Title: Functional approximations with Stein's method of exchangeable pairs
Authors: Mikolaj Kasprzak - University of Luxembourg (Luxembourg) [presenting]
Abstract: Functional limit theorems are an important class of results describing convergence in distribution of Markov chains to diffusion processes. The convergence occurs in a space of functions (the Skorokhod space of cadlag paths) and after a proper rescaling of parameters, such that, in the limit, the chain is forced to make jumps infinitely often and its paths become continuous. Such results prove useful whenever the processes one wishes to model are discrete in nature, but it is more convenient to describe them using SDEs. This might be the case, for instance, if the conclusions one wishes to draw are easily obtainable using stochastic analysis or if one does not want changes in the local details to affect the model significantly. Researchers using functional approximations, for instance in population biology (where one may often switch from a discrete model to a continuous one after a proper rescaling and letting the population size go to infinity), are interested in measuring the error they make when doing so. Stein's method of exchangeable pairs turns out to be particularly useful in this case and provides powerful upper bounds on distances between the discrete and the limiting continuous processes. The theoretical setup for infinite-dimensional Stein's method, an abstract approximation theorem and concrete models it can be used for, will be considered. Among the examples, certain classes of U-processes and a graph-valued process will be discussed.