  This paper investigates the asymptotic expansion for the maximum rate of
fixed-length codes over a parallel Gaussian 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 of the
channel, i.e., capacity, in the asymptotic expansion, and the closed-form
expression of the capacity can be obtained by the well-known water-filling
algorithm. The main contribution of this paper is a self-contained proof of an
upper bound on the second-order asymptotics of the parallel Gaussian channel
with feedback. The proof techniques involve developing an information spectrum
bound followed by using Curtiss' theorem to show that a sum of dependent random
variables associated with the information spectrum bound converges in
distribution to a sum of independent random variables, thus facilitating the
use of the usual central limit theorem. Combined with existing achievability
results, our result implies that the presence of feedback does not improve the
second-order asymptotics.
