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Let $\left\{X_{1}, X_{2}, \ldots X_{k}\right\}$ denote a set of $k$ IID $\operatorname{Bern}(p)$ random variables. Also, I have a set of $k$ non-negative integer weights denoted by $\left\{a_{1}, a_{2}, \ldots a_{k}\right\}$ such that $\sum_{i} a_{i}=k$.

As given here, the probability mass function (PMF) of $Y=\sum_{i} a_{i} X_{i}$ approximately using the Lyapunov CLT is $$ Y \stackrel{\text { Approx }}{\sim} \mathrm{N}\left(p k, p(1-p) \sum_{j} a_{j}^{2}\right) $$

My aim is to find the condition on weights $\{a_i\}$ so that Lyapunov CLT can be applied. I think that it follows from the below condition:

For some $\delta >0$. $$\lim _{k \rightarrow \infty} \frac{1}{s_{k}^{2+\delta}} \sum_{i=1}^{k} \mathbb{E}\left[\left|X_{i}-\mu_{i}\right|^{2+\delta}\right]=0$$

Here, $s_{k}^{2}=\sum_{i=1}^{k} \sigma_{i}^{2}$.

Can someone help me simplify this condition to find the constraint on weights $\{a_i\}$ so that Lyapunov CLT holds.

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1 Answer 1

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If $\delta = 1$ then $\text{Var}(a_i X_i) = a_i^2 p (1-p)$ and $$ \mathbb{E}\left[\left|X_{i}-\mu_{i}\right|^{2+\delta}\right] = a_i^3p(1-p)[p^2 + (1-p)^2] $$ so your condition simplifies to $$ \left( \sum_i a_i^3 \right)^2 = o\left[ \left( \sum_i a_i^2 \right)^3 \right] $$ Intuitively this is a requirement that your weights aren't too far from uniform. This makes sense because the Lyapunov CLT generalizes CLTs that require independence and identicalness to situations where your data are still independent but not necessarily identical.

This example is similar to Example 2.7.2 in https://www.amazon.com/Elements-Large-Sample-Theory-Springer-Statistics/dp/0387985956 However, your weights are a little different than their example's statistic's.

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