. Finally, Inequalities (30)-(33) lead to the desired result

M. Amewou-atisso, S. Ghosal, J. K. Ghosh, and R. V. Ramamoorthi, Posterior consistency for semiparametric regression problems, Bernoulli, vol.9, pp.291-312, 2003.
DOI : 10.3150/bj/1068128979
URL : https://doi.org/10.3150/bj/1068128979

J. Arbel, G. Gayraud, and J. Rousseau, Bayesian optimal adaptive estimation using a sieve prior, Scand. J. Stat, vol.40, pp.549-570, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01203280

A. K. Banerjee and G. K. Bhattacharyya, Testing hypotheses in a two-state semi-Markov process, Sankhya: The Indian Journal of Statistics, Series A, vol.38, issue.4, pp.340-356, 1976.

V. S. Barbu and N. Limnios, Semi-Markov chains and hidden semiMarkov models toward applications. Their use in reliability and DNA analysis, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00530330

B. R. Bath and S. K. Deshpande, Testing for Markov process VS semi-Markov process, Commun. Statist.-Theor. Meth, vol.15, pp.2375-2382, 1986.

L. Birgé, Robust testing for independent non identically distributed variables and Markov chains. Specifying Statistical Models, pp.134-162, 1983.

L. Birgé, Approximation dans les espaces métriques et théorie de l'estimation, Z. Wahrsch. Verw. Gebiete, vol.65, pp.181-237, 1983.

L. Birgé, Robust tests for model selection, IMS Collections, From Probability to Statistics and Back: High-Dimensional Models and Processes -A Festschrift in Honor of Jon A. Wellner 9, pp.47-64, 2013.

P. Bulla and P. Muliere, Bayesian nonparametric estimation for reinforced Markov renewal processes, Stat. Inference Stoch. Process, vol.10, issue.3, pp.283-303, 2007.
DOI : 10.1007/s11203-006-9000-x

I. Chang, Y. Chuang, and C. A. Hsiung, A class of nonparametric K-sample tests for semi-Markov counting processes, Statistica Sinica, vol.9, pp.211-227, 1999.

I. Chang, Y. Chuang, and C. A. Hsiung, Goodness-of-fit tests for semi-Markov and Markov survival models with one intermediate state. Scand, J. Stat, vol.28, issue.3, pp.505-525, 2001.

N. Choudhuri, S. Ghosal, and A. Roy, Bayesian estimation of the spectral density of a time series, J. Amer. Statist. Assoc, vol.99, pp.1050-1059, 2004.

E. Ç-inlar, Markov renewal theory, Adv Appl Probab, vol.1, pp.123-187, 1969.

T. Economou, T. C. Bailey, and Z. Kapelan, MCMC implementation for Bayesian hidden semi-Markov models with illustrative applications, Stat Comput, vol.24, issue.5, pp.739-752, 2014.
DOI : 10.1007/s11222-013-9399-z
URL : https://ore.exeter.ac.uk/repository/bitstream/10871/8326/5/MCMC%20implementation%20for%20Bayesian%20hidden%20semi-Markov%20models%20with%20an%20illustrative%20application%20to%20recurrent%20event%20data.pdf

I. Epifani, L. Ladelli, and A. Pievatolo, Bayesian estimation for a parametric Markov renewal model applied to seismic data, Electron J Stat, vol.8, issue.2, pp.2264-2295, 2014.
DOI : 10.1214/14-ejs952
URL : https://doi.org/10.1214/14-ejs952

E. Gassiat and J. Rousseau, About the posterior distribution in hidden Markov models with unknown number of states, Ann Stat, vol.35, issue.1, pp.192-223, 2013.

S. Ghosal, . Van-der, and A. Vaart, Convergence rates of posterior distributions for noniid observations, Ann Stat, vol.35, issue.1, pp.192-223, 2007.

S. Ghosal, . Van-der, and A. Vaart, Fundamentals of Nonparametric Bayesian Inference, 2017.
DOI : 10.1017/9781139029834

S. Ghosal and A. Roy, Posterior consistency of Gaussian process prior for nonparametric binary regression, Ann Stat, vol.34, issue.5, pp.2413-2429, 2006.
DOI : 10.1214/009053606000000795
URL : https://doi.org/10.1214/009053606000000795

S. Ghosal, J. K. Ghosh, and R. V. Ramamoorthi, Posterior consistency of Dirichlet mixtures in density estimation, Ann Stat, vol.27, issue.1, pp.143-158, 1999.

S. Ghosal, J. K. Ghosh, . Van-der, and A. Vaart, Convergence rates of posterior distributions, Ann Stat, vol.28, issue.2, pp.500-531, 2000.
DOI : 10.1214/aos/1016218228
URL : https://doi.org/10.1214/aos/1016218228

J. K. Ghosh and R. V. Ramamoorthi, Bayesian Nonparametrics, 2003.

V. S. Korolyuk and N. Limnios, Stochastic systems in merging phase space, 2005.

L. Cam and L. M. , Asymptotic Methods in Statistical Decision Theory, 1986.
DOI : 10.1007/978-1-4612-4946-7

L. Cam and L. M. , On local and global properties in the theory of asymptotic normality of experiments, Stochastic processes and related topics, M.Puri, 1975.

L. Cam and L. M. , Convergence of estimates under dimensionality restrictions, Ann Stat, vol.1, pp.38-53, 1973.

P. Lévy, Processus semi-markoviens, Proc. Int. Cong. Math. (Amsterdam, pp.416-426, 1954.

N. Limnios and G. Opri?an, Semi-Markov Processes and Reliability, 2001.

V. K. Malinovskii, Asymptotic optimality of criteria in the problem of testing hypotheses for a recurrent semi-Markov process, Journal of Soviet Mathematics, vol.59, issue.4, pp.955-959, 1992.

D. Pati, D. Dunsony, and S. Tokdary, Posterior consistency in conditional distribution estimation, J Multivariate Anal, vol.116, pp.456-472, 2013.

M. J. Phelan, Bayes estimation from a Markov renewal process, Ann Stat, vol.18, issue.2, pp.603-616, 1990.

R. Pyke, Markov renewal processes: definitions and preliminary properties, Ann. Math. Stat, vol.32, pp.1231-1242, 1961.
DOI : 10.1214/aoms/1177704863

R. Pyke, Markov renewal processes with finitely many states, Ann. Math. Stat, vol.32, pp.1243-1259, 1961.
DOI : 10.1214/aoms/1177704864

J. Rousseau, N. Chopin, and B. Liseo, Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process, Ann Stat, vol.40, issue.2, pp.964-995, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00504969

W. L. Smith, Regenerative stochastic processes, Proc. Roy. Soc. London, Ser. A, vol.232, pp.6-31, 1955.

X. Shen and L. Wasserman, Rates of convergence of posterior distributions, Ann Stat, vol.29, pp.687-714, 2001.

L. Takács, Some investigations concerning recurrent stochastic processes of a certain type, Magyar Tud. Akad. Mat. Kutato Int. Közl, vol.3, pp.115-128, 1954.

Y. Tang and S. Ghosal, Posterior consistency of Dirichlet mixtures for estimating a transition density, J. Statist. Plann. Inference, vol.137, pp.1711-1726, 2007.

W. Tsai, Rank tests for a class of semi-Markov models with censored matched pairs, Stat Probabil Lett, vol.3, issue.5, pp.281-286, 1985.

L. Wasserman, Asymptotic properties of nonparametric Bayesian procedures, Lecture Notes in Statistics, vol.133, pp.293-304, 1998.