information:filtering, stylistic:style_transfer
Infinite divisibility and Cramér's theorem

For any positive integer n {\textstyle {\text{n}}}, any normal distribution with mean μ {\textstyle \mu } and variance σ 2 {\textstyle \sigma ^{2}} is the distribution of the sum of n {\textstyle {\text{n}}} independent normal deviates, each with mean μ n {\textstyle {\frac {\mu }{n}}} and variance σ 2 n {\textstyle {\frac {\sigma ^{2}}{n}}}. This property is called infinite divisibility.

Conversely, if X 1 {\textstyle X_{1}} and X 2 {\textstyle X_{2}} are independent random variables and their sum X 1 + X 2 {\textstyle X_{1}+X_{2}} has a normal distribution, then both X 1 {\textstyle X_{1}} and X 2 {\textstyle X_{2}} must be normal deviates.

This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.