The Linnik class I 0 {\displaystyle I_{0}} is one of the central concepts in the arithmetic of probability distributions. Following Yuri Vladimirovich Linnik, denote by I 0 {\displaystyle I_{0}} the class of distributions that have no indecomposable divisors. Examples of distributions belonging to the class I 0 {\displaystyle I_{0}} include normal distributions (Cramér's decomposition theorem), Poisson distributions (Raikov's theorem), as well as their convolutions (Linnik's theorem).
Properties distributions of the Linnik class Let μ ∈ I 0 {\displaystyle \mu \in I_{0}} . Then, according to Khinchin's second theorem, the distribution μ {\displaystyle \mu } is infinitely divisible. Hence, by the Lévy formula, its characteristic function can be represented in the form
where β ∈ R {\displaystyle \beta \in \mathbb {R} } , σ ≥ 0 {\displaystyle \sigma \geq 0} , and F {\displaystyle F} is a Borel measure on R ∖ { 0 } {\displaystyle \mathbb {R} \setminus \{0\}} satisfying the condition
∫ R ∖ { 0 } x 2 1 + x 2 d F ( x ) < ∞ . {\displaystyle \int _{\mathbb {R} \backslash \{0\}}{x^{2} \over {1+x^{2}}}dF(x)<\infty .} The measure F {\displaystyle F} is called the Lévy spectral measure of the infinitely divisible distribution μ {\displaystyle \mu } . The central problem in the arithmetic of probability distributions is to determine conditions on σ {\displaystyle \sigma } and F {\displaystyle F} that are necessary and sufficient for an infinitely divisible distribution μ {\displaystyle \mu } to belong to the class I 0 {\displaystyle I_{0}} . Denote by L {\displaystyle {\mathfrak {L}}} the class of infinitely divisible distributions μ {\displaystyle \mu } having the property that the Lévy spectral measure F {\displaystyle F} in (1) is discrete and concentrated on a set of the form
… excerpt ends here. Continue reading the full article.
