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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.
Cramer's rule is used in the Ricci calculus in various calculations involving the Christoffel symbols of the first and second kind. [14] In particular, Cramer's rule can be used to prove that the divergence operator on a Riemannian manifold is invariant with respect to change of coordinates. We give a direct proof, suppressing the role of the ...
The central limit theorem can provide more detailed information about the behavior of than the law of large numbers. For example, we can approximately find a tail probability of M N {\displaystyle M_{N}} – the probability that M N {\displaystyle M_{N}} is greater than some value x {\displaystyle x} – for a fixed value of N {\displaystyle N} .
The Cramér–Rao bound is stated in this section for several increasingly general cases, beginning with the case in which the parameter is a scalar and its estimator is unbiased.
In mathematics, the Cramér–Wold theorem [1] [2] or the Cramér–Wold device [3] [4] is a theorem in measure theory and which states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. [5] [6] [7] It is used as a method for proving joint convergence results.
Cramér's theorem may refer to Cramér’s decomposition theorem, a statement about the sum of normal distributed random variable; Cramér's theorem (large deviations), a fundamental result in the theory of large deviations; Cramer's theorem (algebraic curves), a result regarding the necessary number of points to determine a curve
The proof follows a similar approach to the other Chernoff bounds, but applying Hoeffding's lemma to bound the moment generating functions (see Hoeffding's inequality). Hoeffding's inequality. Suppose X 1, ..., X n are independent random variables taking values in [a,b]. Let X denote their sum and let μ = E[X] denote the sum's expected value.
Cramér’s decomposition theorem for a normal distribution is a result of probability theory. It is well known that, given independent normally distributed random variables ξ 1 , ξ 2 , their sum is normally distributed as well.