Large deviations

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

The theory of large deviations is concerned about the probabilities of rare events. Consider, for example, tossing a fair die n times. There are 6 possible outcomes per toss and a total of 6 n possible outcomes. What is the probability of the average of the throws being close to 1 + 2 + 3 + 4 + 5 + 6 2 = 3.5 ? This is a small deviation event for large n, since by the law of large numbers, the probability is close to 1 and, in average, each face of the die appears about n/6 times. What is the probability of the average of the throws being about 4, or the probability of getting each of faces 1 to 5 with about 1 percent of the throws and face 6 with about 95 percent of the throws? Both are rare or large deviation events with vanishing probabilities as n increases. Although the probabilities of such rare events can be computed precisely, given knowledge of the probability distribution, such exact calculations are usually complex and render little insights into the problem. It is often of interest to examine how fast the probability of a rare event decreases with increasing n. Large deviation techniques offer simple and insightful ways to compute the decaying rate of rare event probabilities, and have found numerous applications in communication and computer networks, sensing systems, computational biology, statistical mechanics, and risk analysis.

Original languageEnglish
Title of host publicationMathematical Foundations for Signal Processing, Communications, and Networking
Pages295-315
Number of pages21
ISBN (Electronic)9781439855140
DOIs
StatePublished - 1 Jan 2017

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