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 proved in a landmark 1967 paper of Richard M. Dudley; Dudley himself credited Volker Strassen for making the connection between entropy and regularity.
Statement of the theorem
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} ]}. \,](../I/m/104400ff731b3cd5858614dc79bfa426.png)
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.](../I/m/4c268b5b429baac36fa02f6f9fe71547.png)
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).
Non-uniqueness
The solution is highly non-unique. There are infinitely many representation of the 0 function, and any of these can be added to a representation to obtain another representation.[1]
References
- ↑ Steele, J. Michael. Stochastic calculus and financial applications. Vol. 45. Springer, 2001.
- Dudley, Richard M. (1967). "The sizes of compact subsets of Hilbert space and continuity of Gaussian processes". J. Functional Analysis 1: 290–330. doi:10.1016/0022-1236(67)90017-1. MR 0220340.
- Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015. (See chapter 11)