Markov chain theorem to prove positive recurrence

Question

I need help with this problem that uses the following result:

Let $P$ and $P'$ be probability transition matrices of irreducible chains that are identical except for a finite number of terms in the first row. So $p_{0j} \neq p'_{0j}$ for some $j$, $j\leq k$, $k < \infty$. Then all the states of $P$ are positive recurrent if and only if all the states in $P$ are positiev recurrent.

This is my problem I want to solve

Use this result and what you know about another Markov chain to find conditions under which the following chain is positive recurrent:

$$\begin{bmatrix} .2 & .4 & .1 & .3 & &\cdots \\ q & 0 & p & 0 & & \cdots\\ 0 & q & 0 & p & & &\\ & \ddots & & \ddots & \end{bmatrix}$$

Here $pq > 0$ and $ p + q = 1$.

I tried many things like changing the first row to just one $1$ but I guess my thinking process is not great for positive recurrence. I guess I need to change the first row to make it positive recurrent but I do not know what conditions I will get.

Also I think this is similar to gambler's ruin problem matrix.

Answer 1

Hint 1:

The theorem you cite allows you to manipulate the first row, or at least its first four entries. How might you do that to obtain something familiar?

Hint 2:

What familiar chain would you have if the first row was $[0 \quad 1 \quad 0 \quad 0 \quad \dots]$?

Hint 3:

With the change referenced in Hint 2, you'd have a $p \uparrow, q \downarrow$ random walk on the nonnegative integers. What do you know about positive recurrence of that chain?

Solution:

The $p \uparrow, q \downarrow$ random walk on the nonnegative integers is positive recurrent if, and only if, $q > p$. (If $q = p$, then the chain is null recurrent, and if $q < p$, then the chain is transient.)