Cramér’s decomposition theorem
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, has been proved by Harald Cramér. 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).
The precise statement of the theorem
Let a random variable ξ be normally distributed and admits a decomposition as a sum ξ=ξ1+ξ2 of two independent random variables. Then the summands ξ1 and ξ2 are normally distributed as well.
A proof of Cramér's decomposition theorem uses the theory of entire functions.
- Lévy, Paul (1935). "Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées". J. Math. Pures Appl. 14: 347–402.
- Cramer, Harald (1936). "Über eine Eigenschaft der normalen Verteilungsfunktion". Mathematische Zeitschrift. 41 (1): 405–414. doi:10.1007/BF01180430.
- Linnik, Yu. V.; Ostrovskii, I. V. (1977). Decomposition of random variables and vectors. Providence, R. I.: Translations of Mathematical Monographs, 48. American Mathematical Society.