Il teorema di Cramer (grandi deviazioni)

Il teorema di Cramer (grandi deviazioni) 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: {displaystyle Lambda (t)=log operatorname {e} [esp(tX_{1})].} Permettere {stile di visualizzazione X_{1},X_{2},punti } be a sequence of iid real random variables with finite logarithmic moment generating function, per esempio. {displaystyle Lambda (t)nome operatore {e} [X_{1}].} In the terminology of the theory of large deviations the result can be reformulated as follows: Se {stile di visualizzazione X_{1},X_{2},punti } is a series of iid random variables, then the distributions {stile di visualizzazione a sinistra({matematico {l}}({tfrac {1}{n}}somma _{io=1}^{n}X_{io})Giusto)_{nin mathbb {N} }} satisfy a large deviation principle with rate function {displaystyle Lambda ^{*}} .

References Klenke, Achim (2008). Probability Theory. Berlino: Springer. pp. 508. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6. "Cramér theorem", Enciclopedia della matematica, EMS Press, 2001 [1994] Categorie: Large deviations theoryProbability theorems