On conistent hypothesis testing

We explore natural links between different types of consistency: consistency, uniform consistency and pointwise consistency. On the base of these results we provide both sufficient conditions and necessary conditions for existence of different types of consistent tests for the problems of hypothesis testing on a probability measure of independent sample, on a mean measure of Poisson process, on a solution of linear ill-posed problem in Gaussian noise, on a solution of deconvolution problem and for a signal detection in Gaussian white noise. In the last three cases we show that necessary conditions and sufficient conditions coincide.