  In this paper, we consider single- and multi-user Gaussian channels with
feedback under expected power constraints and with non-vanishing error
probabilities. In the first of two contributions, we study asymptotic
expansions for the additive white Gaussian noise (AWGN) channel with feedback
under the average error probability formalism. By drawing ideas from Gallager
and Nakibo\u{g}lu's work for the direct part and the meta-converse for the
converse part, we establish the $\varepsilon$-capacity and show that it depends
on $\varepsilon$ in general and so the strong converse fails to hold.
Furthermore, we provide bounds on the second-order term in the asymptotic
expansion. We show that for any positive integer $L$, the second-order term is
bounded between a term proportional to $-\ln_{(L)} n$ (where $\ln_{(L)}(\cdot)$
is the $L$-fold nested logarithm function) and a term proportional to
$+\sqrt{n\ln n}$ where $n$ is the blocklength. The lower bound on the
second-order term shows that feedback does provide an improvement in the
maximal achievable rate over the case where no feedback is available. In our
second contribution, we establish the $\varepsilon$-capacity region for the
AWGN multiple access channel (MAC) with feedback under the expected power
constraint by combining ideas from hypothesis testing, information spectrum
analysis, Ozarow's coding scheme, and power control.
