“Monte-Carlo method” Consider the function \(f(x_1, \ldots, x_n)\) defined in \(V = \{-1 \leqslant x_i \leqslant 1,\ i = 1, \ldots, n\}\) and bounded from below and above, \(|f(x_1, \ldots, x_n)| \leqslant C\). Define a random variable \(\eta = f(\xi_1, \ldots, \xi_n)\), where \(\xi_i\) are distributed uniformly between \(-1\) and \(1\).
Show that \(\mathbf{E}\eta = I\), where \(I = \int_V f(x_1, \ldots, x_n)\, d^n x\). Hence the random variables can be used to compute the high-dimensional integrals with a given precision.
Consider the series of \(N\) random variables \(\eta_i = f(\xi_{i1}, \ldots \xi_{in})\), \(i = 1, \ldots, N\) where all \(\xi_{ij}\) are distributed uniformly in \([-1, 1]\). Then the quantity \(\tilde{I} = \frac{1}{N}(\eta_1 + \ldots + \eta_N)\) for large \(N\) approaches \(I\) with high enough probability. Estimate how large \(N\) should be to ensure that \[ P(|\tilde{I} - I| < \Delta) \geqslant 1 - \alpha, \tag{53} \] where \(\Delta\) and \(\alpha\) are given small numbers.