  This paper considers a massive connectivity setting in which a base-station
(BS) aims to communicate sources $(X_1,\cdots,X_k)$ to a randomly activated
subset of $k$ users, among a large pool of $n$ users, via a common downlink
message. Although the identities of the $k$ active users are assumed to be
known at the BS, each active user only knows whether itself is active and does
not know the identities of the other active users. A naive coding strategy is
to transmit the sources alongside the identities of the users for which the
source information is intended, which would require $H(X_1,\cdots,X_k) +
k\log(n)$ bits, because the cost of specifying the identity of a user is
$\log(n)$ bits. For large $n$, this overhead can be significant. This paper
shows that it is possible to develop coding techniques that eliminate the
dependency of the overhead on $n$, if the source distribution follows certain
symmetry. Specifically, if the source distribution is independent and
identically distributed (i.i.d.) then the overhead can be reduced to at most
$O(\log(k))$ bits, and in case of uniform i.i.d. sources, the overhead can be
further reduced to $O(1)$ bits. For sources that follow a more general
exchangeable distribution, the overhead is at most $O(k)$ bits, and in case of
finite-alphabet exchangeable sources, the overhead can be further reduced to
$O(\log(k))$ bits. The downlink massive random access problem is closely
connected to the study of finite exchangeable sequences. The proposed coding
strategy allows bounds on the relative entropy distance between finite
exchangeable distributions and i.i.d. mixture distributions to be developed,
and gives a new relative entropy version of the finite de Finetti theorem which
is scaling optimal.
