Abstract
We extend the method constructing suitable increasing sequences of partitions from the metric spaces of Chap. 1 to the setting of families of distances. This makes it possible to extend the necessary and sufficient conditions for sample boundedness of Gaussian processes to vastly more general classes of processes.
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Notes
- 1.
The reason why we take r of the type r = 2κ−3 for an integer κ is purely for technical convenience.
- 2.
In [132] the present theorem is stated without assuming this condition but the proof given there is in error. The condition (8.9) is a very mild extra hypothesis, since in the separation condition, we have already implicitly assumed that B j+2(t ℓ,  2n+κ) = B j+2(t ℓ,  (4r)2n+1) is quite smaller than B j+1(t ℓ,  2n+1).
- 3.
Bernoulli process, which can be thought of as the “limiting case p = ∞″, motivated the present investigation.
- 4.
Here ∥X t∥u is the L p norm for p = u.
- 5.
Not to be confused with the interior \(\overset {\circ } A\) of A!
- 6.
A ball A is convex set with non-empty interior and A = −A.
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Talagrand, M. (2021). Partitioning Scheme and Families of Distances. 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_8
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