Cramér's decomposition theorem

{{Short description|Theorem in probability theory}}

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 ξ₁, ξ₂, their sum is normally distributed as well. It turns out that the converse is also true. The latter result, initially announced by Paul Lévy,{{Cite journal|last=Lévy|first=Paul|title=Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées.|journal=J. Math. Pures Appl.|volume=14 |year=1935 |pages=347–402}} has been proved by Harald Cramér.{{Cite journal|last=Cramer|first=Harald|date=1936|title=Über eine Eigenschaft der normalen Verteilungsfunktion|journal=Mathematische Zeitschrift|volume=41 |issue=1 |pages=405–414|doi=10.1007/BF01180430}} This became a starting point for a new subfield in probability theory, decomposition theory for random variables as sums of independent variables (also known as arithmetic of probabilistic distributions).{{Cite book|title=Decomposition of random variables and vectors.|author=Linnik, Yu. V.|author-link=Yuri Linnik|author2=Ostrovskii, I. V.|publisher=Translations of Mathematical Monographs, 48. American Mathematical Society|year=1977|location=Providence, R. I.}}