Topics in probability theory

This thesis focuses on the theoretical aspects of several classes of continuous time, continu- ous state space stochastic processes. Chapter 2 and Chapter 3 consider Levy processes and Levy driven stochastic functional differential equations. Chapter 4 studies a high dimensional factor model and the eigenvalues of the sample auto-covariance matrix. In Chapter 2 we extend the construction of the so-called weak subordination of multivariate Levy processes in [48] to an infinite dimensional setting. More specifically, we give sufficient conditions for the existence of the weak subordination between a Levy processes defined on an arbitrary Hilbert space and a sequence-valued Levy subordinator defined on suitable Banach spaces. As by-products of our main results, we obtain a characterization of subordinators on certain sequence spaces, as well as a characterization of Levy measures on direct sums of Banach spaces with different geometries. Chapter 3 focuses on a Levy driven stochastic delayed differential equation (SDDE) which arises as a continuous time analogue to the discrete time GARCH process. The SDDE was obtained in the recent works [64, 65, 157] as a weak limit in the Skorokhod topology of a sequence of suitably scaled discrete GARCH processes, as the time between observations tends to zero. In our work, we give sufficient conditions for the existence, uniqueness and regularity of the solution to the SDDE. We show that the SDDE can be reformulated as a stochastic functional differential equation and investigate the behaviour of its sample paths. The mean process and the covariance process of the solution are computed and are shown to exhibit similar behaviours to the discrete GARCH process. Chapter 4 focuses on a high dimensional factor model proposed by [102, 103] and subsequently studied in [109] to model time series data. We investigate the asymptotic distribution of leading eigenvalues of the (product symmetrized) sample auto-covariance matrix under a high dimensional regime where the dimension and sample size tend to infinity simultaneously. Utilizing some new developments [51, 108, 109] in high dimensional random matrix theory, we obtain a central limit theorem for the empirical eigenvalues after suitable centering and scaling. It is shown that the correct centerings for the eigenvalues are in general not equal to the population eigenvalues