Abstract
We prove the Bernoulli conjecture.
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Notes
- 1.
A statement that has to be qualified by the fact that I do not read anything!
- 2.
It is in homage to this extraordinary result that I have decided to violate alphabetical order and to call Theorem 6.2.8 the Latała-Bednorz theorem. This is my personal decision, and impressive contributions of Witold Bednorz to this area of probability theory are brought to light in particular in Chap. 16.
- 3.
The simpler proof presented here comes from [17].
- 4.
The resulting information will allow us to remove some of the Bernoulli r.v.s at certain steps of our construction.
- 5.
As usual the ball B(t ℓ, σ) below is the ball for the distance d.
- 6.
- 7.
In words, the points of Ω n(t) are those for which the sequence (π q(t))q≤n+1 does not have big jumps.
- 8.
The reader should notice that the set I′ is not constructed, but is known once w and w′ are known.
- 9.
Please observe what is happening here. The information (10.78) is a consequence of (10.73), not of the fact that we have partitioned the set T. This is coherent with our proof of Latała’s principle. There is only one piece what satisfies (10.77) and (10.78). This piece is what is left of T after we have removed some parts which satisfy (10.76), and we took no action to decrease its diameter in any sense.
- 10.
You can try to visualize things that way: we use one single point w to describe the “position” of T. When we break T into small pieces, this gets easier. Furthermore, the position of some of the small pieces is better described by an other point than w.
- 11.
The real reason to use j + 2 rather than j + 1 will become apparent only later. In short, it is because in (10.98), we absolutely need to have a condition bearing on Δ(D, I, w, k, j + 2), not on the larger quantity Δ(D, I, w, k, j + 1).
- 12.
Actually we have F(0) = b(T).
References
Bednorz, W., Latała, R.: On the boundedness of Bernoulli processes. Ann. Math. (2) 180(3), 1167–1203 (2014)
Latała, R.: On the boundedness of Bernoulli processes on thin sets. Elec. Comm. Probab. 13, 175–186 (2008)
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Talagrand, M. (2021). Proof of the Bernoulli Conjecture. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Cham. https://doi.org/10.1007/978-3-030-82595-9_10
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