# Cramér-Edgeworth Expansions

• Rabi Bhattacharya
• Manfred Denker
Chapter
Part of the DMV Seminar book series (OWS, volume 14)

## Abstract

Let Q be a probability measure on ( k ,B k ), B k denoting the Borel sigmafield on k . Assume that the s — th absolute moment of Q is finite,
$${\rho_s}: = \int {{{\left\| x \right\|}^s}Q(dx) < \infty }$$
(1.1)
, for some integer s ≥ 3, and that Q is normalized,
$$\int {{x^{{(i)}}}Q(dx) = 0(1 \leqslant i \leqslant k)}, \int {{x^{{(i)}}}{x^{{(j)}}}Q(dx) = {\delta_{{ij}}}(1 \leqslant i,j \leqslant k)}$$
(1.2)
. Then the characteristic function of Q satisfies ∣(ξ) - 1 ∣ ≤ 1/2 for ∥ ξ ∥≤ 1, and has continuous derivatives of all orders ν = (ν(1),ν(2),..., ν(k) such that ∣ν∣:=∑ν(i)s. Hence the (principal branch of the) logarithm of has the expansion
$$\log \hat{Q}(\xi ) = \sum\limits_{{2 \leqslant \left| v \right| \leqslant s - 1}} {\frac{{{\xi^v}}}{{v!}}({D^v}\log \hat{Q})(0)}$$
$$\begin{gathered} + s\sum\limits_{{\left| v \right| = s}} {\frac{{{\xi^v}}}{{v!}}\int\limits_0^1 {{{(1 - u)}^{{s - 1}}}({D^v}\log \hat{Q})(u\xi )du,(\left\| \xi \right\| \leqslant 1)} } . \hfill \\ ({\xi^v}: = {({\xi^{{(1)}}})^{{{v^{{(1)}}}}}}...{({\xi^{{(k)}}})^{{{v^{{(k)}}}}}},D: = grad = ({D_1},...,{D_k}), \hfill \\ {D^v}: = D_1^{{{v^{{(1)}}}}}...D_k^{{v(k)}},v!: = {v^{{(1)}}}!...{v^{{(k)}}}!) \hfill \\ \end{gathered}$$
(1.3)
. Here, and elsewhere, D j denotes differentiation with respect to the j—th coordinate. The letter ν denotes a multi-index.

## Keywords

Probability Measure Linear Independence Continuous Component Borel Measurable Function Principal Branch
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