Аннотация: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.