Théorème de Cramer (grandes déviations)

Théorème de Cramer (grandes déviations) 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: {style d'affichage Lambda (t)=log operatorname {E} [exp(tX_{1})].} Laisser {style d'affichage X_{1},X_{2},des points } be a sequence of iid real random variables with finite logarithmic moment generating function, par exemple. {style d'affichage Lambda (t)nom de l'opérateur {E} [X_{1}].} In the terminology of the theory of large deviations the result can be reformulated as follows: Si {style d'affichage X_{1},X_{2},des points } is a series of iid random variables, then the distributions {style d'affichage à gauche({mathématique {L}}({tfrac {1}{n}}somme _{je=1}^{n}X_{je})droit)_{nin mathbb {N} }} satisfy a large deviation principle with rate function {displaystyle Lambda ^{*}} .

References Klenke, Achim (2008). Probability Theory. Berlin: Springer. pp. 508. est ce que je:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. "Cramér theorem", Encyclopédie des mathématiques, Presse EMS, 2001 [1994] Catégories: Large deviations theoryProbability theorems