Abstract
We study order 2 Gaussian chaos and the tails of higher-order chaos.
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Notes
- 1.
It would also be natural to consider processes where the size of the “increments” X s − X t is controlled by a distance d in a different manner, e.g., for all u > 0, P(|X s − X t|≥ ud(s, t)) ≤ ψ(u), for a given function ψ, see [143]. This question has received considerably less attention than the condition (16.1).
- 2.
If one works a tad harder, one may get B rather than 2B in the next inequality.
- 3.
The condition α j ≥ 1∕400 simply ensures that α j stays away from 0.
- 4.
We remind the reader that in particular \(\mathsf {E} Y^j_t=0\).
- 5.
Again, no skill whatsoever is required there.
- 6.
And, as we look at the structure at increasingly finer scale, these are the first gaps which will appear, and the gaps inside each block are much smaller.
- 7.
One may use in particular that since the elements of U k,ℓ are \(\leq 2^{-2^k}\), it should be obvious that one can achieve \(b_{k,\ell }\geq \beta _\ell b_k-2^{-2^k}\geq (1-2^{-k})\beta _\ell b_k\).
- 8.
I won’t dare attempting a literal translation, but roughly this distinguishes between questions of self-evident importance and more arbitrary questions one may ask.
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Talagrand, M. (2021). Convergence of Orthogonal Series: Majorizing Measures. 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_16
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