Two Problems in Applied Stochastic Processes: Scaling Limits of the Multiple Range in 1D Random Walk and Uncertainty Quantification in Satellite Imagery

In this thesis, we discuss two problems in applied stochastic processes. The first is to show novel scaling limits for the multiple range of random walk in an interval. The second is uncertainty quantification of satellite imagery via deep learning. We first consider a symmetric simple random walk moving on the one dimensional interval $\{0, ..., N\}$. Let $\tau_N$ be the time when the random walk reaches either 0 or N for the first time. The $p$-multiple-range $R^{(p)}_N$ is the number of locations that the random walk visits exactly $p$ times. It turns out that the expected orders of $R^{(p)}_N$ is $O(\log(N))$. We compute, by the method of moments, that the joint scaled limits of $(R^{(1)}_N, ... R^{(m)}_N)/\log(N)$ is of the form $(Z, ... Z)$ where $Z$ is an exponential(2) random variable. In particular, the limit is totally correlated over its components. We then consider the binary segmentation of buildings and roads in Sentinel-2 satellite imagery using deep learning models. We evaluate three models: a convolutional neural network (CNN), a Bayesian convolutional neural network (BCNN), and a Monte Carlo dropout neural network (MCDN). Our primary aim is to assess both the segmentation accuracy and the uncertainty quantification capabilities of these models. We leverage Sentinel-2 data, which consists of 13 multispectral bands at varying spatial resolutions, to examine the impact of different band subsets on segmentation performance and uncertainty quantification. The models are evaluated using standard segmentation metrics, such as the $\F$ score, along with novel approaches we have developed for quantifying uncertainty for binary image segmentation.