Asymptotic Expansion of Distribution in Central Limit Theorem for Non-Identically Distributed Random Variables

My question is related to the following theorem (e.g. Section XVI.4 of Feller's 1971 book): Let $Z_i$ $(i=1,\cdots,n)$ be independent and identically distributed random variables with mean zero, variance $\sigma^2$ and third absolute moment $\mu_3$. Let $F_n(z)$ be the distribution function of $\frac{\sum_{i}Z_i}{\sqrt{n\sigma^2}}$. If the $Z_i$ are non-lattice variables, then $$F_n(z) - \Phi(z) = \frac{\mu_3}{6\sigma^3\sqrt{n}}(1-z^2)\phi(z) + o\Big(\frac{1}{\sqrt{n}}\Big)$$ where $\Phi$ and $\phi$ are, respectively, the distribution function and density of an $N(0,1)$ random variable. The $o(n^{-1/2})$ term is uniform in $z$. A similar (slightly more involved) expansion hods for the lattice case.

I would like to know whether a similar theorem holds for independent and non-identically distributed random variables. I am OK with assuming that all moments are finite, the number of different distributions is finite, and the number of each such distributions in the sum grows linearly. That is, I am considering summations of the form $\sum_{k=1}^{K}\sum_{i=1}^{n_k}Z_{k,i}$ (for a given $k$ the $Z_{k,i}$ are identically distributed), where $\sum_{k}n_k=n$, each $n_k$ grows linearly in $n$, and $K$ is independent of $n$. I would like to be able to handle both lattice and non-lattice variables.

In a book chapter by Petrov, it is stated that such expansions exist (see Page 8 of http://books.google.co.uk/books?id=Tffi5NtKw5IC&printsec=frontcover#v=onepage&q&f=false, 6 lines before Section 1.5). However, no reference is given.

Edgeworth expansions may be relevant here, but they only seem to be valid under more restrictive conditions (e.g. when the variables have a density).

V.V.Petrov also wrote two extremely useful books on summation theory. In his 1975 book "Sums of independent random variables", Section 4 of Chapter VI is specifically devoted to asymptotic expansions in CLT for non-identically distributed r.v.'s.

• Thanks for the answers - Petrov's book looks quite useful, especially Section VII.1 for the lattice case. For the non-lattice case, I'm not sure that Section VI.4 is what I'm after - it looks mainly suited to random variables with a density. In particular, on p173 there is a condition $$\sqrt{n} \int_{|t|>\varepsilon}\frac{1}{|t|}\prod_{j=1}^{n}|v_j(t)|dt \to 0$$ for all $\varepsilon>0$, where $v_j$ is the characteristic function of the $j$-th variable in the sum. If I am not mistaken, this will generally fail for discrete non-lattice variables (please correct me if I am wrong). – jmscarlett Sep 7 '13 at 15:47
• After more searching, I didn't find exactly what I asked for in the question, but I did find "A Local Limit Theorem and Recurrence Conditions for Sums of Independent Non-Lattice Random Variables" (Mineka, Silverman), which turned out to be sufficient for what I need. – jmscarlett Sep 7 '13 at 18:03
• It looks like you are right. – Yuri Bakhtin Sep 7 '13 at 19:55

There is a large literature, too long to review here. Deheuvels 1989 and Chistyakov 1996 come to mind. But also try: Google search and consider the relevant hits. Hope this helps.

• Dear @ofer zeitouni: I have two suggestions. Please do not use url shorteners, as they aid link-rot and prevent people from seeing the destination of the link. Further, I suggest against linking to lmgtfy.com (with or without a url shortener). Thank you for your attention. – Ricardo Andrade Dec 3 '13 at 13:45
• Yes, it was a not so successful joke. I replaced the link – ofer zeitouni Dec 3 '13 at 14:03
• No worries, @ofer zeitouni. By the way, you can make the link look shorter by using the syntax [link-description](http://link-url) which gives this: link-description. That looks shorter and has the advantage of having a (short) text description. – Ricardo Andrade Dec 3 '13 at 14:05
• @Ricardo Andrade: actually, the link you provide does not work for me. :) – ofer zeitouni Dec 3 '13 at 15:49
• It was just an example where you need to replace "link-url" with the actual url. More explicitly, the code [this is MathOverflow](http://mathoverflow.net) produces the following link: this is MathOverflow. I apologize in case I missed a joke in your previous comment. :) – Ricardo Andrade Dec 3 '13 at 17:52