From Wikipedia, $ \sum_1^k{Z_i^2}$ is Chi-squared distributed( $Z_i$ is a standard normal random variable) Also, it is followed by that Pearson's cumulative test statistic $ \sum_1^n{(O_i-E_i)^2 \over E_i}$ approaches to Chi-squared distribution. ( $O_{i}$ = the number of observations of type $i$ , $E_{i}$ = the expected (theoretical) frequency of type $i$ ) I have been searching for the proof that $ \sum_1^n{(O_i-E_i)^2 \over E_i}$ approaches to $ \sum_1^k{Z_i^2}$ , but I could not find it anywhere. Is there anyone to show the proof?

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