  Two--sided bounds are constructed for a probability density function of a
weighted sum of chi-square variables. Both cases of central and non-central
chi-square variables are considered. The upper and lower bounds have the same
dependence on the parameters of the sum and differ only in absolute constants.
The estimates obtained will be useful, in particular, when comparing two
Gaussian random elements in a Hilbert space and in multidimensional central
limit theorems, including the infinite-dimensional case.
