Multivariable goodness tests and approximation of the residues of quadratic forms

Consideration was given to the omega square Cramer–von Mises tests intended to verify the goodness hypothesis about the distribution of the observed multivariable random vector with the distribution in the unit cube. The limit distribution of the statistics of these tests was defined by the distribution of an infinite quadratic form in the normal random variables. For convenience of computing its distribution, the residue of the quadratic form was approximated by a finite linear combination of the $\chi^2$-distributed random variables. Formulas for determination of the residue parameters were established.