arXiv subjects

Christopher Corley

Publications and source records attributed to Christopher Corley.

3 recordsLinked to original sources

arXiv2021

The complex zeros of random orthogonal polynomials

We utilize Cauchy's argument principle in combination with the Jacobian of a holomorphic function in several complex variables and the first moment of a ratio of two correlated complex normal random variables to prove explicit formulas for the density and the mean distribution of complex zeros of random polynomials spanned by orthogonal polynomials on the unit circle and on the unit disk. We then inquire into the consequences of their asymptotical evaluations.

math.PR↗

arXiv2020

The level crossings of random sums

Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent and identically distributed complex normal random variables with mean zero and variances $\{\sigma_{j}^{2}\}_{j = 0}^{N}$. Let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of holomorphic functions that are real-valued on the real line. The purpose of the present study is that of examining the number of times that the random sum $\sum_{j = 0}^{N} \eta_{j} f_{j} (z)$ crosses the complex level $\boldsymbol{K} = K_{1} + i K_{2}$, where $K_{1}$ and $K_{2}$ are constants independent of $z$. More specifically, we establish an exact formula for the expected density function for the complex zeros. We then reformulate the problem in terms of successive observations of a Brownian motion. We further answer the basic question about the expected number of complex zeros for coefficients of nonvanishing mean values.

math.PR↗

arXiv2019

The density of complex zeros of random sums

Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of given analytic functions that are real-valued on the real number line. We prove an exact formula for the expected density of the distribution of complex zeros of the random equation $\sum_{j = 0}^{N} \eta_{j} f_{j} (z) = \mathbf{K}$, where $\mathbf{K} \in \mathds{C}$. The method of proof employs a formula for the expected absolute value of quadratic forms of Gaussian random variables. We then obtain the limiting behaviour of the density function as $N$ tends to infinity and provide numerical computations for the density function and empirical distributions for random sums with certain functions $f_{j} (z)$. Finally, we study the case when the $f_{j} (z)$ are polynomials orthogonal on the real line and the unit circle.

math.PR↗