Long time asymptotics of some weakly interacting particle systems and higher order asymptotics of generalized fiducial distribution

In probability and statistics limit theorems are some of the fundamental tools that rigorously justify a proposed approximation procedure. However, typically such results fail to explain how good is the approximation. In order to answer such a question in a precise quantitative way one needs to develop the notion of convergence rates in terms of either higher order asymptotics or non-asymptotic bounds. In this dissertation, two different problems are studied with a focus on quantitative convergence rates.;In first part, we consider a weakly interacting particle system in discrete time, approximating a nonlinear dynamical system. We deduce a uniform in time concentration bound for the Wasserstein-1 distance of the empirical measure of the particles and the law of the corresponding deterministic nonlinear Markov process that is obtained through taking the particle limit. Many authors have looked at similar formulations but under a restrictive compactness assumption on the particle domain. Here we work in a setting where particles take values in a non-compact domain and study several time asymptotics and large particle limit properties of the system. We establish uniform in time propagation of chaos along with a rate of convergence and also uniform in time concentration estimates. We also study another discrete time system that models active chemotaxis of particles which move preferentially towards higher chemical concentration and themselves release chemicals into the medium dynamically modify the chemical field. Long time behavior of this system is studied.;Second part of the dissertation is focused on higher order asymptotics of Generalized Fiducial inference. It is a relevant inferential procedure in standard parametric inference where no prior information of unknown parameter is available in practice. Traditionally in Bayesian paradigm, people propose posterior distribution based on the non-informative priors but imposition of any prior measure on parameter space is contrary to the ``no-information" belief (according to Fisher's philosophy). Generalized Fiducial inference is one such remedy in this context where the proposed distribution on the parameter space is only based on the data generating equation. In this part of dissertation we established a higher order expansion of the asymptotic coverage of one-sided Fiducial quantile. We also studied further and found out the space of desired transformations in certain examples, under which the transformed data generating equation yields first order matching Fiducial distribution.