Expected number of real zeros of random hyperbolic polynomial
There are many known results concerning the mathematical expectation of the number of real zeros of the random algebraic polynomial \(\sum_{k=0}^{n}g_k t^k\) where \(\{g_k\}_{k=0}^n\) is a sequence of independent identically distributed normal standard random variables. However, in comparison, very little is known about the polynomials in the form of \(f_n(t)= \sum_{k=0}^n g_k \cosh kt\). The analysis involved with the latter polynomial becomes complicated and therefore less information is known. The asymptotic value for the expected number of real zeros of \(f_n(t)\) denoted by \(EN_n\) is obtained, by various authors, and known to be \((1/\pi)\log n\). The error term involved in the calculation of these asymptotic values, however, is large and generally is an increasing function of \(n\). This paper shows that this error term in fact is \(O(1)\). The author gives a very detailed estimate of the terms involved in calculation of \(EN_n\) and proves that \(EN_n\sim (1/\pi)\log n+0.038...+o(1)\). The result is obtained by dividing the intervals \((-\infty, \infty)\) to several subinterval and using different methods in each. These methods should be of interest to those who would like to generalize the known results for \(f_n(t)\) to other cases already developed for the random algebraic polynomials. For earlier results on the subject see \textit{A. T. Bharucha-Reid} and \textit{M. Sambandham} [``Random polynomials, probability and mathematical statistics'' (1986; Zbl 0615.60058)] and the reviewer [``Topics in random polynomials'' (1998; Zbl 0949.60010)] and their references.
scientific article; zbMATH DE number 5130369
scientific article