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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel 1990

Authors and Affiliations

  • Rabi Bhattacharya
    • 1
  • Manfred Denker
    • 2
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Institut für MathematischeStochastikGöttingenGermany

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