Limit theorems for the number of solutions of a system of random equations (Q2752968)

The author investigates the number and the set structure of the solutions of a consistent system of random equations of the form NEWLINE\[NEWLINE \varphi_t(x_{s_1(t)},\dots, x_{s_{d(t)}(t)})=a_t,\quad t=1, \dots,T, NEWLINE\]NEWLINE with respect to the variables \(x_1,\dots, x_n\in\{0,\dots, q-1\}\), \(q\geq 2\), where the indices \(s_1(t),\dots, s_{d(t)}(t)\) are selected at random and independently for different \(t\) according to the equiprobable selection procedure without replacement. Conditions are found under which the distribution of the number of solutions of the system converges to the distribution of a random variable of the form \(A\cdot 2^{\eta_1}\cdots q^{\eta_{q}-1}\), where \(A\) is the order of the group of permutations \(g: \{0,\dots, q-1\}\rightarrow\{0,\dots, q-1\}\) satisfying the conditions \(\varphi_t(y_1,\dots, y_{d(t)})\equiv\varphi_t(g(y_1),\dots g(y_{d(t)}))\), \(t=1,\dots, T\), and \(\eta_1, \dots,\eta_{q-1}\) are independent Poisson random variables with parameters \(\lambda_1,\dots,\lambda_{q-1}\), respectively. Explicit expressions for the parameters \(\lambda_1,\dots,\lambda_{q-1}\) are given. These results essentially generalize analogous theorems proved by the author [Theory Probab. Appl. 40, No. 2, 376-383 (1995); translation from Teor. Veroyatn. Primen. 40, No. 2, 430-437 (1995; Zbl 0847.60049)] and \textit{V. G. Mikhajlov} [ibid. 41, No. 2, 265-274 (1996); resp. ibid. 41, No. 2, 272-283 (1996; Zbl 0881.60089)] for the case \(q=2\).