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CENTRAL LIMIT THEOREMS AND THE GEOMETRY OF POLYNOMIALS

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Let X ∈ {0, . . . , n} be a random variable with standard deviation σ and let f_X be its probability generating function. Pemantle conjectured that if σ is large and f_X has no roots close to 1 in the complex plane then X must approximate a normal distribution. In this talk, I will discuss a complete resolution of Pemantle’s conjecture. I shall also mention a how these ideas can be used to prove a multivariate central limit theorem for strong Rayleigh distributions, thereby resolving a conjecture of Gosh, Liggett and Pemantle. This talk is based on joint work with Marcus Michelen.

This talk is part of the Probability series.

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