EPSRC Reference: 
EP/W010607/1 
Title: 
Poisson 2021 
Principal Investigator: 
Novak, Dr S 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Faculty of Science & Technology 
Organisation: 
Middlesex University 
Scheme: 
Standard Research  NR1 
Starts: 
01 June 2022 
Ends: 
31 May 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: 

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 wellknown 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 socalled 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 longstanding 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 