Owhadi, Houman and Scovel, Clint (2015) Conditioning Gaussian measure on Hilbert space. . (Submitted) https://resolver.caltech.edu/CaltechAUTHORS:20160224-065740350
![]() |
PDF
- Submitted Version
See Usage Policy. 227Kb |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20160224-065740350
Abstract
For a Gaussian measure on a separable Hilbert space with covariance operator C, we show that the family of conditional measures associated with conditioning on a closed subspace S^⊥ are Gaussian with covariance operator the short S(C) of the operator C to S. We provide two proofs. The first uses the theory of Gaussian Hilbert spaces and a characterization of the shorted operator by Andersen and Trapp. The second uses recent developments by Corach, Maestripieri and Stojanoff on the relationship between the shorted operator and C-symmetric oblique projections onto S^⊥. To obtain the assertion when such projections do not exist, we develop an approximation result for the shorted operator by showing, for any positive operator A, how to construct a sequence of approximating operators A^n which possess A^n- symmetric oblique projections onto S^⊥ such that the sequence of shorted operators S(A^n) converges to S(A) in the weak operator topology. This result combined with the martingale convergence of random variables associated with the corresponding approximations C^n establishes the main assertion in general. Moreover, it in turn strengthens the approximation theorem for shorted operator when the operator is trace class; then the sequence of shorted operators S(A^n) converges to S(A) in trace norm.
Item Type: | Report or Paper (Discussion Paper) | ||||||
---|---|---|---|---|---|---|---|
Related URLs: |
| ||||||
ORCID: |
| ||||||
Additional Information: | (Submitted on 13 Jun 2015 (v1), last revised 1 Sep 2015 (this version, v2)). September 2, 2015. The authors gratefully acknowledge this work supported by the Air Force Office of Scientific Research under Award Number FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators). | ||||||
Funders: |
| ||||||
Record Number: | CaltechAUTHORS:20160224-065740350 | ||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20160224-065740350 | ||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||
ID Code: | 64709 | ||||||
Collection: | CaltechAUTHORS | ||||||
Deposited By: | Ruth Sustaita | ||||||
Deposited On: | 24 Feb 2016 18:18 | ||||||
Last Modified: | 03 Oct 2019 09:40 |
Repository Staff Only: item control page