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On the clustering of Padé zeros and poles of random power series

Stamatis Dostoglou, Petros Valettas

Abstract

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 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 $[m,n]$-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.

On the clustering of Padé zeros and poles of random power series

Abstract

We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the -Padé approximant of a random power series for independent, with finite first moment, and Lévy function satisfying . Under the same assumptions we show that almost surely has infinitely many zeros in the unit disc, with the unit circle serving as a natural boundary for . For the radius of the largest disc containing at most zeros of , a deterministic result of Edrei implies that in our setting the poles of the -Padé approximant almost surely cluster uniformly at the circle of radius as and stays fixed, and we provide almost sure rates of converge of these 's to . We also show that our results on the clustering of the zeros hold for log-concave vectors with not necessarily independent coordinates.
Paper Structure (16 sections, 22 theorems, 139 equations)

This paper contains 16 sections, 22 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Padé, Toeplitz, and Erdős -Turán
  3. Padé approximants and formulas
  4. Random Toeplitz determinants
  5. Clustering of zeros
  6. Erdős-Turán for Padé numerators and denominators
  7. Non-asymptotic clustering of roots
  8. Asymptotic clustering of poles
  9. Natural boundary
  10. Random zeros in the unit disc
  11. Clustering of poles
  12. Logarithmically concave data
  13. Anti-concentration for coordinates of log-concave vectors
  14. Large deviation estimates for projections of log-concave vectors
  15. Determinant of Toeplitz matrices with log-concave symbol
  16. ...and 1 more sections

Key Result

Lemma 2.2

For any (deterministic) monic polynomial $P$ with $\deg P=d$, for any random variable $\xi$, and for any $\varepsilon >0$, we have

Theorems & Definitions (38)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3: quantitative invertibility
  • proof
  • Definition 2.4: clustering
  • Proposition 2.5
  • Proposition 2.6: Jensen Formula
  • Proposition 2.7: Erdős-Turán
  • Proposition 2.8
  • ...and 28 more