Modern Probabilistic Methods With Applications

In this thesis, we investigate a stochastic model describing the spread of an epidemic among a population of n individuals. A significant strand of the research considers the development of a quantum random walk description for the SIS model using quantum flow and the lamplighter group. On the theoretical side, this entails proving a new algorithm to determine the hidden state of the lamplighter, as well as examine the dynamic of the walk. A second strand of the research is to apply a modern method to the model under investigation. This led us to introduce the associated discrete case of superstatistics and develop new learning algorithm. The superstatistics have been investigated using the saddlepoint approximation, the Markov chain Monte Carlo methods, and a Hamiltonian. Furthermore, we study the superstatistics of stochastic processes and apply its technique to a well-known model in the epidemic literature known as SIR. No other scholars, at least to the best of our knowledge, have studied SIR via superstatistics. The contribution made by this thesis has been published in [1]–[5]