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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 3, Pages 518–533 (Mi tvp2662)  

On estimation of the error of Monte-Carlo technique caused by imperfections of the distribution of random numbers

G. A. Kozlov

Moscow

Abstract: An approach to estimation of the Monte-Carlo technique error caused by imperfections of the distribution of random numbers is proposed. The approach is illustrated by an example of the simple integral $\overline\varphi=\int_0^1\varphi(x) dx$ calculation by the method of indeopendent tests. The error is estimated by
$$ S=\sup U(\varphi),\quad\varphi\in G,\quad U(\varphi)=(\int_0^1\varphi(x) dF(x)-\overline\varphi)/\sqrt{\int_0^1(\varphi(x)-\overline\varphi)^2 dx}, $$
where $F$ is the distribution function of random numbers in the interval $[0,1]$, $G$ is the class of functions with finite “standartized variation”:
$$ G=\{\varphi\colon\bigvee_0^1\varphi/\sqrt{\int_0^1(\varphi(x)-\overline\varphi)^2 dx}\le v\}. $$

It is shown that the problem of determining the value $S$ can be reduced to a variational problem of finding the function that minimizes the functional $U(\varphi)=\int_0^1\varphi dF$ under the following restrictions:
$$ \int_0^1\varphi dx=0,\quad\int_0^1\varphi^2 dx=1\quadand\quad\bigvee_0^1\varphi\le v $$
A solution of this variational problem is given.

Full text: PDF file (865 kB)

English version:
Theory of Probability and its Applications, 1973, 17:3, 493–509

Bibliographic databases:

Received: 24.03.1970

Citation: G. A. Kozlov, “On estimation of the error of Monte-Carlo technique caused by imperfections of the distribution of random numbers”, Teor. Veroyatnost. i Primenen., 17:3 (1972), 518–533; Theory Probab. Appl., 17:3 (1973), 493–509

Citation in format AMSBIB
\Bibitem{Koz72}
\by G.~A.~Kozlov
\paper On estimation of the error of Monte-Carlo technique caused by imperfections of the distribution of random numbers
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 3
\pages 518--533
\mathnet{http://mi.mathnet.ru/tvp2662}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=309262}
\zmath{https://zbmath.org/?q=an:0261.62017}
\transl
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 3
\pages 493--509
\crossref{https://doi.org/10.1137/1117057}


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