  Convergence of order $O(1/\sqrt{n})$ is obtained for the distance in total
variation between the Poisson distribution and the distribution of the number
of fixed size cycles in generalized random graphs with random vertex weights.
The weights are assumed to be independent identically distributed random
variables which have a power-law distribution. The proof is based on the
Chen--Stein approach and on the derived properties of the ratio of the sum of
squares of random variables and the sum of these variables. These properties
can be applied to other asymptotic problems related to generalized random
graphs.
