讲座题目：Randomized urn models with irreducible and reducible replacement policy

讲 座 人：张立新教授 浙江大学数学科学学院 

讲座时间：2018年10月18日下午3点-4点

讲座地点：浙江工商大学综合楼601会议室

主讲人简介：

张立新，浙江大学求是特聘教授。1995年获复旦大学理学博士学位， 1997年晋升为教授，2001-2012年担任浙江大学统计学研究所副所长、常务副所长、所长，2009-2015年任浙江大学数学系副主任。现任浙江大学数学科学学院副院长、杭州市钱江特聘专家、IMS-China委员会委员、中国现场统计研究会理事、浙江省现场统计研究会理事长。主要从事概率极限理论、相依数据模型、临床试验自适应随机化设计、随机过程轨道性质等领域的研究，发表了学术论文140多篇，于1998年入选"浙江省跨世纪151人才工程"， 2008年入选教育部“新世纪优秀人才支持计划”、2010年获得浙江省自然科学基金杰出青年基金，2012年获得国家自然科学基金杰出青年基金。

摘要：

Generalized Friedman urn is one of the simplest and most useful models considered in probability theory. Since Athreya and Ney (1972) showed the almost sure convergence of urn proportions in a randomized urn model with irreducible replacement matrix under the $L\log L$ moment assumption, this assumption has been regarded as the weakest moment assumption, but the necessary has never been shown. In this talk, we will present the strong convergence of generalized randomized Friedman urns. It is proved that, when the random replacement matrix is irreducible in probability, the sufficient and necessary moment assumption for the almost sure convergence of the urn proportions is that the expectation of the replacement matrix is finite, which is less stringent than the $L\log L$ moment assumption, and when the replacement is reducible, the $L\log L$ moment assumption is a weakest sufficient condition. The rate of convergence and the central limit theorem are also discussed.