  We prove the compact law of the iterated logarithm for stationary and ergodic
differences of (reverse or not) martingales taking values in a separable
$2$-smooth Banach space (for instance a Hilbert space). Then, in the martingale
case, the almost sure invariance principle is derived from a result of Berger.
From those results, we deduce the almost sure invariance principle for
stationary processes under the Hannan condition and the compact law of the
iterated logarithm for stationary processes arising from non-invertible
dynamical systems. Those results for stationary processes are new, even in the
real valued case. We also obtain the Marcinkiewicz-Zygmund strong law of large
numbers for stationary processes with values in some smooth Banach spaces.
Applications to several situations are given.
