  We introduce a new characterization of the Cauchy distribution and propose a
class of goodness-of-fit tests to the Cauchy family. The limit distribution is
derived in a Hilbert space framework under the null hypothesis and under fixed
alternatives. The new tests are consistent against a large class of
alternatives. A comparative Monte Carlo simulation study shows that the test is
competitive to the state of the art procedures, and we apply the tests to
log-returns of cryptocurrencies.
