Interpreting the Lindeberg's condition

Question

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).

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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: $$ Z_n:=\frac{\sum_{k=1}^n(X_k-\mu_k)}{s_n}\overset{L}{\to}N(0,1). $$

Answer 1

Let's interpret the limiting average $Z_n$ as a "vote" in which each $X_k$ casts its ballot for the result of $Z$.

To me, Lindeberg's condition can be seen as ensuring that this vote is democratic; no minority of voters dominate and prevent a majority from coming to the CLT conclusion.

As a trivial non-example, the sequence $X, 0, 0, \ldots$ does not average to a Normal distribution.

Another: let $U_n$ be Uniform variables. The sum (before normalizing for variance)

$$ Z_N = \sum_{n=1}^N \frac{U_n}{2^n} $$

has a finite variance but is almost surely bounded as $N\to \infty$, and as such is most certainly not Normal.

Answer 2

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