Rohan Shenoy

Optimal transport provides an inherently geometric and highly structured framework for studying spaces of probability measures, supplying a rich theoretical toolkit for contemporary statistics, machine learning, and generative modelling. In applications, however, the measures of interest are almost never known precisely, calling for a theory of optimal transport that accounts for statistical uncertainty. We construct such a framework, lifting the classical theory to the setting of \textit{random} probability measures. We introduce the $L^2$ over Wasserstein space $L^2_W$, establishing that it inherits the formal Riemannian structure of the Wasserstein space by characterising distances and geodesic geometry. The structure induces random flows with Wasserstein gradient flow sample paths, making it the natural extension of the Wasserstein space which allows for random gradient flow dynamics. We ensemble statistical convergence results of the optimal transport machinery using the empirical measure within the $L^2_W$ framework. Moreover, in the setting of Bayesian non-parametrics, we refine Schwartz's consistency theorem to the Wasserstein topology and deduce posterior convergence of the same machinery in $L^2_W$. We demonstrate that the growing theory of random token sampling for transformer models using self-attention flow paths can be embedded into the $L^2_W$ framework. The results provide a unified treatment of random optimal transport and its consequences for principled inference and generative modelling under the statistical uncertainty of random sampling.

In the context of time-subordinated Brownian motion models, Fourier theory and methodology are proposed to modelling the stochastic distribution of time increments. Gaussian Variance-Mean mixtures and time-subordinated models are reviewed with a key example being the Variance-Gamma process. A non-parametric characteristic function decomposition of subordinated Brownian motion is presented. The theory requires an extension of the real domain of certain characteristic functions to the complex plane, the validity of which is proven here. This allows one to characterise and study the stochastic time-change directly from the full process. An empirical decomposition of S&P log-returns is provided to illustrate the methodology.

The purpose of this note is to prove the celebrated Discrete Renewal Theorem in a common special case. We use only very elementary methods from real analysis, rather than markov chain theory, complex analysis, or generating functions. Provided is an introduction to a 1d discrete renewal process via a board game example, our proof the discrete renewal theorem, as well as background and history of the Erdǒs-Feller-Pollard Theorem.