Given an irreducible Markov chain with period $k$, I would like to show that $p^{(n)}(x,x)$ does not converge to a limit for a recurrent state $x$.
For the proof, I will find two subsequences that for one $p^{(n_l)}(x,x) \rightarrow 0$, and for the other one $p^{(n_i)}(x,x) \rightarrow c>0$
First since the period of $x$ is $k$, pick any number $l$ that is coprime to $k$, then we have $p^{(l^n)}(x,x) = 0 \rightarrow 0$ as $n\rightarrow \infty.$
Now, look at the Markov chain with transition matrix (possibly infinite size) given by $P^k$ where $P$ is the original transition matrix. There exists a stationary distribution $\pi$ on the closed irreducible subset containing $x$. Also in this new Markov chain, $x$ has period 1, and $x$ is also recurrent because $$\sum_{i=1}^\infty \tilde p^{(n)}(x,x) = \sum_{i=1}^\infty p^{(nk)}(x,x) = \sum_{i=1}^\infty p^{(n)}(x,x) = \infty.$$ By the MC convergence theorem, we know $p^{(nk)}(x,x) \rightarrow \pi(x) > 0$.