Markov chains and their distribution ruins

Question

you chain.

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I'm having trouble wrapping my head around finding the stationary distribution when the state space is infinite. Anyone have any tips/advice or a solution? Thanks!

Answer 1

_{Note: the following answer was posted after the OP posted a nearly correct answer in the comment section above.}

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From $\boldsymbol{\pi}^\top = \boldsymbol{\pi}^\top {\bf P} $, eventually, we obtain

$$ \pi_k = \color{blue}{\left(\frac{p}{1-p}\right)^k \pi_0}$$

for all $k \in \Bbb N$. Since $\boldsymbol{\pi}$ is a probability mass function (PMF), $ \sum\limits_{k=0}^\infty \pi_k = 1 $, which yields

$$\pi_0 = \color{blue}{\dfrac{1-2p}{1-p}}$$

which is positive because $p < \frac12$.