A family of multivariate $\chi^2$-statistics

We consider a sequence of independent trials, under the hypothesis $H_0$ each of them is a realisation of a certain polynomial scheme. We introduce a family of multivariate $\chi^2$-statistics where the outcomes are equipped with weights depending on the ordinal of the trial. Earlier results on families of multivariate $\chi^2$-statistic are extended to this family.
This research was supported by grant 1758.2003.1 of President of Russian Federation for support of leading scientific schools.