  We present several new phenomena about almost sure convergence on homogeneous
chaoses that include Gaussian Wiener chaos and homogeneous sums in independent
random variables. Concretely, we establish the fact that almost sure
convergence on a fixed finite sum of chaoses forces the almost sure convergence
of each chaotic component. Our strategy uses "{\it extra randomness}" and a
simple conditioning argument. These ideas are close to the spirit of
\emph{Stein's method of exchangeable pairs}. Some natural questions are left
open in this note.
