On the Clustering of Padé Zeros and Poles of Random Power Series

We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé 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 Lévy function ${\mathcal L}(a_{j}, \varepsilon ) =\sup _{t\in \mathbb R} \mathbb{P}(|a_{j} -t|<\varepsilon )$ satisfying ${\mathcal L}(a_{j}, \varepsilon ) \leq K\varepsilon$. 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 $f$, the poles of the $[m,n]$-Padé 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.