On one generalization of Chernov's distance

The variable $\rho(\mathbf{p};A,B)$ is introduced to characterize, for a given vector $\mathbf{p}$, the distance between finite sets $A$ and $B$ of vectors of probabilities of outcomes in polynomial schemes of trials having a common set of outcomes. In the case of singletons $A=\{\mathbf{a}\}$, $B=\{\mathbf{p}\}$ the value of $\rho(\mathbf{p};A,B)$ coincides with the Chernov distance between $\mathbf{p}$ and $\mathbf{a}$. We indicate the probabilistic sense of the generalized Chernov distance $\rho(\mathbf{p};A,B)$ and establish some of its properties. For distinguishing between $m$ polynomial distributions $(n,\mathbf{p}_1),\dots,(n,\mathbf{p}_m)$ we consider a Bayesian decision rule, where the proper distribution is found in $k\in\{1,\dots,m-1\}$ most plausible variants. For this rule, we find explicit and asymptotic (as $n\to\infty$) estimates of probabilities of errors depending on at most $C_{m-1}^k$ generalized Chernov distances and, moreover, establish, in a sense, its optimality.