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The paper contains a new and elementary proof of the fact that if α ∈ (0,1] then every scale mixture of a symmetric α-stable probability measure is infinitely divisible. This property is known to be a consequence of Kelker's result for the Cauchy distribution and some nontrivial properties of completely monotone functions. It is known that this property does not hold for α = 2. The problem discussed in the paper is still open for α ∈ (1,2).
Grażyna Mazurkiewicz. "On the infinite divisibility of scale mixtures of symmetric α-stable distributions, α ∈ (0,1]." Banach Center Publications 90.1 (2010): 79-82. <http://eudml.org/doc/281601>.
@article{GrażynaMazurkiewicz2010, abstract = {The paper contains a new and elementary proof of the fact that if α ∈ (0,1] then every scale mixture of a symmetric α-stable probability measure is infinitely divisible. This property is known to be a consequence of Kelker's result for the Cauchy distribution and some nontrivial properties of completely monotone functions. It is known that this property does not hold for α = 2. The problem discussed in the paper is still open for α ∈ (1,2).}, author = {Grażyna Mazurkiewicz}, journal = {Banach Center Publications}, keywords = {stable distribution; scale mixture; variance mixture}, language = {eng}, number = {1}, pages = {79-82}, title = {On the infinite divisibility of scale mixtures of symmetric α-stable distributions, α ∈ (0,1]}, url = {http://eudml.org/doc/281601}, volume = {90}, year = {2010}, }
TY - JOUR AU - Grażyna Mazurkiewicz TI - On the infinite divisibility of scale mixtures of symmetric α-stable distributions, α ∈ (0,1] JO - Banach Center Publications PY - 2010 VL - 90 IS - 1 SP - 79 EP - 82 AB - The paper contains a new and elementary proof of the fact that if α ∈ (0,1] then every scale mixture of a symmetric α-stable probability measure is infinitely divisible. This property is known to be a consequence of Kelker's result for the Cauchy distribution and some nontrivial properties of completely monotone functions. It is known that this property does not hold for α = 2. The problem discussed in the paper is still open for α ∈ (1,2). LA - eng KW - stable distribution; scale mixture; variance mixture UR - http://eudml.org/doc/281601 ER -