1 research outputs found
On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua
Linear combinations of chi square random variables occur in a wide range of
fields. Unfortunately, a closed, analytic expression for the pdf is not yet
known. As a first result of this work, an explicit analytic expression for the
density of the sum of two gamma random variables is derived. Then 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).Comment: 21 pages, 19 figures. 3rd versio