A family of asymptotically independent statistics in polynomial scheme containing the Pearson statistic

We consider a polynomial scheme with $N$ outcomes. The Pearson statistic is the classical one for testing the hypothesis that the probabilities of outcomes are given by the numbers $p_1,\ldots,p_N$. We suggest a couple of $N-2$ statistics which along with the Pearson statistics constitute a set of $N-1$ asymptotically jointly independent random variables, and find their limit distributions. The Pearson statistics is the square of the length of asymptotically normal random vector. The suggested statistics are coordinates of this vector in some auxiliary spherical coordinate system.