Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 42 / 6 / i

An Edgeworth expansion refines the central limit theorem by incorporating higher cumulants. For iid centred observations with variance ${\sigma }^2$, standardized sum $Z_n={\sum }_i{X}_i/(\sigma \sqrt{n})$, standardized third cumulant ${\gamma }_1={\kappa }_3/{\sigma }^3$ and fourth cumulant ${\gamma }_2={\kappa }_4/{\sigma }^4$, its central-region density expansion isunder sufficient moment and smoothness/nonlattice conditions. Here $H_3=z^3-3z$, $H_4=z^4-6z^2+3$, $H_6=z^6-15z^4+45z^2-15$ are probabilists' Hermite polynomials. Expanding the characteristic exponent gives the cubic and quartic terms, while squaring the cubic term in the exponential gives the $H_6$ correction. Integrating the first correction, using $(\varphi H_{k-1}{)}^{\prime }=-\varphi H_k$, givesThus skewness determines the first departure from the normal approximation; inversion gives a Cornish-Fisher expansion for quantiles. These are asymptotic expansions, not guaranteed nonnegative densities in extreme tails. Lattice data require continuity/lattice corrections, and moment assumptions alone do not justify a uniform smooth-density expansion.