Local limit theorem for Bernoulli sums

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

• Can't one do this by Stirling's formula? It seems to me that $k-np$ should be $o(n^{2/3})$ for the approximation to hold. – Lucia Sep 2 '13 at 8:31
• I am sorry, my bad. I will correct the conditions - in fact I wanted something more general. – TOM Sep 2 '13 at 10:26

The distribution of $X$ is called the Poisson-Binomial distribution and searching on that phrase will find a large amount of literature on it.