On a class of random walks in simplexes
On a class of random walks in simplexes
复制标题
关于一类单纯形中的随机游走
DOI：
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发表时间：
2017
影响因子：
1
通讯作者：
S. Volkov
中科院分区：
文献类型：
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作者：
Tuan;S. Volkov
我们研究了一类随机游动模型的极限行为，其取值于$d$维单位标准单纯形，$d\ge 1$，定义如下。从一个内部点z$开始，该过程选择单纯形的d+1$个顶点之一，概率取决于z$，然后粒子随机跳到连接z$和所选顶点的线段上的一个新位置z '$。在某些特殊情况下，利用Beta分布的性质，我们证明了马尔可夫链的极限分布实际上是Dirichlet分布。我们还考虑了一个相关的历史依赖的随机游动模型在$[0，1]$的基础上的瓮型计划。我们表明，这种随机游走收敛分布的反正弦定律。
We study the limit behaviour of a class of random walk models taking values in the $d$-dimensional unit standard simplex, $d\ge 1$, defined as follows. From an interior point $z$, the process chooses one of the $d+1$ vertices of the simplex, with probabilities depending on $z$, and then the particle randomly jumps to a new location $z'$ on the segment connecting $z$ to the chosen vertex. In some specific cases, using properties of the Beta distribution, we prove that the limiting distributions of the Markov chain are, in fact, Dirichlet. We also consider a related history-dependent random walk model in $[0,1]$ based on an urn-type scheme. We show that this random walk converges in distribution to the arcsine law.