The Lévy-Khintchine Formula for Standard Measures

Abstract

In this section we study infinitely divisible experiments E = (X, A, (P_θ)P_{θ ∈ θ}) for a finite set θ which is always assumed to be θ = {1,...,k}, k ≧ 2. We then write S_k: = S _θ. Our first aim is to classify the distributions of the log-likelihood processes of infinitely divisible experiments. We show that E is infinitely divisible if and only if each distribution of the log-likelihood process is infi-nitely divisible on the semigroup ([ −∞, ∞)^k−1, +). The different processes are not easy to compare on [−∞, ∞)^k−1. Therefore we first consider the P _l -distributions (l = 1,...,k) of the statistics

$$ T~=~{{({{T}_{i}})}_{_{i=~1,...,k}}},~{{T}_{i}}~=~\frac{d{{p}_{i}}}{d~\sum\limits_{j=1}^{k}{{{P}_{j}}}} $$

which leads to the theory of standard measures on the simplex S_k. The standard measures of infinitely divisible experiments are described.