On consistency of nonparametric tests

For $\chi^2-$tests with increasing number of cells, Cramer-von Mises tests, tests generated ${L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients, we find necessary and sufficient conditions of consistency and inconsistency for sequences of alternatives having a given rate of convergence to hypothesis in ${L}_2$-norm. We provide transparent interpretations of these conditions allowing to understand the structure of such consistent sequences. We show that, if set of alternatives is bounded closed center-symmetric convex set $U$ with "small" $L_2$ – ball removed, then compactness of set $U$ is necessary condition for existence of consistent tests.