This dissertation is comprised of two separate projects. The first concerns a Markov chain called the Random Logistic Model. For r in (0,4] and x in [0,1] the logistic map fr(x) = rx(1 - x) defines, for positive integer t, the dynamical system xr(t + 1) = f(xr(t)) on [0,1], where xr(1) = x. The interplay between this dynamical system and the Markov chain xr,N(t) defined by perturbing the logistic map by truncated Gaussian noise scaled by N-1/2, where N -> infinity, is studied. A natural question is whether one can quantify this interplay via probabilistic limit theorems for xr,N(t). There are two possible limits: the vanishing-noise limit N -> infinity for fixed t and the ergodic limit t -> infinity followed by the vanishing-noise limit. Both lead to a set of probabilistic limit theorems where the underlying deterministic dynamics take over. A particular case of interest is for r = 4. In the second project we perform an asymptotic analysis of Bayesian parallel density estimators which are based on logspline density estimation. The parallel estimator we introduce is in the spirit of a kernel density estimator presented in recent studies. We provide a numerical procedure that produces the density estimator itself in place of the sampling algorithm. We derive an error bound for the mean integrated squared error for the full data posterior estimator and investigate the parameters that arise from the logspline density estimation and the numerical approximation procedure. Our investigation leads to the choice of parameters that result in the error bound scaling appropriately in relation to them.