PreprintCC BY 4.0arXiv
Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots
Abstract
Let X1,X2,… be i.i.d. complex-valued random variables with arbitrary Borel probability law μ, and set Pn(z)=∏j=1n(z−Xj). For every deterministic sequence kn=o(n), we prove that the empirical zero measure of the kn-th derivative Pn(kn) converges weakly to μ almost surely.
Related
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BibTeX
@misc{zhu2026convergence,
title = {{Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots}},
author = {Hongkai Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15984},
eprint = {2609.15984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15984},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convergence for Small-Order Derivatives of Random Polynomials with Independent Roots
AU - Hongkai Zhu
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15984
UR - https://arxiv.org/abs/2609.15984
AB - Let X_1,X_2,… be i.i.d. complex-valued random variables with arbitrary Borel probability law μ, and set P_n(z)=∏_j=1^n(z−X_j). For every deterministic sequence k_n=o(n), we prove that the empirical zero measure of the k_n-th derivative P_n^(k_n) converges weakly to μ almost surely.
ER -