The density of complex zeros of random sums

A random polynomial is a polynomial whose coefficients follow some probability distribution. The fundamental questions that need to be studied are the distribution and correlations between zeros, pairing between zeros and critical points, distribution values, and nodal surfaces. The computation of the average distribution of real zeros of random polynomials was studied by Bloch and Pólya, Littlewood and Offord, Erdős, Kac and others. For standard normally distributed coefficients, the expected density of real zeros is given by Kac’s exact formula. The famous result due to Hammersley asserts that, when the coefficients are complex independent standard normal random variables, the zeros of a random complex polynomial largely tend towards the unit circle as the degree approaches infinity. For complex zeros, the expected density was dealt with by Shepp and Vanderbei for real independent and identically distributed normal coefficients. Their technique exploits the argument principle and Cholesky factorization to reduce the question to the evaluation of a holomorphic function of four correlated normal random variables. Their results were generalized by Ibragimov and Zeitouni to a wide class of distribution of coefficients. Recently, Vanderbei extended the results he obtained with Shepp to random sums with holomorphic functions that are real-valued on the real line as the basis functions. Our interest in this dissertation is to refine the techniques of random fields pioneered by Rice in his treatment of the questions on real zeros to obtain exact formulas for the expected density of the distribution of complex zeros of a family of random sums, such as truncated random trigonometric series and random orthogonal polynomials on the unit circle. We further study the level crossings and answer the question about the expected number of complex zeros for coefficients with nonvanishing mean values and distinct variances