Editing Normal distribution (section)

=== Infinite divisibility and Cramér's theorem ===
For any positive integer {{mvar|n}}, any normal distribution with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math> is the distribution of the sum of {{mvar|n}} independent normal deviates, each with mean <math display=inline>\frac{\mu}{n}</math> and variance <math display=inline>\frac{\sigma^2}{n}</math>. This property is called [[infinite divisibility (probability)|infinite divisibility]].<ref>{{harvtxt |Patel |Read |1996 |loc=[2.3.6] }}</ref>

Conversely, if <math display=inline>X_1</math> and <math display=inline>X_2</math> are independent random variables and their sum <math display=inline>X_1+X_2</math> has a normal distribution, then both <math display=inline>X_1</math> and <math display=inline>X_2</math> must be normal deviates.<ref>{{harvtxt |Galambos |Simonelli |2004 |loc=Theorem&nbsp;3.5 }}</ref>

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.<ref name="Bryc 1995 35">{{harvtxt |Bryc |1995 |p=35 }}</ref>