Long and Short Memory Stationary Processes: Prediction and Estimation

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The project focuses on prediction and estimation problems for second order discrete - or continuous-parameter stationary random processes. These processes may display short, intermediate or long memory.The project investigates two problems: (1) The prediction problem. Having observed part of the past, one wishes to predict the future. The goal is to describe the rate of decrease of the prediction error as the number of past observations increases. This rate will depend on the dependence structure of the underlying random process and the smoothness properties of its spectral density function. (2) The estimation problem: Under suitable assumptions on the underlying random process, the goal is to investigate the statistical properties of unknown estimators parameters characterizing the process.Long memory processes are observed in many areas of applications, such as long term global temperature records, financial asset prices and many others. The study of the probabilistical and statistical properties of such process is a chanllenging problem. However, such a study will provide better understanding of the underlying phenominon and provide better prediction of the future -- all very important tasks.