Equivalence of starting point cutoff and the concentration of hitting times on a general state space

In this article we extend a result of Martinez and Ycart from their paper “Decay Rates and Cutoff for Convergence and Hitting Times of Markov Chains with Countably Infinite State Space” [7] to show that for a regular Markov chain on a general state space, the existence of a cutoff phenomenon for total variation distance to stationarity as the starting point tends to infinity is equivalent to the concentration of hitting times for any fixed regular set as the starting points tend to infinity. We apply this result to show that all random walks on the half-line with bounded steps exhibit starting point cutoff.