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Expected Real Roots

Hard · Market Microstructure · Quant Trader interview question · probability, expected-value, asymptotics, kac-polynomial

A high-frequency trading firm uses polynomial models to predict market movements. You are tasked with analyzing the expected number of times a price trajectory, modeled as a polynomial, crosses its average value. Assume the price trajectory is represented by a polynomial of degree $n$ with coefficients independently drawn from a standard normal distribution. What is the asymptotic expected number of real roots of this polynomial as $n$ becomes very large? In other words, consider a polynom