Interpreting the Lindeberg's condition I know the Lindeberg's CLT but I don't have a good grasp of the intuition behind the Lindeberg's condition. Could you please give some intuition behind said condition via an example (or, perhaps, via 2 related examples, one satisfying the condition and one not satisfying).

For completeness and to fix notation, I reproduce the wiki's statement of the Lindeberg's CLT and condition below. Please note that wiki provides an intuition based on a consequence of the Lindeberg's condition. I don't this intuition helpful.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and $X_k:\Omega\to\mathbb{R}, k\in\mathbb{N}$, be independent random variables defined on that space. Assume the expected values $E(X_k)=\mu_k$ and variances $\text{Var}(X_k)=\sigma_k^2$ exist and are finite. Also let $s_n^2\equiv\sum_{k=1}^n\sigma_k^2$. If this sequence of independent variables $X_k$ satisfies Lindeberg's condition:
  $$
\lim_{n\to\infty}\frac{1}{s^2_n}\sum_{k=1}^nE[(X_k-\mu_k)^2\cdot1_{|X_k-\mu_k|>\epsilon s_n|}=0
$$
  for all $\epsilon>0$, where $1_{\{\cdots\}}$ is the indicator function, then the central limit theorem holds:
  $$
Z_n:=\frac{\sum_{k=1}^n(X_k-\mu_k)}{s_n}\overset{L}{\to}N(0,1).
$$

 A: I'm not sure about an informative example which does not satisfy the Lindeberge condition. But here is one which does which I find particularly insightful.
Let $\xi_i$ be a sequence of zero mean, variance 1 iid random variables and $a_i$ a non-random sequence satisfying:
$$\max_i^n \frac{|a_i|}{\|a_i\|_2} \rightarrow 0$$
Now, define the normalized elements of the linear combination:
$$X_{n,i} = \frac{a_i \xi_i}{\|a\|_2}$$
which satisfies the Lindeberge condition.:
$$\sum_i^n \mathbb E \left [ \left | X_i\right |^2 1(|X_i| > \epsilon)\right ] \leq \sum_i^n \mathbb E \left [ \left | X_i\right |^2 1 \left(|\xi_i| > \epsilon \frac{\|a\|_2}{\max_i^n |a_i|} \right)\right ] = 
\mathbb E \left [ \left | \xi_i\right |^2 1 \left(|\xi_i| > \epsilon \frac{\|a\|_2}{\max_i^n |a_i|} \right)\right ]$$
but $\mathbb \xi_i^2$ is finite so by DCT and the condition on the $a_i$ we have that this goes to $0$ for every $\epsilon > 0$.
See https://people.stat.sc.edu/gregorkb/STAT_824_sp_2021/STAT_824_Lec_11_supplement.pdf
