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    Estimating large deviation rate functions


    Duffy, Ken R. and Williamson, Brendan D. (2015) Estimating large deviation rate functions. Working Paper. arXiv.

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    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:

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