  We present some new results about the distribution of a random walk whose
independent steps follow a $q-$Gaussian distribution with exponent
$\frac{1}{1-q}; q \in \mathbb{R}$. In the case $q>1$ we show that a stochastic
representation of the point reached after $n$ steps of the walk can be
expressed explicitly for all $n$. In the case $q<1,$ we show that the random
walk can be interpreted as a projection of an isotropic random walk, i.e. a
random walk with fixed length steps and uniformly distributed directions.
