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Let $\{X(t):t\in[0,\infty)\}$ be a Markov process with state space $(-\infty;+\infty)$, having continuous sample paths and a transition p.d.f. given by $p(y;s;x;t), t<s$. How to show that if we have: $$\begin{cases}\displaystyle (i)'\lim_{\Delta t \to 0^{+}}\frac{1}{\Delta t}\int_{-\infty}^{\inf...

Chapter $2$ Exercise $7$ Question (a) Page $85$ Linda J. S. Allen $2010$ $7$. Verify the following statement. Assume the period of state $i$ in a DTMC model is $\rm{d}(i)=0$. Then the set $\{i\}$ is a communication class in the Markov chain. for all $i \in S$, $d(i)=\rm{gcd}\{n/ p_{ii}^{(n)}>0...

I learnt the concept of the period of state i for DTMC and I'm wondering why we have $p_{ii}^{(2n)}=1$ and $p_{ii}^{(2n+1)}=0$? if $d(i)=p$ then $p_{ii}^{(pn)}=1$ and $p_{ii}^{(pn+1)}=0$?

I know that ; the little oh landau notation $$f(\Delta t)=o(\Delta t) \iff \lim_{\Delta t \to 0}\frac{f(\Delta t)}{\Delta t}=0$$ but I can't figure out why $$\lim_{\Delta t \to 0}p_{0}(t)\frac{o(\Delta t)}{\Delta t}=0\; \rm{and}\;\lim_{\Delta t \to 0}\frac{o(\Delta t)}{\Delta t}=0 $$ Do we have...

Exercise 2 Question (a) Page 84 Textbook: An Introduction to Stochastic Processes with Applications to Biology 2nd Edition Linda J. S. Allen 2010 Exercise Suppose $P$ is an $N\times N$ stochastic Matrix (column sums equal one), $P= \begin{pmatrix} p_{11} & p_{12} & \ldots & p_{1N} \\ p_{21} & p...