Abstract
We present a simple formula for the expected number of times that a complex‐valued Gaussian stochastic process has a zero imaginary part and the absolute value of its real part is bounded by a constant value M. We show that only some mild conditions on the stochastic process are needed for our formula to remain valid. We further apply this formula to a random algebraic polynomial with complex coefficients. We show how the above expected value in the case of random algebraic polynomials varies for different behaviour of M.
Highlights
There is a significant amount of work concerning the expected number of zeros of stationary normal processes
Random polynomials are special cases of the latter processes and, their properties are of special interest
Random polynomials with complex coefficients will introduce a new dimension to the study of nonstationary stochastic processes, caused by real and imaginary parts of polynomials
Summary
There is a significant amount of work concerning the expected number of zeros of stationary normal processes. Random polynomials are special cases of the latter processes and, their properties are of special interest. Random polynomials with complex coefficients will introduce a new dimension to the study of nonstationary stochastic processes, caused by real and imaginary parts of polynomials. We initially develop some properties of a complex-valued stochastic process in order to apply the result to random polynomials with complex coefficients. In [5], the average number E{N(α,β;M)} of real M-almost zeros of a complex-valued stochastic process H(t) = ζ(t) + ıξ(t) in the interval t ∈ (α,β) is given as β. The polynomial coefficients aj + ıbj have real and imaginary parts aj and bj forming sequences of independent normal random variables. We will show that the case of arbitrary mean and variance is amenable to a similar, albeit more technically involved, asymptotic analysis
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