Let be the infinitesimal transition matrix of a Markov chain, and suppose maxi i.

Let Λ be the infinitesimal transition matrix of a Markov chain, and suppose µ ≥ maxi λi. If R = I + 1

µΛ, prove that R has nonnegative entries and that S(t) = ∞

i=0 e−µt (µt)i i! Ri coincides with P(t). (Hint: Verify that S(t) satisfies the same defining differential equation and the same initial condition as P(t).)