  Star-formation rates (SFR) of disk galaxies strongly correlate with stellar
mass, with a small dispersion in SSFR at fixed mass, sigma~0.3 dex. With such
small scatter this star-formation main sequence (SFMS) has been interpreted as
deterministic and fundamental. Here we demonstrate that it is a simple
consequence of the central limit theorem. Our derivation begins by
approximating in situ stellar mass growth as a stochastic process, much like a
random walk (where the expectation of SFR at any time is equal to the SFR at
the previous time). We then derive expectation values for median SSFR of
star-forming disks and their scatter over time. We generalize the results for
stochastic changes in SFR that are not independent of each other but are
correlated over time. For unbiased samples of (disk) galaxies, we derive an
expectation that <SSFR> should be independent of mass, decline as 1/T, and have
a relative scatter that is independent of mass and time. The derived SFMS and
its evolution matches published data to z=10 with sufficient accuracy to
constrain cosmological parameters. The framework reproduces several important
observables, including: the scatter in SSFR at fixed mass; the SFHs of nearby
dwarf galaxies and the Milky Way; and the scatter in the Tully-Fisher relation.
The evolution of the mass function is less well reproduced and we discuss ways
to generalize the framework to include other sources of stellar mass such as
mergers. The predicted dispersion in SSFR has consequences for the
classification of quiescent galaxies, as such galaxies have heterogeneous
formation histories, and many may only be temporarily diminished in their
star-formation activity. The implied dispersion in SFHs, and the SFMS's
insensitivity to timescales of stochasticity, thus substantially limits the
ability to connect massive galaxies to their progenitors over long cosmic
baselines. [TRUNC.]
