Let \(\{(\xi_ t,\eta_ t)\}^ n_{t=1}\) be a sequence of independent random vectors taking on values \((m,\ell)\), \(m=1,2,...,N\), \(\ell =0,1,...,M-1\), with the unknown probabilities \(p_ mp_{m\ell}\) where \(p_ m=P\{\xi_ t=m\}\), \(p_{m\ell}=P\{\eta_ t=\ell | \xi_ t=m\}\), \(t=1,2,...,N\). The author investigates asymptotic properties of separable statistics based on maximum likelihood estimators of \(p_ m\) and \(p_{m\ell}.\) The obtained results provide generalizations of the author's paper: 'Randomized separable statistics', in Theory and Practice of Mathematical Methods and Means of Computational Techniques, Kiev Inst. Kibern. Akad Nauk SSR, 67-75 (1981) and of the author and \textit{S. A. Lapin}, Mat. Zametki 34, No.5, 745-756 (1983; Zbl 0536.62007); English translation in Math. Notes 34, 864-870 (1983).