An equation of the Kolmogorov-Feller type in a factorized probability space (Q2711122)

Let \(\Omega_0=\{\omega_k\}\) be a finite or countable set and \(P_k\) be the probability of the elementary event \(\omega_k\). Let further, \(\Omega_N=\{\omega_{{\mathbf k}}: \omega_{k_1}\otimes\cdots\otimes\omega_{k_N}\), \(k_i\in (1,2,\ldots, |\Omega_0|)\}\) be the direct product of \(N\) copies of \(\Omega_0\) factorized with respect to the permutation group of indices \(k_1,\ldots, k_N\). Let, finally, for \({\mathcal M}=(M_1,M_2,\ldots, M_{|\Omega_0|})\) the set \(\Omega_N({\mathcal M})\) consist of all the elements \(\omega_{{\mathbf k}}\in\Omega_N\) such that the number \(N_k\) of occurrences of the element \(\omega_k\in\Omega_0\) in \(\omega_{{\mathbf k}}\) is less than or equal to the respective \(M_k.\) The initial distribution \(P_k\) on \(\Omega_0\) generates a probability measure \({\mathcal P}_N\) on \(\Omega_N({\mathcal M})\) in a natural way. The authors study the properties of some random variables defined on the probability space \((\bigoplus_{N=0}^{\infty}\Omega_N({\mathcal M}),\Sigma, {\mathcal P}^{\infty})\) where \(\Sigma\) is the \(\sigma\)-algebra of all subsets of \(\bigoplus_{N=0}^{\infty}\Omega_N({\mathcal M})\) and \({\mathcal P}^{\infty}\) is a measure constructed by means of \({\mathcal P}_N\), \(N=0,1,\ldots.\)