  Probabilistic amplitude shaping (PAS) is a coded modulation strategy in which
constellation shaping and channel coding are combined. PAS has attracted
considerable attention in both wireless and optical communications. Achievable
information rates (AIRs) of PAS have been investigated in the literature using
Gallager's error exponent approach. In particular, it has been shown that PAS
achieves the capacity of the additive white Gaussian noise channel (B\"ocherer,
2018). In this work, we revisit the capacity-achieving property of PAS and
derive AIRs using weak typicality. Our objective is to provide alternative
proofs based on random sign-coding arguments that are as constructive as
possible. Accordingly, in our proofs, only some signs of the channel inputs are
drawn from a random code, while the remaining signs and amplitudes are produced
constructively. We consider both symbol-metric and bit-metric decoding.
