Conditional Expectations and Pouring Water

Student: Gaitsgori Georgy

Educational Programme: Mathematics (Bachelor)

Year of Graduation: 2020

About ten years ago A. Cherny and P. Grigoriev obtained the following striking result: for any $\epsilon > 0$ and any two random variables $X$ and $Y$ with the same distribution, there is a sequence of sigma-algebras $F_n$ such that $||X_n - Y||_\infty < \epsilon$, where $X_1 = E(X|F_1), \dots, X_n = E(X_{n-1}|F_n)$. In this paper we provide the transparent interpretation of this problem, show the optimal selection of such sequence, and give the exact first term of the asymptotic behavior of $\epsilon = \epsilon_n$ when $n$ tends to infinity.

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