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 Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 1, Pages 52–72 (Mi tvp324)

Limit theorems for the number of solutions of a system of random equations

V. A. Kopyttsev

Abstract: We investigate the number and the set structure of the solutions of a consistent system of random equations of the form
$$\varphi_t(x_{s_1(t)},\ldots,x_{s_{d(t)}(t)})=a_t,\quad t=1,\ldots, T,$$
with respect to the variables $x_1,\ldots, x_n\in\{0,\ldots,q-1\}$, $q\ge 2$, where the indices $s_1(t),\ldots,s_{d(t)}(t)$ are selected randomly 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 $g: \{0,\ldots,q-1\}{\longleftrightarrow}\{0,\ldots,q-1\}$ satisfying the conditions $\varphi_t(y_1,\ldots y_{d(t)})\equiv\varphi_t(g(y_1),\ldots, g(y_{d(t)}))$, $t=1,\ldots,T$, and $\eta_1,\ldots,\eta_{q-1}$ are independent Poisson random variables with parameters $\lambda_1,\ldots,\lambda_{q-1}$, respectively. Explicit expressions for the parameters $\lambda_1,\ldots\lambda_{q-1}$ are given. These results essentially generalize analogous theorems proved for the case $q=2$ in [V. A. Kopytsev, Theory Probab. Appl., 40 (1995), pp. 376–383] and [V. G. Mikhailov, Theory Probab. Appl., 41 (1996), pp. 265–274].

Keywords: systems of random equations, true solution, vicinity of a true solution, the total number of solutions, permutation groups, Poisson distribution.

DOI: https://doi.org/10.4213/tvp324

Full text: PDF file (820 kB)

English version:
Theory of Probability and its Applications, 2001, 45:1, 51–68

Bibliographic databases:

Citation: V. A. Kopyttsev, “Limit theorems for the number of solutions of a system of random equations”, Teor. Veroyatnost. i Primenen., 45:1 (2000), 52–72; Theory Probab. Appl., 45:1 (2001), 51–68

Citation in format AMSBIB
\Bibitem{Kop00}
\by V.~A.~Kopyttsev
\paper Limit theorems for the number of solutions of a system of random equations
\jour Teor. Veroyatnost. i Primenen.
\yr 2000
\vol 45
\issue 1
\pages 52--72
\mathnet{http://mi.mathnet.ru/tvp324}
\crossref{https://doi.org/10.4213/tvp324}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1810974}
\zmath{https://zbmath.org/?q=an:0982.60060}
\transl
\jour Theory Probab. Appl.
\yr 2001
\vol 45
\issue 1
\pages 51--68
\crossref{https://doi.org/10.1137/S0040585X9797804X}

• http://mi.mathnet.ru/eng/tvp324
• https://doi.org/10.4213/tvp324
• http://mi.mathnet.ru/eng/tvp/v45/i1/p52

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This publication is cited in the following articles:
1. V. G. Mikhailov, “The Poisson limit theorem for the number of noncollinear solutions of a system of random equations of a special form”, Discrete Math. Appl., 11:4 (2001), 391–400
2. V. G. Mikhailov, “Izuchenie predelnogo povedeniya chisla reshenii sistem uravnenii so sluchainym vkhozhdeniem neizvestnykh”, Matem. vopr. kriptogr., 1:3 (2010), 27–43
3. V. A. Kopyttsev, “O porogovom effekte dlya srednego chisla reshenii sistemy sluchainykh uravnenii”, Matem. vopr. kriptogr., 10:3 (2019), 67–80
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