Cramér's decomposition theorem

Cramér's decomposition theorem states that if a sum of independent random variables has a normal distribution, then every summand also has a normal distribution.

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Characteristic function: A function that encodes a probability distribution through the expected value of a complex exponential. The theorem's proof factors the sum's characteristic function into those of its independent summands.
Random variable: A measurable quantity whose value is determined by the outcome of a random experiment. The theorem concerns sums of random variables rather than arbitrary numerical decompositions.
Harald Cramér: A Swedish mathematician and statistician known for foundational work in probability and mathematical statistics. Cramér established the decomposition result that bears his name.
Lévy's decomposition theorem: A theorem characterizing the possible convolution decompositions of infinitely divisible distributions. It studies decompositions across infinitely divisible laws, rather than only Gaussian sums.
Independence (probability theory): A relationship in which knowing one random variable's value does not change the distribution of another. Independence makes the characteristic function of the sum a product.
Convolution (probability): The probability distribution of a sum of independent random variables, formed by convolving their distributions. A Gaussian sum means the summands' distributions convolve to a Gaussian distribution.
Paul Lévy: A French mathematician whose work shaped modern probability theory, including the study of stable distributions. Lévy's investigations of sums and limit laws helped frame decomposition questions.
Raikov's theorem: A theorem stating that independent summands of a Poisson-distributed sum must themselves be Poisson-distributed under its hypotheses. It is a discrete analogue of Gaussian decomposition rigidity.
Gaussian distribution: A continuous probability distribution with a bell-shaped density determined by its mean and variance. The theorem identifies Gaussian summands from a Gaussian sum.
Normal distribution: A probability distribution whose density has a symmetric bell shape specified by a mean and variance. The theorem's conclusion and hypothesis both depend on this distribution family.