastic fixed-point equations appear. Such recursions admit so-called special endogenous solutions constructed on a random weighted tree formed from i.i.d. copies of the inputs to the recursion. We are interested in the tail behavior of the endogenous solution a max-type recursion that arises in the analysis of the branching random walk and also as the limiting waiting time distribution on parallel queueing networks with synchronization. The particular form of this equation allows us to analyze the tail behavior of the solutions by extending classical change-of-measure and renewal-theoretic techniques to random trees. Our techniques offer a formulation of this tail behavior that allows for efficient simulation of tail probabilities via an importance sampling algorithm.

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**Adam Waterbury****– UNC-Chapel Hill
Weak Limits of Quasi-Stationary Distributions**

A question of great importance in ecology is what conditions must be met in order for a population of interacting and, possibly, competing species to coexist with one another over long time spans. In reality if the population is finite, then after a large enough amount of time has passed, one or more of the species are sure to face extinction. However, the time that it takes for extinction to occur can be quite large, so it is natural to consider whether the population can sustain any long-term coexistence before any of the species are extinct. If the dynamics of a population are modeled by a Markov process, then such metastability is captured in the notion of a quasi-stationary distribution. In this work we analyze the limiting behavior of quasi-stationary distributions of a family of Markov chains that model the evolution of interacting biological populations. In particular, we show that under some large deviations assumptions, the support of weak limit points of the quasi-stationary distributions can be described in terms of dynamical properties of the law of large numbers limit of the processes.