A Non-Random Sampling Method (Classic Reprint) - Tapa blanda

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Excerpt from A Non-Random Sampling Method

Unfortunately, the theorems referred to are not broad enough to be applicable to practical Monte Carlo problems because of the restriction to k l in Ostrowski's theorem and to very smooth integrands in the theorems of the writer and of Peck. A new approach to the problem is therefore taken, in Section 4, by reintroducing statistical considerations. If, at the beginning of a problem, the numbers gk are chosen at random from some continuous distribution, then, with exception of choices having probability zero, they will indeed turn out to be linearly independent irrational numbers. We can therefore regard the gk as random variables (they will be taken to have a uniform distribution in the unit cube, since there is no loss of generality in this choice); for given N (and a given function IN is then a function of the random variables gk, and we may discuss its statistics.

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