  We consider the sum of two self-similar centred Gaussian processes with
different self-similarity indices. Under non-negativity assumptions of
covariance functions and some further minor conditions, we show that the
asymptotic behaviour of the persistence probability of the sum is the same as
for the single process with the greater self-similarity index.
  In particular, this covers the mixed fractional Brownian motion introduced in
Cheridito (2001) and shows that the corresponding persistence probability
decays asymptotically polynomially with persistence exponent $1-\max(1/2,H),$
where $H$ is the Hurst parameter of the underlying fractional Brownian motion.
