  This paper considers delay-limited communication over quasi-static fading
channels under a long-term power constraint. A sequence of length-$n$
delay-limited codes for a quasi-static fading channel is said to be
capacity-achieving if the codes achieve the delay-limited capacity, which is
defined to be the maximum rate achievable by delay-limited codes. The
delay-limited capacity is sometimes referred to as the zero-outage capacity in
wireless communications. The delay-limited capacity is the appropriate choice
of performance measure for delay-sensitive applications such as voice and video
over fading channels. It is shown that for any sequence of capacity-achieving
delay-limited codes with vanishing error probabilities, the normalized relative
entropy between the output distribution induced by the length-$n$ code and the
$n$-fold product of the capacity-achieving output distribution, denoted by
$\frac{1}{n}D(p_{Y^n}\|p_{Y^n}^*)$, converges to zero. Additionally, we extend
our convergence result to capacity-achieving delay-limited codes with
non-vanishing error probabilities.
