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How many Laplace transforms of probability measures are there?
Fuchang Gao1, Wenbo V Li, Jon A Wellner
1Department of mathematics, University of Idaho, fuchang@uidaho.edu.
Proceedings of the American Mathematical Society. American Mathematical Society
|September 28, 2011
Summary
Researchers derived a metric entropy bound for Laplace transforms using small deviation probabilities of Gaussian processes. These findings offer new insights into the behavior of these specific smooth Gaussian processes.
Area of Science:
- Probability Theory
- Information Theory
- Stochastic Processes
Background:
- Laplace transforms are fundamental in analyzing probability measures.
- Small deviation probabilities provide insights into the tail behavior of random variables.
- Gaussian processes are widely used models in various scientific fields.
Purpose of the Study:
- To establish a bracketing metric entropy bound for Laplace transforms of probability measures on [0, ∞).
- To explore the connection between metric entropy and small deviation probabilities of smooth Gaussian processes.
Main Methods:
- Utilizing the relationship between bracketing metric entropy and small deviation probabilities.
- Analyzing a specific class of smooth Gaussian processes.
Main Results:
- A novel bracketing metric entropy bound was derived for the specified class of Laplace transforms.
- The study highlights the utility of small deviation analysis in bounding entropy.
Conclusions:
- The derived bound provides a valuable tool for understanding Laplace transforms of probability measures.
- The results concerning the smooth Gaussian process may have independent applications.
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