Stable Convergence in Statistical Inference and Numerical Approximation of Stochastic Processes

Stable convergence is a type of convergence of random variables, which is stronger than weak convergence but weaker than convergence in probability. It has been used in asymptotic theory of statistics and probability since Renyi originated his work (cf. [91]) in 1963. In this thesis, we study applications of stable convergence in two aspects. First, we show how to estimate the asymptotic (conditional) covariance matrix, which appears in many central limit theorems for stable laws in high-frequency estimation. We employ the idea of subsampling to provide a positive semi-definite estimator for this matrix. We show that our estimator is consistent in both noiseless models and models with additive microstructure noise. This estimate together with stable convergence theorems allow us to make some statistical inferences such as constructing confidence intervals or doing hypothesis testing. Moreover, we provide a decomposition of the leading error terms, from which we are able to get some insights about how to configure the subsampler by optimally choosing its tuning parameters (e.g., the number of subsamples). This leads to a rate of convergence for the subsampler. Second, we apply stable convergence theorems to show a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in [32]. In the original work of [32], the authors propose to use Fourier transformation to embed a given one dimensional Levy semi-stationary process into a two-parameter stochastic field. For the latter, they use a simple iteration procedure and study the strong approximation error (in L^2 sense) of the resulting numerical scheme given that the volatility process is fully observed. In this work, we give a more precise assessment of the numerical error associated with the Fourier method. We complement their study by analyzing the weak limit of the error process in the framework of Brownian semi-stationary processes, where the drift and the volatility processes need to be numerically simulated