On one update for the limit formula of the normalized difference between asymptotically optimal tests

In the paper, in terms of general statistical experiment, we prove a general theorem on the formula for the limit of the normalized difference between the powers of the asymptotically optimal test and the most powerful test. The limit allows to calculate the deficiency of the asymptotically optimal test. In contrast to previous papers, the theorem works when the difference between the powers of tests is of any order (not only of order $n^{-1}$) and the Cramér condition (C) does not hold for the characteristic function of the statistic of the asymptotically optimal test.