Көйлексіздіктің қалыпты функциясы және оның қолданылуы

Probability function
In mathematics — specifically, in large deviations theory — a rate function is a function used to quantify the probabilities of rare events. Such functions are used to formulate large deviation principles. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities. A rate function is also called a Cramér function, after the Swedish probabilist Harald Cramér.

A natural question to ask, given the somewhat abstract setting of the general framework above, is whether the rate function is unique. This turns out to be the case: given a sequence of probability measures (μδ)δ>0 on X satisfying the large deviation principle for two rate functions I and J, it follows that I(x) = J(x) for all x ∈ X.

It is possible to convert a weak large deviation principle into a strong one if the measures converge sufficiently quickly. If the upper bound holds for compact sets F and the sequence of measures (μδ)δ>0 is exponentially tight, then the upper bound also holds for closed sets F. In other words, exponential tightness enables one to convert a weak large deviation principle into a strong one.

The notion of a rate function emerged in the 1930s with the Swedish mathematician Harald Cramér's study of a sequence of i. i. d. random variables (Zi)i∈. Namely, among some considerations of scaling, Cramér studied the behavior of the distribution of the average as n→∞. He found that the tails of the distribution of Xn decay exponentially as e−nλ(x) where the factor λ(x) in the exponent is the Legendre–Fenchel transform (a. k. a. the convex conjugate) of the cumulant generating function For this reason this particular function λ(x) is sometimes called the Cramér function. The rate function defined above in this article is a broad generalization of this notion of Cramér's, defined more abstractly on a probability space, rather than the state space of a random variable.