CLT for triangular array of finite uniformly distributed variables I am interested in central limit theorems for the following rather simple setup of finite uniform distributions:
Let $X_{ni}$ for $i \leq n$ and $n \in \{1,2,3,\ldots\}$ be independent discrete random variables such that $X_{ni}$ is uniformly distributed on the interval $[-a_{ni},a_{ni}] \subset \mathbb{Z}$.
Let $s_n^2 = \mathbb{V}(X_n) = \sum_i \mathbb{V}(X_{ni})$ be the variance of $X_n = \sum_i X_{ni}$.

Are there necessary and sufficient conditions on the array $(a_{ni})$ implying a central limit $\frac{X_n}{s_n} \rightarrow N(0,1)$ ?

It is well-known that under the assumption $\frac{1}{s_n^2}\max_i \mathbb{V}(X_{ni}) \rightarrow 0$ for $n \rightarrow \infty$, the Lindeberg condition is both necessary and sufficient.
My question thus can be given in two parts:
Assuming $\frac{1}{s_n^2}\max_i \mathbb{V}(X_{ni}) \rightarrow 0$.

Is it known how the Lindeberg condition translate to a (hopefully simple) condition that can be directly written in terms of the parameters $(a_{ni})$ ?

Assuming $\frac{1}{s_n^2}\max_i \mathbb{V}(X_{ni}) \not\rightarrow 0$.

Is it anyways possible for $X_{ni}$ to satisfy a central limit?

I finally add that I have very little background on probability theory. References to books/papers answering these or related questions are highly appreciated.
 A: 
Is it known how the Lindeberg condition translate to a (hopefully simple) condition...

Yes. In your situation the assumption $$
\frac{1}{s_n^2}\max_i \mathbb{V}(X_{ni}) \rightarrow 0, n\to\infty, \tag{1}
$$ (equivalently, $\frac{\max_i a_{ni}^2}{\sum_{i} a_{ni}^2}\to 0,n\to\infty$) implies the Lindeberg condition. Try to prove this (hint: for any $\varepsilon>0$, the expression in question is zero for $n$ large enough). 

Assuming $\frac{1}{s_n^2}\max_i \mathbb{V}(X_{ni}) \not\rightarrow 0$,
  is it anyways possible for $X_{ni}$ to satisfy a central limit?

No. (Provided the meaning of "central limit" is standard, which is normal distribution.) You can write characteristic to check that the uniform smallness assumption $(1)$ is necessary in your case. 
A: This is an attempt to solve the first part of my question assuming $\frac{max_i \mathbb{V}(X_{ni})}{s_n^2} \rightarrow 0$.
Since resorting doesn't change $X_n$, we also use w.l.o.g. that $a_{n1} \leq \dots \leq a_{nn}$ for any $n$.
Claim:
The Lindeberg condition holds. This is, for any $\epsilon > 0$,
$$\frac{1}{s_n^2}\sum_{i=1}^n\mathbb{E}\big(X_{ni}^2\cdot I\big\{|X_{ni}| \geq \epsilon s_n\big\}\big) \rightarrow 0.$$
Proof:
The support of $X_{ni}$ is bounded by $a_{ni}$. By this, I mean
$$|x| > a_{ni} \Rightarrow Prob(X_{ni} = x) = 0.$$
The variance is
$$\mathbb{V}(X_{ni}) = \tfrac{1}{3} a_{ni}(a_{ni}+1), \quad s_n^2 = \frac{1}{3} \sum_{i=1}^n a_{ni}(a_{ni}+1)$$
For any $k$ consider the sequence in $n$ given by $a_{n,n-k}$ for $n > k$.
Since the $a_{ni}$ are sorted in $i$, the sequence $a_{nn}$ grows at least as fast as any of the sequences $a_{n,n-k}$. This is,
$$ a_{n,n-k} \in \mathcal{O}(a_{nn})$$
for any $k$.
The assumed condition $\frac{\mathbb{V}(X_{nn})}{s_n^2} \rightarrow 0$ says that $\mathbb{V}(X_{nn})$ grows strictly slower than $s_n^2$.
In symbols,
$$\mathbb{V}(X_{nn}) \in \mathcal{o}(s_n^2).$$
Since $\mathbb{V}(X_{nn}) \sim a_{nn}^2$ this implies that
$$a_{nn} \sim \sqrt{\mathbb{V}(X_{nn})} \in \mathcal{o}(s_n).$$
Therefore, it exists a global integer $N$ such that $\epsilon s_n > a_{nn}$ for all $n \geq N$.
I finally want to conclude that
$$\mathbb{E}\big(X_{ni}^2\cdot I\big\{|X_{ni}| \geq \epsilon s_n\big\}\big) = 0$$
for $n \geq N$ because the support of $X_{ni}$ is completely contained in the excluded interval.
Thus, the sum equals zero for large enough $n$ implying the Lindeberg condition.
