Cramér's theorem (large deviations)

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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.


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 :


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 .


  • Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 508. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
  • Hazewinkel, Michiel, ed. (2001) [1994], "Cramér theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4