Raikov’s theorem, named for Russian mathematician
Dmitrii Abramovich Raikov
Dmitrii Abramovich Raikov (Russian: , born 11 November 1905 in Odessa; died 1980 in Moscow) was a Russian mathematician who studied functional analysis.
Raikov studied in Odessa and Moscow, graduating in 1929. He was secretary of the Komsomol a ...
, is a result in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. It is well known that if each of two
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
random variables ξ
1 and ξ
2 has a
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 or space if these events occur with a known co ...
, then their sum ξ=ξ
1+ξ
2 has a Poisson distribution as well. It turns out that the converse is also valid.
Statement of the theorem
Suppose that a random variable ξ has Poisson's distribution and admits a decomposition as a sum ξ=ξ
1+ξ
2 of two independent random variables. Then the distribution of each summand is a shifted Poisson's distribution.
Comment
Raikov's theorem is similar to
Cramér’s decomposition theorem. The latter result claims that if a sum of two independent random variables has normal distribution, then each summand is normally distributed as well. It was also proved by
Yu.V.Linnik that a convolution of normal distribution and Poisson's distribution possesses a similar property ().
An extension to locally compact Abelian groups
Let
be a locally compact Abelian group. Denote by
the convolution semigroup of probability distributions on
, and by
the degenerate distribution concentrated at
. Let
.
The Poisson distribution generated by the measure
is defined as a shifted distribution of the form
One has the following
Raikov's theorem on locally compact Abelian groups
Let
be the Poisson distribution generated by the measure
. Suppose that
, with
. If
is either an infinite order element, or has order 2, then
is also a Poisson's distribution. In the case of
being an element of finite order
,
can fail to be a Poisson's distribution.
References
{{reflist
Characterization of probability distributions
Probability theorems
Theorems in statistics