  We derive explicit distance bounds for Stratonovich iterated integrals along
two Gaussian processes (also known as signatures of Gaussian rough paths) based
on the regularity assumption of their covariance functions. Similar estimates
have been obtained recently in [Friz-Riedel, AIHP, to appear]. One advantage of
our argument is that we obtain the bound for the third level iterated integrals
merely based on the first two levels, and this reflects the intrinsic nature of
rough paths. Our estimates are sharp when both covariance functions have finite
1-variation, which includes a large class of Gaussian processes.
  Two applications of our estimates are discussed. The first one gives the a.s.
convergence rates for approximated solutions to rough differential equations
driven by Gaussian processes. In the second example, we show how to recover the
optimal time regularity for solutions of some rough SPDEs.
