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In this work, we interest to the estimation of the steady-state probabilities of the two processes (the hidden Markov chain and the emission process) composing a standard Hidden Markov Model. Since the estimation of steady-state Markov chain is well known, more attention will be given to the estimation of the steady-state of the emission process. Among different methods, we will focus more on the technique based on the regenerative notion, often used in steady-state Markov chain simulation. Numerical Monte Carlo simulations are carried to show the usefulness of our proposal and to appreciate the quality of different proposed estimators. [ABSTRACT FROM AUTHOR]