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Details of Grant 

EPSRC Reference: EP/W010607/1
Title: Poisson 2021
Principal Investigator: Novak, Dr S
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Faculty of Science & Technology
Organisation: Middlesex University
Scheme: Standard Research - NR1
Starts: 01 June 2022 Ends: 28 February 2023 Value (£): 45,726
EPSRC Research Topic Classifications:
Mathematical Analysis Statistics & Appl. Probability
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
02 Aug 2021 EPSRC Mathematical Sciences Small Grants Panel August 2021 Announced
Summary on Grant Application Form
Probability Theory is a branch of science dealing with random events. It provides mathematical tools for evaluating chances, forecasting possible outcomes, and suggesting optimal choices in the presence of uncertainty.

The theory of sums of random variables lies at the heart of research in Probability Theory. It is truly fundamental when dealing with the aggregate effect of random events, e.g., when observing the combined effect of a large number of small or rare random contributions.

The work on the theory of sums of random variables started in the 17th century, and continued ever since. Among famous contributors are Bernoulli, de Moivre, Laplace, and other well-known mathematicians. Their results have shaped modern Probability Theory.

The actual distribution of a sum of random variables is typically complex, and one would prefer using a simpler and more tractable approximate distribution (e.g., normal, Poisson or compound Poisson). However, one can only substitute a complex actual distribution by an approximate one if there is a sharp estimate of the accuracy of approximation indicating the error would be "small".

As a result, a lot of work in Probability Theory has been devoted to evaluating the accuracy of approximation. In particular, work on the accuracy of normal approximation to the distributions of sums of random variables started in the late 19th century, and still continues.

However, in situations where one deals with rare events a natural approximating distribution is Poisson (or, more generally, compound Poisson).

The class of compound Poisson laws is so general that the class of all possible limit laws to the distributions of sums of asymptotically "small" random variables (the class of so-called infinitely divisible distributions) coincides with the class of weak limits of compound Poisson distributions.

Poisson and compound Poisson approximations naturally arise when one deals with the number of long head runs is discrete random sequences, the number of long match patterns in DNA sequences, aggregate claims to a (re)insurance company, the number of exceedances of high thresholds in extreme value theory, etc..

The approximation of a complex actual distribution by a Poisson and compound Poisson one is only justified if there is a sharp estimate of the accuracy of approximation indicating the error is "small". Hence the need of sharp estimates of the accuracy of approximation.

The proposed research will concentrate on establishing sharp estimates of the accuracy of Poisson and compound Poisson approximation to the distribution of a sum of random variables. In particular, we aim to address a long-standing open question concerning establishing an estimate of the accuracy of (compound) Poisson approximation with a correct (the best possible) constant at the leading term.
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Organisation Website: http://www.mdx.ac.uk