  We consider channel simulation protocols between two communicating parties,
Alice and Bob. First, Alice receives a target distribution $Q$, unknown to Bob.
Then, she employs a shared coding distribution $P$ to send the minimum amount
of information to Bob so that he can simulate a single sample $X \sim Q$. For
discrete distributions, Harsha et al. (2009) developed a well-known channel
simulation protocol -- greedy rejection sampling (GRS) -- with a bound of
${D_{KL}[Q \,\Vert\, P] + 2\ln(D_{KL}[Q \,\Vert\, P] + 1) + \mathcal{O}(1)}$ on
the expected codelength of the protocol. In this paper, we extend the
definition of GRS to general probability spaces and allow it to adapt its
proposal distribution after each step. We call this new procedure Adaptive GRS
(AGRS) and prove its correctness. Furthermore, we prove the surprising result
that the expected runtime of GRS is exactly $\exp(D_\infty[Q \,\Vert\, P])$,
where $D_\infty[Q \,\Vert\, P]$ denotes the R\'enyi $\infty$-divergence. We
then apply AGRS to Gaussian channel simulation problems. We show that the
expected runtime of GRS is infinite when averaged over target distributions and
propose a solution that trades off a slight increase in the coding cost for a
finite runtime. Finally, we describe a specific instance of AGRS for 1D
Gaussian channels inspired by hybrid coding. We conjecture and demonstrate
empirically that the runtime of AGRS is $\mathcal{O}(D_{KL}[Q \,\Vert\, P])$ in
this case.
