Conditions for recurrence and transience of a Markov chain on $\mathbb{Z}^+$ and estimation of a geometric success probability
非负整数马尔可夫链的常返与暂留条件及几何成功概率估计
Let $Z$ be a discrete random variable with support $\Z^+ = \{0,1,2,\dots\}$. We consider a Markov chain $Y=(Y_n)_{n=0}^\infty$ with state space $\Z^+$ and transition probabilities given by $P(Y_{n+1} = j|Y_n = i) = P(Z = i+j)/P(Z \geq i)$. We prove that convergence of $\sum_{n=1}^\infty 1/[n^3 P (Z=n)]$ is sufficient for transience of $Y$ while divergence of $\sum_{n=1}^\infty 1/[n^2 P (Z \geq n)]$ is sufficient for recurrence. Let $X$ be a $\mbox{Geometric}(p)$ random…