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 $$\mathbb{P}(X=k)=(1+o(1))\mathbb{P}(Z\in (k,k+1))$$
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2$\begingroup$ 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. $\endgroup$– LuciaSep 2, 2013 at 8:31
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$\begingroup$ I am sorry, my bad. I will correct the conditions - in fact I wanted something more general. $\endgroup$– TOMSep 2, 2013 at 10:26
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2 Answers
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There are a lot of results along these lines in
V. V. Petrov, Sums of independent random variables, Springer-Verlag, 1975
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The distribution of $X$ is called the Poisson-Binomial distribution and searching on that phrase will find a large amount of literature on it.
answered Sep 2, 2013 at 12:05