Estimation of necessary sample size for testing simple close hypotheses

Let $F_{1_n}$ and $F_{2_n}$ be the $n$-times direct products of distributions $F_1$ and $F_2$ correspondingly. The problem of estimation of necessary sample size for testing hypothesis $F_1$ against $F_2$ is represented as the problem of estimation $\nu=\min\{n\colon\operatorname{var}(F_{1_n},F_{2_n})\ge u=\mathrm{const}\}$. The upper and lower bounds for $\nu$ are given and, supposing $\operatorname{var}(F_{1_n},F_{2_n})\to0$, the asymptotically equivalent estimations for $\nu$ are described in terms of semigroups of limit distributions of $L=\sum\ln[dF_2(X_i)/dF_1(X_i)]$.