In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, Raikov’s theorem, named after Dmitry Raikov, states that if the sum of two independent random variables X and Y has a Poisson distribution

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

, then both X and Y themselves must have the Poisson distribution. It says the same thing about the Poisson distribution that Cramér's theorem

Cramér's theorem

In mathematical statistics, Cramér's theorem is one of several theorems of Harald Cramér, a Swedish statistician and probabilist.- Normal random variables :...

 says about the normal distribution. It can readily be shown by mathematical induction

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

that the same is true of the sum of more than two independent random variables.