Gaussian and related processes: Lower tail probability and application

In this dissertation we focus on the lower tail probabilities for Gaussian and iterated processes, consisting of two parts. In the first part, we study the lower tail probabilities for Gaussian processes. Let X be a real-valued Gaussian process indexed by an index set S with mean zero. The lower tail probability studies the asymptotic rate of convergence of the probability of X at time t less than x for all t between 0 and 1, as x converges to 0. A survey of general theory, classic results and new examples is provided. In the second part, we present general results on the lower tail probability for the iterated processes. Let X be a two-sided stochastic process, Y a continuous stochastic process independent of X. An iterated process Z is defined as Z(t)=X(Y(t)). The asymptotic rate of the probability of Z at time t less than x for all t between 0 and 1, as x converges to 0, is calculated with X and Y under certain regular conditions.