Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, but I'm not sure if I got the concept entirely. Here is the transition matrix I found:
$$P=\left(\begin{array}{cccc}p_1&p_2&...&p_n\\p_1&p_2&...&p_n\\...&...&...&...\\ p_1&p_2&...&p_n\end{array}\right)$$
The questions that I have are: 1. Is the transition matrix correct? 2. How to show that the Markov chain is ergodic?
Thank you!
In regards to question $(2)$:
We have that $p_{i,j} > 0\ \forall i,j$, consider $$p_{i,j}^{(2)} = \sum_{k}p^{}_{i,k}p_{k,j} > 0,\text{ since }p > 0 \ \forall i,j$$ With an induction step, we can show that $p^{(n)}_{i,j} > 0 \ \forall i,j,n$.
Let us assume that $p^{(n)}_{i,j} > 0\ \forall i,j$ for some $n$, we will show that $p^{(n+1)}_{i,j} > 0\ \forall i,j$. $$p_{i,j}^{(n+1)} = \sum_{k}p^{(n)}_{i,k}p_{k,j} > 0 \text{, since }p_{k,j}, p^{(n)}_{i,k} > 0 \ \forall i,j,k$$
So, we have shown that $\forall i,j,n:\ \ p_{i,j}^{(n)} > 0$, because of this, it is possible for all states to reach all other states in a finite number of steps, and we can conclude that this markov chain is ergodic.