  We develop a systematic matrix-analytic approach, based on intertwinings of
Markov semigroups, for proving theorems about hitting-time distributions for
finite-state Markov chains -- an approach that (sometimes) deepens
understanding of the theorems by providing corresponding
sample-path-by-sample-path stochastic constructions. We employ our approach to
give new proofs and constructions for two theorems due to Mark Brown, theorems
giving two quite different representations of hitting-time distributions for
finite-state Markov chains started in stationarity. The proof, and
corresponding construction, for one of the two theorems elucidates an
intriguing connection between hitting-time distributions and the interlacing
eigenvalues theorem for bordered symmetric matrices.
