In this work we establish a sequence of asymptotic statistical results for important quantities in Topological Data Analysis (TDA). Furthermore, we investigate the use of topological methods to analyze protein coexpression data from a structural perspective. We give an initial introduction to the techniques of topological data analysis, and outline existing statistical results in the large-sample setting.We first investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. Specific statistics considered include the persistent Betti numbers of Cech and Vietoris-Rips complexes over point sets in Euclidean space, along with Euler characteristics, and the total edge length of the k-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS).Next we study approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.Finally, we examine the problem of network analysis from the topological perspective, using the techniques of persistence homology to extract meaningful structural features from protein coexpression data. We propose novel topological separation metrics in an optimization framework, and make an application to the protein coexpression for three-spine stickleback (gasterosteus aculeatus)
