Cramér's theorem (large deviations)

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

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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, i.e.   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

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  • Klenke, Achim (2008). Probability Theory. Berlin: Springer. pp. 508. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  • "Cramér theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]