Dudley's theorem
In probability theory, Dudley's theorem is a result relating the expected upper bound and regularity properties of a Gaussian process to its entropy and covariance structure.
History
The result was first stated and proved by V. N. Sudakov, as pointed out in a paper by Richard M. Dudley.[1] Dudley had earlier credited Volker Strassen with making the connection between entropy and regularity.
Statement
Let (Xt)t∈T be a Gaussian process and let dX be the pseudometric on T defined by
- [math]\displaystyle{ d_{X}(s, t) = \sqrt{\mathbf{E} \big[ | X_{s} - X_{t} |^{2} ]}. \, }[/math]
For ε > 0, denote by N(T, dX; ε) the entropy number, i.e. the minimal number of (open) dX-balls of radius ε required to cover T. Then
- [math]\displaystyle{ \mathbf{E} \left[ \sup_{t \in T} X_{t} \right] \leq 24 \int_0^{+\infty} \sqrt{\log N(T, d_{X}; \varepsilon)} \, \mathrm{d} \varepsilon. }[/math]
Furthermore, if the entropy integral on the right-hand side converges, then X has a version with almost all sample path bounded and (uniformly) continuous on (T, dX).
References
- ↑ Dudley, Richard (2016). "V. N. Sudakov's work on expected suprema of Gaussian processes". High Dimensional Probability. VII. pp. 37–43. https://link.springer.com/chapter/10.1007/978-3-319-40519-3_2.
- Dudley, Richard M. (1967). "The sizes of compact subsets of Hilbert space and continuity of Gaussian processes". Journal of Functional Analysis 1 (3): 290–330. doi:10.1016/0022-1236(67)90017-1.
- Ledoux, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. (See chapter 11)
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