Expected Real Roots - Quant Trader Interview Question
Difficulty: Hard
Category: Market Microstructure
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Topics: probability, expected-value, asymptotics, kac-polynomial
Problem Description
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
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