The asymptotic behavior of the Pearson statistic (Q2752969)

A polynomial distributed random vector of dimension \(s\) on a certain probability space is considered. The common Pearson statistic (chi-square statistic) created by this vector is studied from the viewpoint of a convergence to \(N(0,1)\). The convergence of the distribution of the Pearson statistic to \(\Phi(0,1)\) is generalized for the Pearson statistic reduced to a subset \(W\) of the set \(\{1,2,\dots, s\}\). The convergence of the Pearson statistic for a certain Wiener process is also proved. The probability that the maximum partial sum exceeds a given level is estimated. The random broken line is a random process. Its distribution \(P_n\) is a probability measure defined on the Borel sigma-algebra of subsets of a certain Banach space. A Wiener measure \(P_0\) can be defined on this sigma-algebra. The sequence \(\{P_n\}\) weakly converges under certain conditions to the Wiener measure \(P_0\). The weak convergence for some functionals of partial sums of the Pearson statistics is proved.