We prove Central limit theorem with characteristic function. If we know the $X_i$ are independent but not identically distributed, is there any weaker condition which still yields the convergence to normal distribution?
For example, suppose $X_i$ are independent with $E X_i = 0$, $\text{Var}(X_i) = \sigma_i^2$, and $$\lim_{n \to \infty} \frac{1}{\sigma(n)^3} \sum_{i=1}^n E[|X_i|^3] = 0$$ where $$\sigma(n)^2 = \text{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \sigma_i^2$$ Then $\displaystyle \frac{1}{\sigma(n)} \sum_{i=1}^n X_i$ converges in distribution to a standard normal random variable.