  We extend two results about the ordinary continued fraction expansion to best
simultaneous Diophantine approximations of vectors or matrices. The first is
Levy-Khintchin Theorem about the almost sure growth rate of the denominators of
the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about
the almost sure limit distribution of the sequence of products $q_n
d(q_n\theta, Z)$ where the $q_n$'s are the denominators of the convergents
associated with the real number $\theta$ by the ordinary continued fraction
algorithm. Beside these two main results, we show that when $d\ge2$, for almost
all vectors $\theta\in R^d$, $\liminf_{n\to\infty} q_{n+k}d(q_n\theta, Z^d)=0$
for all positive integers $k$, where $(q_n)_{n\in N}$ is the sequence of best
approximation denominators of $\theta$.
