An unknown signal θ being received in the presence of a Markovian symmetric Cauchy random noise whose probability law is known, we examine the problem of identifying θ with any given signal θ_o. We use two quite different methods to test the hypothesis “θ = θ_o”against an alternative hypothesis as large as possible. Several situations are presented in which the problem under consideration can, or cannot, be solved. In the first case, consistent tests are given.