Editing Legendre transformation (section)

===Probability theory===
In [[large deviations theory]], the ''rate function'' is defined as the Legendre transformation of the logarithm of the [[moment generating function]] of a random variable. An important application of the rate function is in the calculation of tail probabilities of sums of [[Independent and identically distributed random variables|i.i.d. random variables]], in particular in [[Cramér's theorem (large deviations)|Cramér's theorem]].

If <math>X_n</math> are i.i.d. random variables, let <math>S_n=X_1+\cdots+X_n</math> be the associated [[random walk]] and <math>M(\xi)</math> the moment generating function of <math>X_1</math>.  For <math>\xi\in\mathbb R</math>, <math>E[e^{\xi S_n}] = M(\xi)^n</math>.  Hence, by [[Markov's inequality]], one has for <math>\xi\ge 0</math> and <math>a\in\mathbb R</math>
<math display="block">P(S_n/n > a) \le e^{-n\xi a}M(\xi)^n=\exp[-n(\xi a - \Lambda(\xi))]</math>
where <math>\Lambda(\xi)=\log M(\xi)</math>.  Since the left-hand side is independent of <math>\xi</math>, we may take the infimum of the right-hand side, which leads one to consider the supremum of <math>\xi a - \Lambda(\xi)</math>, i.e., the Legendre transform of <math>\Lambda</math>, evaluated at <math>x=a</math>.