“Cramér's theorem (large deviations)”的意思、由来-开放百科全书

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Cramér's theorem (large deviations)

释义

Statement
References

Cramér's theorem is a fundamental result in the theory of large deviations, a subdiscipline of probability theory. It determines the rate function of a series of iid random variables.

A weak version of this result was first shown by Harald Cramér in 1938.

Statement

The logarithmic moment generating function (which is the cumulant-generating function) of a random variable is defined as:

Let be a sequence of iid real random variables with finite logarithmic moment generating function, e.g. for all .

Then the Legendre transform of :

satisfies,

for all

In the terminology of the theory of large deviations the result can be reformulated as follows:

If is a series of iid random variables, then the distributions satisfy a large deviation principle with rate function .

References

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6|pages=508 }}
{{springer|title=Cramér theorem|id=p/c027000}}

2 : Large deviations theory|Probability theorems

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