Long and Short Memory Stationary Processes: Prediction and Estimation

该项目的重点是二阶离散或连续参数平稳随机过程的预测和估计问题。这些过程可能表现为短记忆、中记忆和长记忆。本项目研究两个问题：（1）预测问题。在观察了过去的一部分之后，人们希望预测未来。我们的目标是描述随着过去观测次数的增加，预测误差的下降率。该速率将取决于基础随机过程的相关结构及其谱密度函数的平滑特性。(2)估计问题：在对潜在随机过程的适当假设下，目标是研究表征该过程的未知估计参数的统计性质。长记忆过程在许多应用领域都有观察到，如长期全球温度记录，金融资产价格等。对这类过程的概率统计性质的研究是一个具有挑战性的问题。然而，这样的研究将提供对潜在现象的更好理解，并提供对未来的更好预测-所有这些都是非常重要的任务。

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.