arXiv subjects

David Hertz

Publications and source records attributed to David Hertz.

2 recordsLinked to original sources

arXiv2026

Algorithm to Compute a Kharitonov-Type Sector Containing All Roots of Hurwitz Interval Polynomials

This paper presents a Kharitonov-type algorithm for complex interval Hurwitz polynomials that determines whether all roots of a given interval polynomial lie within a prescribed angular sector of the complex plane. The method requires evaluating a finite set of additional Kharitonov polynomials. For complex coefficient uncertainty, up to sixteen such polynomials are sufficient, while in the real-coefficient case up to eight are needed. A bisection-based refinement procedure is introduced to compute a containing sector that encloses the angles of all roots. The algorithm progressively tightens the sector bounds and can achieve arbitrarily small accuracy. In the real-coefficient case, the symmetry of the construction allows the real Kharitonov result to be derived directly from the complex case. Numerical experiments suggest that the minimal containing sector coincides with the sector determined by the vertex polynomials, or possibly by a subset of them.

math.GM↗

arXiv2020

Improved Hoeffding's Lemma and Hoeffding's Tail Bounds

The purpose of this letter is to improve Hoeffding's lemma and consequently Hoeffding's tail bounds. The improvement pertains to left skewed zero mean random variables $X\in[a,b]$, where $a<0$ and $-a>b$. The proof of Hoeffding's improved lemma uses Taylor's expansion, the convexity of $\exp(sx), s\in {\bf R}$ and an unnoticed observation since Hoeffding's publication in 1963 that for $-a>b$ the maximum of the intermediate function $τ(1-τ)$ appearing in Hoeffding's proof is attained at an endpoint rather than at $τ=0.5$ as in the case $b>-a$. Using Hoeffding's improved lemma we obtain one sided and two sided tail bounds for $P(S_n\ge t)$ and $P(|S_n|\ge t)$, respectively, where $S_n=\sum_{i=1}^nX_i$ and the $X_i\in[a_i,b_i],i=1,...,n$ are independent zero mean random variables (not necessarily identically distributed). It is interesting to note that we could also improve Hoeffding's two sided bound for all $\{X_i: a_i\ne b_i,i=1,...,n\}$. This is so because here the one sided bound should be increased by $P(-S_n\ge t)$, wherein the left skewed intervals become right skewed and vice versa.

math.PR↗