The density of complex zeros of random sums
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Let {η_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 ∑_j = 0^Nη_j f_j (z) = K, where K∈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.
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