  We study non-trivial translation-invariant probability measures on the space
of entire functions of one complex variable. The existence (and even an
abundance) of such measures was proven by Benjamin Weiss. Answering Weiss
question, we find a relatively sharp lower bound for the growth of entire
functions in the support of such measures. The proof of this result consists of
two independent parts: the proof of the lower bound and the construction, which
yields its sharpness. Each of these parts combines various tools (both
classical and new) from the theory of entire and subharmonic functions and from
the ergodic theory. We also prove several companion results, which concern the
decay of the tails of non-trivial translation-invariant probability measures on
the space of entire functions and the growth of locally uniformly recurrent
entire and meromorphic functions.
