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Lecture "Transforming i.d. Random Variables into each other with Conditional Expectations"

In October 2 Georgy Gaitsgori (HSE) delivered a lecture «Transforming i.d. random variables into each other with conditional expectations»

About ten years ago A. Cherny and P. Grigoriev obtained the following striking results: for any 𝜖 > 0 and any two random variables 𝑋 and 𝑌 with the same distribution, there is a sequence of sigma-algebras 𝐹𝑛 such that ||𝑋𝑛−𝑌||∞ <𝜖, where 𝑋1 =𝐸(𝑋|𝐹1),...,𝑋𝑛 = 𝐸(𝑋𝑛−1|𝐹𝑛). This result is highly connected to Risk Theory or more precisely to law invariant risk measures.

In this talk some background of the problem and its transparent interpretation were provided. The theorem was proved, the optimal selection of such sequence was shown, and the exact first term of the asymptotic behavior of 𝜖 = 𝜖𝑛, when 𝑛 tends to infinity was given. 
Gaitsgori Georgy.pdf