Lindeberg's condition

Remarks

The Lindeberg central limit theorem states that under this condition $(S_n - \sum_{k \le n} \mu_k) / s_n$ converges in distribution to a standard normal variable, where $S_n = X_1 + \cdots + X_n$.

Example: an i.i.d. sequence with variance $\sigma^2 \in (0, \infty)$ satisfies the condition. Here $s_n^2 = n \sigma^2$, and the sum equals $\sigma^{-2} E[(X_1 - \mu_1)^2 \mathbf{1}_{\set{\abs{X_1 - \mu_1} > \varepsilon \sigma \sqrt{n}}}]$. It tends to $0$ by dominated convergence, since $E[(X_1 - \mu_1)^2] < \infty$.

Non-example: let $X_1$ be standard normal and $X_k = 0$ for $k \ge 2$. Then $s_n = 1$ for all $n$, and the sum equals $E[X_1^2 \mathbf{1}_{\set{\abs{X_1} > \varepsilon}}] > 0$, which does not tend to $0$. In this case one variable carries all the variance.

The condition implies Feller's condition $\max_{k \le n} \sigma_k^2 / s_n^2 \to 0$. The Lyapunov condition, $s_n^{-2-\delta} \sum_{k \le n} E[\abs{X_k - \mu_k}^{2+\delta}] \to 0$ for some $\delta > 0$, implies Lindeberg's condition and is often easier to check.