On the efficient calculation of a linear combination of chisquare random variables with an application in counting string vacua
Abstract
Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the probability density function is not yet known. Starting out from an analytic expression for the density of the sum of two gamma variables, a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of terms compared to the exponential growth of most other methods. It provides a vast improvement in accuracy and shows only logarithmic growth in the required precision. In addition, it is applicable to a much greater number of terms and currently the only way of computing the distribution for hundreds of terms. As an application, the exponential dependence of the eigenvalue fluctuation probability of a random matrix model for 4D supergravity with N scalar fields is found to be of the asymptotic form exp(0.35N).
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 December 2013
 DOI:
 10.1088/17518113/46/50/505202
 arXiv:
 arXiv:1208.2691
 Bibcode:
 2013JPhA...46X5202B
 Keywords:

 Mathematics  Probability;
 High Energy Physics  Theory;
 15A18;
 60G50;
 81T30;
 83E30;
 83E50
 EPrint:
 21 pages, 19 figures. 3rd version