Stochastic processes with heavy tails and temporal dependence: modeling, probabilistic properties and statistical inference

概率论和统计学中的一个经典假设是随机变量（数据）是独立且近似高斯分布的。另一方面，通常可以观察到，在金融、环境科学或网络流量中出现的许多类型的数据实际上远非独立和高斯分布。这促使我们使用时间序列和复杂的随机过程对这些数据进行建模，这些过程可以捕获重尾、弱依赖性或强依赖性以及自相似性。在这个项目中，我们将研究应用概率和应用统计中出现的离散和连续时间序列和随机过程的概率性质和统计推断。我们将重点关注以下几个方面：a)在离散和连续时间序列模型中建模尾巴和记忆。b)网络流量模型的概率性质和统计分析。

A classical assumption in probability and statistics is that random variables (data) are independent andapproximately Gaussian. On the other hand, it is commonly observed that many types of data that occur infinance, environmental sciences or network traffic are in fact far away from being independent and Gaussian.This motivates modeling of such data using time series and complicated stochastic processes that can capture heavy tails, weak or strong dependence, and self-similarity.In this project we will study probabilistic properties and statistical inference for discrete and continuous timeseries and stochastic processes that arise in applied probability and applied statistics. We will focus on thefollowing aspects:a) Modeling tails and memory in discrete and continuous time series models.b) Probabilistic properties and statistical analysis of network traffic models.