  We study the asymptotic behaviour of the time-changed stochastic process
$\vphantom{X}^f\!X(t)=B(\vphantom{S}^f\!S (t))$, where $B$ is a standard
one-dimensional Brownian motion and $\vphantom{S}^f\!S$ is the (generalized)
inverse of a subordinator, i.e. the first-passage time process corresponding to
an increasing L\'evy process with Laplace exponent $f$. This type of processes
plays an important role in statistical physics in the modeling of anomalous
subdiffusive dynamics. The main result of the paper is the proof of the mixing
property for the sequence of stationary increments of a subdiffusion process.
We also investigate various martingale properties, derive a generalized
Feynman-Kac formula, the laws of large numbers and of the iterated logarithm
for $\vphantom{X}^f\!X$.
