Abstract
We describe the basic tools bearing on Bernoulli processes, which parallel those of the Gaussian case. A Bernoulli process is a family of random variables (r.v.s), each of which is a different linear combination of the same independent random signs. There are two obvious ways Bernoulli process may be bounded: First, it may happen that the corresponding Gaussian process (where the random signs are replaced by standard independent Gaussian r.v.s) is already bounded. Second, it may happen that the sum of the absolute values of the coefficients of the random signs is already bounded. We state the Bernoulli conjecture, which asserts that every situation is a combination of these.
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Notes
- 1.
- 2.
But this method is not always sharp.
- 3.
You may find the notation silly, since T 1 is controlled by the ℓ 2 norm and T 2 by the ℓ 1 norm. The idea underlying my notation, here and in similar situations, is that T 1 denotes what I see as the main part of T, whereas T 2 is more like a perturbation term.
- 4.
Please see Footnote 2 on page 2 concerning the name I give to this result.
- 5.
We assume here that I is infinite, leaving the necessary simple modifications of the argument when I is finite to the reader.
- 6.
In particular Bernstein’s inequality suffices to perform the chaining.
- 7.
A good path to get a feeling for this theorem is to study Exercise 14.2.3 in due time.
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Talagrand, M. (2021). Bernoulli Processes. 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_6
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