RARE EVENTS IN SPATIAL RANDOM NETWORKS

<p dir="ltr">This thesis is concerned with rare events in random geometric graphs, the most fundamental random graph model of spatial networks, in two different frameworks---large deviations theory and extreme value theory---and their statistical applications. In the former part of the thesis, we develop the large deviations theory for the number of vertices of degree at most $k$ in random geometric graphs whose vertices are drawn by a homogeneous Poisson or a binomial point process on the unit cube. Considering the relationship between the degree of a vertex and its <i>k</i>-nearest neighbor distance, we develop the theory for the point process associated with the Euclidean volume of <i>k</i>-nearest neighbor balls centered around the vertices. The large deviations behavior is investigated in two different types. The first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. On the other hand, when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence, we examine large deviations based on the notion of $\mathcal{M}_0$-topology. In the latter part of the thesis, we propose an estimator for the tail exponent of a heavy-tailed distribution. This estimator, referred to as the layered Hill estimator, is a generalization of the traditional Hill estimator and is built upon a layered structure formed by subgraphs of extreme vertices in a random geometric graph. We prove that the layered Hill estimator exhibits desirable asymptotic properties such as consistency and asymptotic normality for the tail exponent, and argue that our estimator provides a robust alternative to the traditional estimator, particularly when a portion of the extreme data is missing.</p>