  In this note we (in particular) prove an almost sure invariance principle
(ASIP) for non-stationary and uniformly bounded sequences of random variables
which are exponentially fast $\phi$-mixing. The obtained rate is of order
$o(V_n^{\frac14+\del})$ for an arbitrary $\del>0$, where $V_n$ is the variance
of the underlying partial sums $S_n$. For certain classes of inhomogeneous
Markov chains we also prove a vector-valued ASIP with similar rates.
