Why intervals? A simple limit theorem that is similar to limit theorems from statistics

What is the set of p~, ib le s'aines ~d" a n~asnremem error? In the majority of practical applications, an error is caused m~ 1~, a single cause; it is camaL.d by. a large mtmber of inclependent causes, each of which adds a small O.nlxment to the t(aal error. This fact is widely used in v, atistics: Namely, since it is known that the distlilxltitm ~d" the snm o( many independent stnali random variables is close to one of the s/~called intinitely divisible arars (a class that includes the wen-known Gaussian distriixlthm), we can safely a~s,nne that the distrii~,tlon of the tmal error is infinitely divisible. This asa~mlxion is reed in the maj.ri ty of the statislicai appGcations. In this paper, we prove a similar resuk for the set (g possible vah~es of an error. Namely. if an error equals the stun ,d" many an.all independent ctnnponents, then its se~ at" lxmiide vahg.s is close to an interval; the smaller the omqxments, the ckmer this set m an interval. This res~ih provides ~me im~re jnslifmation for ~ n g intervals in data pn,cessing.