A spectral decomposition for a simple mutation model

Electronic Communications in Probability

Martin Möhle¹ 

关键词: eigenvalues;    eigenvectors;    empirical measure process;    finite Markov chain;    first passage time;    mixing time;    mutation model;    potential theory;    product chain;    random walk on the hypercube;    spectral analysis;   

DOI  :  10.1214/19-ECP222

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

We consider a population of $N$ individuals. Each individual has a type belonging to some at most countable type space $K$. At each time step each individual of type $k\in K$ mutates to type $l\in K$ independently of the other individuals with probability $m_{k,l}$. It is shown that the associated empirical measure process is Markovian. For the two-type case $K=\{0,1\}$ we derive an explicit spectral decomposition for the transition matrix $P$ of the Markov chain $Y=(Y_n)_{n\ge 0}$, where $Y_n$ denotes the number of individuals of type $1$ at time $n$. The result in particular shows that $P$ has eigenvalues $(1-m_{0,1}-m_{1,0})^i$, $i\in \{0,\ldots ,N\}$. Applications to mean first passage times are provided.

【 授权许可】

CC BY