An error of the Monte-Carlo calculation of the integral by means of a~physical generator of random codes

G. A. Kozlov

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 2, pp. 407-415

Résumé

An error of the calculation of a simple integral $\overline\varphi=\int_0^1\varphi\,dx$ by the method of independent tests is estimated in the case when a sequential physical generator of stationary random binary codes with independent digits is used as a source of the random numbers. The imperfection of such a generator can be determined by the value $\varepsilon=P(0)-P(1)$, $P(0)$ and $P(1)$ being the probabilities of 0 and 1 in the code produced. The error mentioned is estimated by the value $$ S(v)=\sup\{\Delta\varphi/\sqrt{\mathbf D\varphi}:\ \varphi\in G(v)\}, $$ where $\Delta\varphi=\int_0^1\varphi\,dF-\overline{\varphi}$, $\mathbf D\varphi=\int_0^1(\varphi-\overline{\varphi})^2\,dx$, $F$ is the actual distribution function of random numbers (if $\varepsilon=0$ then $F(x)=x$, $\Delta\varphi=0$ and $S=0$) and $G(v)=\{\varphi:\bigvee_0^1\varphi/\sqrt{\mathbf D\varphi}\le v\}$ is the class of functions with a finite standartized variation. We prove the relation $\lim_{\varepsilon\to\infty}S(v)/|\,\varepsilon\,|=S^*(v)$ and calculate the function $S^*$. The results may be applied for determining the permissible values of the parameter $\varepsilon$ of the random code generator's imperfection.