Let $p\_i\in (c,1-c)$ for some fixed $c\in(0,1)$ . Consider a sum $X=\varepsilon\_{1}+\cdots+\varepsilon\_{n}$ where $\varepsilon\_{i}$ are independent Bernoulli random variables with parameters $p\_{i}$. Let $Z$ be a normal random variable with the same mean and variance as $X$. I would like to approximate probabilities $\mathbb{P}(X=k)$, where $k$ is "not too far" from the mean. For which $k=k(n)$ can we approximate $\mathbb{P}(X=k)$ by the corresponding normal probability. That is, in which range for $k$ is it true that $$\mathbb{P}(X=k)=(1+o(1))\mathbb{P}(Z\in (k,k+1))$$



Local limit theorem for Bernoulli sums
There are a lot of results along these lines in



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> V. V. Petrov, Sums of independent random variables, Springer-Verlag, 1975
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