## Abstract

We study a family of random Taylor series

F(z)=∑n≥0ζnanzn

with radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients (ζn) ; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that F does not vanish in the disk {|z|⩽r} as r↑1 . Our bounds take different forms according to whether the non-random coefficients (an) grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients (an) display power-law behavior.

F(z)=∑n≥0ζnanzn

with radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients (ζn) ; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that F does not vanish in the disk {|z|⩽r} as r↑1 . Our bounds take different forms according to whether the non-random coefficients (an) grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients (an) display power-law behavior.

Original language | English |
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Pages (from-to) | 377-430 |

Journal | PROBABILITY THEORY AND RELATED FIELDS |

Volume | 171 |

Issue number | 1-2 |

Early online date | 20 May 2017 |

DOIs | |

Publication status | Published - Jun 2018 |