Let $p_i\in (c,1c)$ 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 $$\mathbb{P}(X=k)=(1+o(1))\mathbb{P}(Z\in (k,k+1))$$

2$\begingroup$ Can't one do this by Stirling's formula? It seems to me that $knp$ should be $o(n^{2/3})$ for the approximation to hold. $\endgroup$ – Lucia Sep 2 '13 at 8:31

$\begingroup$ I am sorry, my bad. I will correct the conditions  in fact I wanted something more general. $\endgroup$ – TOM Sep 2 '13 at 10:26
There are a lot of results along these lines in
V. V. Petrov, Sums of independent random variables, SpringerVerlag, 1975
The distribution of $X$ is called the PoissonBinomial distribution and searching on that phrase will find a large amount of literature on it.