  This paper investigates the asymptotic expansion for the size of block codes
defined for the additive white Gaussian noise (AWGN) channel with feedback
under the following setting: A peak power constraint is imposed on every
transmitted codeword, and the average error probability of decoding the
transmitted message is non-vanishing as the blocklength increases. It is
well-known that the presence of feedback does not increase the first-order
asymptotics (i.e., capacity) in the asymptotic expansion for the AWGN channel.
The main contribution of this paper is a self-contained proof of an upper bound
on the asymptotic expansion for the AWGN channel with feedback. Combined with
existing achievability results for the AWGN channel, our result implies that
the presence of feedback does not improve the second- and third-order
asymptotics. An auxiliary contribution is a proof of the strong converse for
the parallel Gaussian channels with feedback under a peak power constraint.
