Logarithmic Potential Theory and Large Deviation

Large deviation principle in logarithmic potential theory

Large Deviation Theory

If we draw a random variable n times from Q, the probability distribution of the sum of the random variables is given by Q. This is the convolution of Q with itself n times. As n → ∞, Q tends to a normal distribution by the central limit theorem. This is shown in Figure 1. The top line is a computed normal distribution with the same mean as Q. However, as shown in Figure 3, when plotted on a lo...

Large deviation theory and applications

Large deviation theory deals with the decay of the probability of increasingly unlikely events. It is one of the key techniques of modern probability, a role which is emphasised by the recent award of the Abel prize to S.R.S. Varadhan, one of the pioneers of the subject. The subject is intimately related to combinatorial theory and the calculus of variations. Applications of large deviation the...

Large deviation theory and extreme waves

The mathematical tools of large deviation theory for rare events are illustrated with some simple examples. These include discrete and continuous Gaussian processes, importance sampling, and evolution equations of the Langevin type. Some of these methods have been used in the study of rogue surface waves but it seems that large deviation theory could have much wider application in geophysical p...

Large Deviation Theory for Stochastic Diierence Equations

The probability density for the solution yn of a stochastic di erence equation is considered. Following Knessl, Matkowsky, Schuss, and Tier [1] it is shown to satisfy a master equation, which is solved asymptotically for large values of the index n. The method is illustrated by deriving the large deviation results for a sum of independent identically distributed random variables and for the joi...