Appendix A: Lemma

 In Sect. 5, the relation (30) for the n-th derivative of a composite function $f\!\circ\! g(x) \equiv f(g(x))$ is used for the derivation of the Edgeworth asymptotic expansion. Here a simplified derivation of Eq. (30) is given. Petrov (1972, 1987) suggests a proof by induction. We note that this derivation is more transparent if one simply considers the Taylor expansion for $f\!\circ\! g$ expressed in terms of the Taylor expansions of f and g - see Bourbaki (1958). We have  

$$f(y) = f(y_0) + {f'\over 1!}\Delta y + {f''\over 2!}\Delta^2 y+ \cdots {f^{(n)} \over n!}\Delta^n y+ \cdots \;$$ (A1)

and  

$$g(x) = g(x_0) + {g'\over 1!}\Delta x + {g''\over 2!}\Delta^2 x+ \cdots {g^{(m)} \over m!}\Delta^m x+ \cdots \; .$$ (A2)

Truncating the expansions at some n and m we find that

$$\begin{eqnarray} \lefteqn{ f\!\circ\! g(x) = f(g(x_0)) } \nonumber \\ \lefteqn{... ... {g^{(m)} \over m!}\Delta^m x+ \cdots \right)^n \; .} \nonumber \end{eqnarray}$$ (A3)

On the other hand, we can write down the Taylor series for the composite function,

$$\begin{eqnarray} \lefteqn{ f\!\circ\! g(x) = f\!\circ\! g(x_0)} \nonumber \\ \l... ...2 x+ \cdots {(f\!\circ\! g)^{(s)} \over s!}\Delta^s x+ \cdots } \end{eqnarray}$$ (A4)

Now using the polynomial theorem,

$$\begin{eqnarray} \lefteqn{ (x_1+x_2+\cdots+x_m)^r } \nonumber \\ \lefteqn{\quad... ...= \sum_{\{k_m\}} r!\prod_{s=1}^m {x_s^{k_s} \over k_s!} \; , } \end{eqnarray}$$ (A5)

where summation extends over all sets of non-negative integers $\{k_m\}$satisfying r=k₁+k₂+ ... +k_s, and comparing the terms $\Delta^{s}x$ with equal s in (A4) and (A4), we obtain

$$\begin{eqnarray} \lefteqn{ {{\rm d}^s \over {\rm d}x^s} f(g(x)) = } \nonumber \\... ...{1 \over k_m!} \left({1 \over m!} g^{(m)}(x)\right)^{k_m} \; .} \end{eqnarray}$$ (A6)

This is the relation (30) which we sought. Here the set $\{k_m\}$ consists of non-negative solutions of the Diophantine equation  

$$k_1+2k_2+ ... +s k_s=s \; ,$$ (A7)

and r=k₁+k₂+ ... +k_s.