**3. Integrals of very large dimensions**

We are coming now to the question of calculation of an integral

$$I(f) = \int\_0^\infty \dots \int\_0^\infty e^{-(\mathbf{x}\_1 + \mathbf{x}\_2 + \dots + \mathbf{x}\_n)} d\mathbf{x}\_1 d\mathbf{x}\_2 \dots d\mathbf{x}\_{10} = \mathbf{1} \tag{20}$$

with the distribution density *p x*ð Þ¼ *<sup>λ</sup><sup>s</sup> e*�*λ*ð Þ *<sup>x</sup>*1þ*x*2<sup>þ</sup> … <sup>þ</sup>*xs* . For *λ*≥2, the estimation variance *η* is infinity. For 0<*λ*<2, the variance will be finite. For *λ*> 1, we obtain *I f*ð Þ¼ *<sup>=</sup><sup>p</sup>* <sup>∞</sup> and *I f* <sup>2</sup> *<sup>=</sup>p*<sup>2</sup> � � <sup>¼</sup> <sup>∞</sup>. However, as seen in **Table 4**, the results of calculations for *N* ¼ 10000 allow us to make the conclusion below. If we have the pseudorandom generator of the high quality and a good *p x*ð Þ then we can calculate the very high dimensional integrals.


**Table 3.**

*The results of numerical calculations for the integral Eq. (19) with the density p x*ð Þ¼ *<sup>i</sup>* 2, 6*x*1,6 *<sup>i</sup> :*


#### **Table 4.**

*The results of numerical calculations for the integral Eq. (20).*
