Exponential families of mixed Poisson distributions

If I = ( I 1 , … , I d ) is a random variable on [ 0 , ∞ ) d with distribution μ ( d λ 1 , … , d λ d ) , the mixed Poisson distribution MP ( μ ) on N d is the distribution of ( N 1 ( I 1 ) , … , N d ( I d ) ) where N 1 , … , N d are ordinary independent Poisson processes which are also independent of I . The paper proves that if F is a natural exponential family on [ 0 , ∞ ) d then MP ( F ) is also a natural exponential family if and only if a generating probability of F is the distribution of v 0 + v 1 Y 1 + ⋯ + v q Y q for some q ⩽ d , for some vectors v 0 , … , v q of [ 0 , ∞ ) d with disjoint supports and for independent standard real gamma random variables Y 1 , … , Y q .