On new conditions for convergence of the distribution of the number of solutions of nonlinear inclusions to the Poisson distribution

V. A. Kopyttsev

Academy of Cryptography of the Russian Federation, Moscow

Abstract: For given subset $B$ of the vector space $K^T$ of dimension $T$ over the field $K=GF(q)$ we study the distribution of the number of solutions $\xi$ of the inclusion system $ x\in K^n\setminus \{0^n\}, A_1x+A_2f(x)\in B, $ where $A_1$ and $A_2$ are the random matrices over the field $K$ of the dimensions $T\times n$ and $T\times m$ with independent elements, $f(x)=$ $(f_1 (x),\ldots ,f_m (x))\colon K^{n}\longrightarrow K^{m}$ is a given nonlinear mapping. The new and more general conditions for the convergence of the distribution of the random variable $\xi$ to the Poisson distribution compared with previously known ones are specified.

Key words: random inclusions, distribution of the number of solutions, Poisson limit theorem.

UDC: 519.719.2+519.226.2

Received 27.I.2025

DOI: 10.4213/mvk502