The main goal of this proposal is to lay the theoretical foundations for the analysis of the stability of Markov chains with heavy-tailed targets, focusing on the processes that underpin many randomised algorithms used in practice. This project is an international collaboration between the universities of Warwick and Stanford, made possible by the EPSRC-NSF Lead
Agency agreement in mathematics. As such, the project presents a unique opportunity to combine the US expertise in theoretical Operations Research with the UK's capability in Computational Statistics, resulting in novel methodology for the analysis of the convergence of Markov chains with heavy-tailed targets.

This theoretical proposal, based on probabilistic methods (e.g. martingale theory, coupling and Lyapunov functions) will have immediate impact within Probability, most notably on the theory of the stability of stochastic systems. In time, however, this work is expected to have impact far beyond applied probability in a number of sub-areas of computational statistics and machine learning where heavy-tailed targets arise