On the Limit Distribution of the Number of Solutions of a~Random Linear System in the Glass of Boolean Functions

I. N. Kovalenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 1, pp. 51-61

Résumé

Let (1) be a system of linear Boolean equations, $a_{ij}$ being independent random variables with distributions given by (2). Let $\nu_n$ denote the number of linearly independent solutions of the system. Condition (3) with some fixed $\delta>0$ implies the convergence of the distributions of $\nu_n$ as $n\to\infty$ to the distribution of a random variable $\nu$ which can be constructed as follows: $$ \nu= \begin{cases} 0\text{if}\quad m+s_{k_0}\le0 \\ m+s_{k_0}\text{if}\quad m+s_{k_0}>0 \end{cases} $$ where die distribution of $s_{k_0}$ is given by (24), (25).