**MSC:**- 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.), See also {30E15}
- 41A63 Multidimensional problems (should also be assigned at least one other classification number in this section)
- 60F10 Large deviations
- 62E17 Approximations to distributions (nonasymptotic)
- 62E20 Asymptotic distribution theory

the cumulative distribution functions of noncentral generalized chi-square distributions,

preferably in the far tails. The basic idea of how to deal with the tail

probabilities consist in first rewriting these probabilities as large parameter values

of the Laplace transform of a suitably defined function f_k, second making a series

expansion of this function and third applying a certain modification of Watsons

lemma.

The function f_k is deduced by applying a geometric representation formula for

spherical measures to the domain of large deviations under consideration. At the so

called dominating point, the largest main curvature of the boundary of this domain

tends to one as the large deviation parameter approaches infinity. Therefore, the

dominating point degenerates asymptotically. For this reason the recent asymptotic

expansion for large deviations in Breitung and Richter (1996) does not

apply.

Assuming a suitable parameterized expansion for the inverse g^{-1} of the negative

logarithm of the density generating function we derive a series expansion for the

function f_k. Note that low order coefficients from the expansion of g^{-1} influence

practically all coefficients in the expansion of the tail probabilities.