Let $(X_t)$ be a discrete time Markov chain on a discrete or continuous state space $S$. The hitting time for a subset $C$ of $S$ is the first time $t \geq 0$ that $X_t \in C$. A "strong $\nu$ time'' is an analogue of a hitting time for a probability measure $\nu$ on $S$ instead of a subset $C$ of $S$.

Suppose $(X_t)$ is equipped with a strong $\nu$ time $T$. How well does the law of $T$ control the convergence rate of $(X_t)$ to its stationary distribution? The answer is not well at all in general, but very well when $(X_t)$ is reversible with nonnegative eigenvalues. In the talk I will explain this result and discuss applications to Markov chain Monte Carlo algorithms as well as cutoff window bounds for birth and death chains.