  We study some SDEs derived from the $q\to 1$ limit of a 2D surface growth
model called the $q$-Whittaker process. The fluctuations are proven to exhibit
Gaussian characteristics that "come down from infinity": After rescaling and
re-centering, convergence to the time-inverted stationary additive stochastic
heat equation holds. The point of view in this paper is a probabilistic
representation of the SDEs by independent sums. By this connection, the normal
and Poisson approximations and the in-between slow decorrelation, all in
particular integrated forms, explain the convergence of the re-centered
covariance functions. With bounds and divergent constants from these
approximations, the proof of the process-level convergence identifies
additional divergent terms in the dynamics and considers cancellation arguments
that treat the independent sums as discrete spin systems.
