I'd like to get a CLT-like approximate distribution (mostly tail behavior) of the sum $X$ of $n$ independent Bernoulli random variables $X_1, \dots, X_n$, with proportions $p_1, \dots, p_n$.

The Lyapunov CLT seems like it could help (as mentioned in CLT can be used for weighted sum of different Bernoulli variables?).

But it's an asymptotic result on sequences of random variables. I have not become very familiar with this condition, but it seems like I could artificially construct $X_{n+1}, \dots$ such that the condition always holds? I'm not entirely sure on this point.

Is there some way to apply it more meaningfully?

If $p_1 = 0.5$ and $p_2, \dots, p_n$ are 0 (or extremely close to zero, like $1/2^{\max(100, n)}$), then clearly I should not be using a normal approximation for the distribution of $X =\sum_{i=1}^n X_i$. My case is not so extreme, but there are some $p_i$ very close to 0/1.

Is there some common rule of thumb to use here? I suppose I could compute the Lyapunov CLT term for $\delta=1$ and compare this to $\frac{1}{\sqrt{n}}$ (when sequence is iid Bernoulli($p$) the term seems to decrease as $\frac{c}{\sqrt{n}}$ where $c$ is a function on $p$). The following is this Lyapunov CLT term -- if it approaches $0$ as $n \to \infty$ then the Lyapunov CLT condition is satisfied. $$\frac{1}{s_n^3}\sum_{i=1}^n E \left[ |X_i - E[X_i]|^3 \right]$$

Alternate idea is to use a Chernoff bound.

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    $\begingroup$ If $p_2,...,p_n$ are very small, a Poisson approximation would be more useful than a Gaussian. You could then do a direct convolution with the first variable to get a good approximation to the combined distribution. If you have some very large $p$'s and some very small $p$'s you could do those as a shifted-Skellam. The general case of approximating a sum of independent Bernoullis with varying $p$ is the Poisson-binomial ... about which there are a number of questions already on site. $\endgroup$
    – Glen_b
    Aug 14, 2020 at 9:38
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    $\begingroup$ Sorry, the link should go here: en.wikipedia.org/wiki/Poisson_binomial_distribution $\endgroup$
    – Glen_b
    Aug 14, 2020 at 9:56