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Dudley's theorem

{{Short description|Concept in probability theory}}

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.{{Cite conference|url=https://link.springer.com/chapter/10.1007/978-3-319-40519-3_2|title=V. N. Sudakov's work on expected suprema of Gaussian processes|conference=High Dimensional Probability|volume=VII|editor-first1 =Christian|editor-last1=Houdré|editor-first2=David |editor-last2=Mason|editor-first3=Patricia|editor-last3 = Reynaud-Bouret|editor-link3 =Patricia Reynaud-Bouret|editor-first4=Jan|editor-last4=Jan Rosiński|first=Richard|last=Dudley|author-link = Richard M. Dudley|date=2016|pages=37–43|doi=10.1007/978-3-319-40519-3_2 }} 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

:d_{X}(s, t) = \sqrt{\mathbf{E} \big[ | X_{s} - X_{t} |^{2} ]}. \,

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

:\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.

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

{{reflist}}

  • {{cite journal

| doi = 10.1016/0022-1236(67)90017-1

| last = Dudley

| first = Richard M.

| authorlink = Richard M. Dudley

| title = The sizes of compact subsets of Hilbert space and continuity of Gaussian processes

| journal = Journal of Functional Analysis

| volume = 1

| year = 1967

| issue = 3

| pages = 290–330

| mr = 0220340

| doi-access =

}}

  • {{ cite book

| last1 = Ledoux

| first1 = Michel

| last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand

| title = Probability in Banach spaces

| publisher = Springer-Verlag

| location = Berlin

| year = 1991

| pages = xii+480

| isbn = 3-540-52013-9

| mr = 1102015

}} (See chapter 11)

Category:Entropy

Category:Theorems about stochastic processes