Fix an aperiodic discrete-time Markov chain $X_0,X_1,\dots$, with $\mathbb P_i$ denoting probabilities according to $X_0=i$. Define the return probability $$p_{ii}^{(n)} = \mathbb P_i[X_n=i].$$
Is it true that $$\sum_{n=0}^\infty |p_{ii}^{(n)}-p_{ii}^{(n+1)}|<\infty\text{?}$$
I am most interested in the null-recurrent case. For transient states there is a positive answer: the return probabilities are themselves summable. I know that for finite state spaces there is exponential convergence to some stationary state, which also implies a positive answer.