The zero divisor problem of multivariable stochastic adaptive control (Q1068048)

The main result of the paper is the following theorem: ''Let \(Q(x_ 1,...,x_ n)\) be a real-valued rational function of n real variables which is not constant. Let \(\{w_ 1,...,w_ n\}\) be n random variables which are jointly absolutely continuous (distributed) with respect to Lebesgue measure. Then \(Q(w_ 1,...,w_ n)\) is also absolutely continuous (distributed) with respect to Lebesgue measure.'' This result is obtained proving that \(\mu_ n(Q^{-1}(E))=0\) for every set \(E\subset R\) such that \(\mu_ 1(E)=0\), where \(\mu_ k\) represents the Lebesgue measure on \(R^ k\). This theorem is applied in the analysis of stochastic adaptive minimum variance control algorithms (the zero divisor problem) for linear discrete-time systems.