Let $mathcal{B}$ be a separable Banach space. In this thesis, it is shown that the Chung's strong law of large numbers holds for a sequence of independent $mathcal{B}$-valued random elements and an array of rowwise independent $mathcal{B}$-valued random elements under some weaker assumptions by using more generalized functions $phi_{n}$'s.

<p>This dissertation establishes limit theory for topological functionals of geometric complexes from a stochastic process viewpoint. Standard filtrations of geometric complexes, such as the Čech and Vietoris-Rips complexes, have a natural parameter <i>r </i>which governs the formation of simplices: this is the basis for persistent homology. However, the parameter <i>r</i> may also be considered the time parameter of an appropriate stochastic process which summarizes the evolution of the filtration.</p><p>Here we examine the stochastic behavior of two of the foremost classes of topological functionals of such filtrations: the Betti numbers and the Euler characteristic. There are also two distinct setups in which the points underlying the complexes are generated, where the points are distributed randomly in <i>R^d</i> according to a general density (the traditional setup) and where the points lie in the tail of a heavy-tailed or exponentially-decaying “noise” distribution (the extreme-value theory (EVT) setup).<br></p><p>These results constitute some of the first results combining topological data analysis (TDA) and stochastic process theory. The first collection of results establishes stochastic process limits for Betti numbers of Čech complexes of Poisson and binomial point processes for two specific regimes in the traditional setup: the sparse regime—when the parameter <i>r </i>governing the formation of simplices causes the Betti numbers to concentrate on components of the lowest order; and the critical regime—when the parameter <i>r</i> is of the order <i>n^-1/d</i> and the geometric complex becomes highly connected with topological holes of every dimension. The second collection of results establishes a functional strong law of large numbers and a functional central limit theorem for the Euler characteristic of a random geometric complex for the critical regime in the traditional setup. The final collection of results establishes functional strong laws of large numbers for geometric complexes in the EVT setup for the two classes of “noise” densities mentioned above.<br></p>

<p>This dissertation establishes a variety of limit theorems pertaining to rare events in stochastic topology, exploiting probabilistic methods to study simplicial complex models. We focus on the filtration of \vc ech complexes and examine the asymptotic behavior of two topological functionals: the Betti numbers and critical faces. The filtration involves a parameter rn>0 that determines the growth rate of underlying Cech complexes. If rn depends also on the time parameter t, the obtained limit theorems will be established in a functional sense.</p> <p>The first part of this dissertation is devoted to investigating the layered structure of topological complexity in the tail of a probability distribution. We establish the functional strong law of large numbers for Betti numbers, a basic quantifier of algebraic topology, of a geometric complex outside an open ball of radius Rn, such that Rn to infinity as the sample size n increases. The nature of the obtained law of large numbers is determined by the decay rate of a probability density. It especially depends on whether the tail of a density decays at a regularly varying rate or an exponentially decaying rate. The nature of the limit theorem depends also on how rapidly Rn diverges. In particular, if Rn diverges sufficiently slowly, the limiting function in the law of large numbers is crucially affected by the emergence of arbitrarily large connected components supporting topological cycles in the limit.</p> <p>The second part of this dissertation investigates convergence of point processes associated with critical faces for a Cech filtration built over a homogeneous Poisson point process in the d-dimensional flat torus. The convergence of our point process is established in terms of the  Mo-topology, when the connecting radius of a Cech complex decays to 0, so slowly that critical faces are even less likely to occur than those in the regime of threshold for homological connectivity. We also obtain a series of limit theorems for positive and negative critical faces, all of which are considerably analogous to those for critical faces.</p>