Pearson's Chi-Square Goodness-of-Fit Asymptotics

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Hypothesis Testing.

Your Task
Given $n$ observations categorized into $k$ mutually exclusive bins with observed counts $O_1, \dots, O_k$ ($\sum O_i = n$) and null probabilities $p_1, \dots, p_k$ ($E_i = n p_i$). Prove that Pearson's test statistic: $\chi^2 = \sum_{i=1}^k \frac{(O_i - E_i)^2}{E_i}$ converges in distribution to $\chi^2(k - 1)$ as $n \to \infty$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.