# LARGE DEVIATIONS THEORYAUTHORMd Ashif RezaMINIPROJECTMA202NAMEMD ASHIF REZAROLL170101023BRANCHCSEINTRODUCTIONIn probability theory the theory of

LARGE DEVIATIONS THEORYAUTHOR:Md Ashif RezaMINI_PROJECT:MA202NAME:MD ASHIF REZAROLL:170101023BRANCH:CSEINTRODUCTIONIn probability theory, the theory of large deviations concerns the asymptotic behaviour ofremote tails of sequences of probability distributions. While some basic ideas of the theorycan be traced to Laplace, the formalization started with insurance mathematics, namely ruintheory with Cram©r and Lundberg. A unified formalization of large deviation theory wasdeveloped in 1966, in a paper by Varadhan. Large deviations theory formalizes the heuristicideas of concentration of measures and widely generalizes the notion of convergence ofprobability measures.

Roughly speaking, large deviations theory concerns itself with the exponential decline of theprobability measures of certain kinds of extreme or tail events.INTRODUCTORY EXAMPLESAn elementary exampleConsider a sequence of independent tosses of a fair coin. The possible outcomes could beheads or tails. Let us denote the possible outcome of the i-th trial by , where we encode headas 1 and tail as 0. Now let denote the mean value after trials, namely? ? =1?€‘ ?=1? ? .Then lies between 0 and 1.

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From the law of large numbers (and also from our experience) weknow that as N grows, the distribution of converges to 0.5 = E [?] (the expectation value ofa single coin toss) almost surely.Moreover, by the central limit theorem, we know that ? ? is approximately normallydistributed for large . The central limit theorem can provide more detailed information aboutthe behavior of ? ? than the law of large numbers. For example, we can approximately finda tail probability of ? ? , ?(? ? > ?) that ? ? is greater than , for a fixed value of . However,the approximation by the CLT may not be accurate if is far from E [? ? ] unless is sufficientlylarge.

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