  The aim of this paper is to propose a methodology for testing general
hypothesis in a Markovian setting with random sampling. A discrete Markov chain
X is observed at random time intervals $\tau$ k, assumed to be iid with unknown
distribution $\mu$. Two test procedures are investigated. The first one is
devoted to testing if the transition matrix P of the Markov chain X satisfies
specific affine constraints, covering a wide range of situations such as
symmetry or sparsity. The second procedure is a goodness-of-fit test on the
distribution $\mu$, which reveals to be consistent under mild assumptions even
though the time gaps are not observed. The theoretical results are supported by
a Monte Carlo simulation study to show the performance and robustness of the
proposed methodologies on specific numerical examples.
