New Developments in Nonparametric Bayesian Inference; Univariate and Multivariate time series with infinite varaince.

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The applicant's research program is concentrated on two major subjects in statistical inference. The first subject is mainly related to Bayesian inference. In the Bayesian paradigm, the statisticians considers having a probabilistic prior knowledge on some unknown parameters. Later, the collected data is used to update their prior knowledge under the chosen model. A nonparametric Bayesian model considers the prior knowledge in a much more general framework with minimal restrictions. The applicant's research area in this field is mostly related to the construction of new processes used in nonparametric Bayesian models and the derivation of fast yet more precise approximation tools for these processes to use them in statistical inference. The applicant also derives efficient simulation techniques for such processes in order to enable practitioners to run their computer programs with faster speed. The applicant's second project investigates the large sample theory for processes which evolve in time. In Statistics, these processes are called time series. Sometimes these time series exhibit unusual and higher spikes or extremes. These spikes and the frequent observation of extremes in time series are modelled with certain random variables with infinite variance. There is extensive research for the univariate infinite variance time series but little is known for multivariate infinite variance time series. In the multivariate case, the size of the spikes in each coordinate may not be of the same type and size. The applicant's research investigates the large sample theory for the infinite variance multivariate time series. The result can be used to model time series in Finance, Telecommunications and many other fields.