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

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).