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Isotropic probability measures in infinite-dimensional spaces.

¹Institute of Geophysics and Planetary Physics, A-025, University of California-San Diego, La Jolla, CA 92093.

Proceedings of the National Academy of Sciences of the United States of America

|December 1, 1987

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Every isotropic probability measure on real sequences is a combination of a point mass at zero and a specific type of measure. These measures are uniquely defined by their finite-dimensional distributions, offering a new approach to inverse problems.

Area of Science:

Probability Theory
Stochastic Processes
Functional Analysis

Background:

Isotropic probability measures on infinite-dimensional spaces are fundamental in probability theory.
Understanding their properties is crucial for modeling complex systems and statistical inference.
Existing methods for linear inverse problems face challenges with certain types of prior information.

Purpose of the Study:

To characterize all isotropic probability measures on the space of real sequences R(infinity).
To establish a unique relationship between these measures and their finite-dimensional marginal distributions.
To explore the implications for linear inverse problems and Bayesian inference techniques.

Main Methods:

Decomposition of isotropic measures into a convex combination involving a measure concentrated at zero and a set I(0)(R(infinity)).

Related Experiment Videos

Characterization of measures in I(0)(R(infinity)) through their finite-dimensional marginal distributions and density functions.

Utilizing the property of complete monotonicity of density functions and Laplace transforms for establishing bijections.

Main Results:

Every isotropic probability measure on R(infinity) is a convex combination of the zero measure and a measure in I(0)(R(infinity)).
Measures in I(0)(R(infinity)) are uniquely determined by their finite-dimensional marginal distributions, which possess completely monotone density functions.
A bijection is established between I(0)(R(infinity)) and the set of probability measures on [0, infinity) via a Laplace transform relationship.
Isotropic measures in I(0)(R(infinity)) assign zero probability to the space of square-summable sequences.

Conclusions:

The study provides a comprehensive characterization of isotropic probability measures on R(infinity).
The established bijection offers a novel way to represent and analyze these measures.
Linear inverse problems with prior information solely based on the inequality Sigma x(i)^2 <= 1 are unsuitable for stochastic inversion and Bayesian inference.