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Cramér's theorem
In mathematical statistics, Cramér's theorem (or Cramér’s decomposition theorem) is one of several theorems of Harald Cramér, a Swedish statistician and probabilist.
== Normal random variables ==
Cramér's theorem is the result that if ''X'' and ''Y'' are independent real-valued random variables whose sum ''X'' + ''Y'' is a normal random variable, then both ''X'' and ''Y'' must be normal as well. By induction, if any finite sum of independent real-valued random variables is normal, then the summands must all be normal.
Thus, while the normal distribution is infinitely divisible, it can ''only'' be decomposed into normal distributions (if the summands are independent).
Contrast with the central limit theorem, which states that the average of independent identically distributed random variables with finite mean and variance is ''asymptotically'' normal. Cramér's theorem shows that a finite average is not normal, unless the original variables were normal.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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