## Abstract

Let μ be a positive Radon measure on ℝ having a Laplace transform L_{μ} and F = F(μ) be a natural exponential family (NEF) generated by μ. A positive Radon measure μ_{λ} with λ > 0 and its associated NEF F_{λ} = F(μ _{λ}) are called the λth convolution powers of μ and F, respectively, if L_{μλ} = L_{μ} ^{λ}. Let f_{α,β}(F) be the image of an NEF F under the affine transformation f_{α,β}: x → αx + β. If for a given NEF F, there exists a triple (α, β, λ) in ℝ^{3} such that f_{α,β}(F) = F _{λ}, we call F a reproducible NEF in the broad sense. In other words, an NEF F is reproducible in the broad sense if a convolution power of F equals an affine transformation of F. Clearly, for λ = 1, F is reproducible in the broad sense if there exists an affinity under which F is invariant. In this paper we obtain a complete classification of the class of reproducible NEF's in the broad sense. We show that this class is composed of infinitely divisible NEF's and that it contains the class of NEF's having exponential and power variance functions as well as NEF's constituting discrete versions of the latter NEF's. We also provide a characterization of the reproducible NEF's in the broad sense in terms of their associated exponential dispersion models.

Original language | English |
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Pages (from-to) | 175-196 |

Number of pages | 22 |

Journal | Journal of Theoretical Probability |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2003 |

## Keywords

- Exponential dispersion models
- Infinitely divisible distributions
- Natural exponential families
- Reproducibility
- Variance functions

## ASJC Scopus subject areas

- Statistics and Probability
- Mathematics (all)
- Statistics, Probability and Uncertainty