Limit distributions of the χ 2 statistic of K. Pearson in a sequence of independent trials

Abstract

We study the χ2 statistic of K. Pearson in a sequence of independent and, generally, inhomogeneous trials with a fixed number of outcomes. It is assumed that the probabilities of occurrence of outcomes of the trials satisfy certain conditions. This problem statement embraces familiar results for the χ2 statistic in the case of multinomial trials. We obtain explicit expressions and estimates for the expectation and the variance of the χ2 statistic. For the χ2 statistic centered and normalized in a suitable way, we find limit distributions (the normal one, the distribution of the sum of the squares of normal random variables and, in particular, the χ2 distribution). Conditions for the convergence to the corresponding limit distributions are given.