  We estimate non-asymptotically the probability of uniform clustering around
the unit circle of the zeros of the $[m,n]$-Pad\'e approximant of a random
power series $f(z) = \sum_{j=0}^\infty a_j z^j$ for $a_j$ independent, with
finite first moment, and L\'evy function satisfying $L(a_j , \varepsilon) \leq
K\varepsilon$. Under the same assumptions we show that almost surely $f$ has
infinitely many zeros in the unit disc, with the unit circle serving as a
natural boundary for $f$. For $R_m$ the radius of the largest disc containing
at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our
setting the poles of the $[m,n]$-Pad\'e approximant almost surely cluster
uniformly at the circle of radius $R_m$ as $n \to \infty$ and $m$ stays fixed,
and we provide almost sure rates of converge of these $R_m$'s to $1$. We also
show that our results on the clustering of the zeros hold for log-concave
vectors $(a_j)$ with not necessarily independent coordinates.
