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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 adetailed checking of the assumptions of [Theorem~2.2,HL13] for the jointdistribution 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 oftransition matrices is given in Section~6 (see [Remark 2.4,HL13]). The basicmaterial 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 theproperties involved in this uniform version.

Additional material on local limit theorem for finite Additive Markov Processes [HL13]