Duffy, Ken R. and Williamson, Brendan D.
(2015)
Estimating large deviation rate functions.
Working Paper.
arXiv.
Abstract
Establishing a Large Deviation Principle (LDP) proves to be a powerful result for a vast
number of stochastic models in many application areas of probability theory. The key object
of an LDP is the large deviations rate function, from which probabilistic estimates of rare
events can be determined. In order make these results empirically applicable, it would be
necessary to estimate the rate function from observations. This is the question we address
in this article for the best known and most widely used LDP: Cramér’s theorem for random
walks.
We establish that even when only a narrow LDP holds for Cram´er’s Theorem, as occurs
for heavy-tailed increments, one gets a LDP for estimating the random walk’s rate function
in the space of convex lower-semicontinuous functions equipped with the Attouch-Wets
topology via empirical estimates of the moment generating function. This result may seem
surprising as it is saying that for Cramér’s theorem, one can quickly form non-parametric
estimates of the function that governs the likelihood of rare events.
| Item Type: |
Monograph
(Working Paper)
|
| Keywords: |
Large Deviation Principle; LDP; probability theory; large deviations rate function; Cramér’s theorem; |
| Academic Unit: |
Faculty of Science and Engineering > Research Institutes > Hamilton Institute |
| Item ID: |
6766 |
| Identification Number: |
arXiv:1511.02295 |
| Depositing User: |
Dr Ken Duffy
|
| Date Deposited: |
11 Jan 2016 17:00 |
| Publisher: |
arXiv |
| URI: |
|
| Use Licence: |
This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available
here |
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