Cramér’s decomposition theorem for a normal distribution is a result of probability theory. It is well known that, given independentnormally distributedrandom 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,[1] has been proved by Harald Cramér.[2] 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).[3]
The precise statement of the theoremedit
Let a random variable ξ be normally distributed and admit 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.
April 08, 2024
cramér, decomposition, theorem, cramér, decomposition, theorem, normal, distribution, result, probability, theory, well, known, that, given, independent, normally, distributed, random, variables, their, normally, distributed, well, turns, that, converse, also,. Cramer s decomposition theorem for a normal distribution is a result of probability theory It is well known that given independent normally distributed random variables 31 32 their sum is normally distributed as well It turns out that the converse is also true The latter result initially announced by Paul Levy 1 has been proved by Harald Cramer 2 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 3 The precise statement of the theorem editLet a random variable 3 be normally distributed and admit a decomposition as a sum 3 31 32 of two independent random variables Then the summands 31 and 32 are normally distributed as well A proof of Cramer s decomposition theorem uses the theory of entire functions See also editRaikov s theorem Similar result for Poisson distribution References edit Levy Paul 1935 Proprietes asymptotiques des sommes de variables aleatoires independantes ou enchainees J Math Pures Appl 14 347 402 Cramer Harald 1936 Uber 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 Retrieved from https en wikipedia org w index php title Cramer 27s decomposition theorem amp oldid 1152609196, wikipedia, wiki, book, books, library,
