Hillier, Grant, Kan, Raymond and Wang, Xiaolu
Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors , London, GB Centre for Microdata Methods and Practice
(CEMMAP Working Paper, CWP14/08).
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Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben, Hillier, Kan, and Wang). Typically, in a recursion of this type the k-th object of interest, dk say, is expressed in terms of all lower-order dj's. In Hillier, Kan, and Wang we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors.
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