This paper proposes additional material to the main statements of [HL13] which are are recalled in Section 2. In particular an application of [Theorem 2.2,HL13] to Renewal Markov Processes is provided in Section 4 and a detailed checking of the assumptions of [Theorem 2.2,HL13] for the joint distribution of local times of a finite jump process is reported in Section 5. A uniform version of [Theorem 2.2,HL13] with respect to a compact set of transition matrices is given in Section 6 (see [Remark 2.4,HL13]). The basic material on the semigroup of Fourier matrices and the spectral approach used in [HL13] is recalled in Section 3 in order to obtain a good understanding of the properties involved in this uniform version.
