报告题目：How heavy-tailed distributions arise in various models with light-tailed input
报告人：Prof.Korshunov Dmitry（Lancaster University）
报告时间：2016年6月1日（周三）15:20-16:20
报告地点：数学楼三楼306
报告摘要：In this talk we discuss three interesting probabilistic models where light-tailed input generates heavy-tailed output. We say that a distribution F is light-tailed if it possesses some positive exponential moment finite. The distribution F is called heavy-tailed if all positive exponential moments are infinite.
The first model is related to the Lamperti problem for Markov chains with asymptotically zero drift. Let 
be a time homogeneous Markov chain on 
of Lamperti type, that is, with asymptotically zero drift, that is,
as i→∞. We assume that m(i) is negative ultimately in space and such that 
is stable with stationary distribution 
. The problem is to describe asymptotic behaviour of
as i→∞. I will explain why stationary distribution of a Markov chain with asymptotically zero drift is usually heavy-tailed even if X has bounded jumps, in contrast to more classical case of asymptotically negative drift where stationary distribution is usually light-tailed. I will also explain how particular regularly varying and Weibull-type distributions arise in this context.
The second model is Gaussian chaos, that is, a polynomial of standard normal variables. The simplest case is given by a product of components of normal random vector with general covariance matrix. The distribution of Gaussian chaos is usually heavy-tailed and we show how does it happen and what kind of tails should we expect.
The third model is related to perpetuities:
where 
is a Markov modulated random walk, that is, is an ergodic Markov chain, and 
are independent identically distributed random variables. Assume 
drifts to minus infinity, so that the perpetuity is well defined. The distribution of the perpetuity is always heavy-tailed. We explain the tail behaviour of this perpetuity under general conditions on 
and 
.