Strong laws for weighted sums of independent identically distributed random variables
The following multidimensional form of Kolmogorov's strong law of large numbers is established. Theorem 1. Let \(H\) be a positive integer. Given an i.i.d. sequence \((X_{1,n})_n\) on the probability measure space \((\Omega_1, F_1, \mu_1)\) satisfying \(E(|X_{1,1}|)<\infty\), it is possible to find a set of full measure \(\widetilde\Omega_1\) such that if \(x_1\in \widetilde\Omega_1\), the following holds: (1) For all probability measure spaces \((\Omega_2, F_2, \mu_2)\) and all i.i.d. sequences \((X_{2,n})_n\) such that \(E(|X_{2,1}|)<\infty\), it is possible to find a set of full measure \(\widetilde\Omega_2\) such that if \(x_2\in\widetilde\Omega_2\) the following holds: (2) \dots \dots (\(H-1\)) For all probability measure spaces \((\Omega_H, F_H, \mu_H)\) and all i.i.d. sequences \((X_{H,n})_n\) such that \(E(|X_{H,1}|)<\infty\), it is possible to find a set of full measure \(\widetilde\Omega_H\) for which, if \(x_H\in\widetilde\Omega_H\) we have \[ \frac{1}{N}\sum_{n=1}^{N} X_{1,n}(x_1)X_{2,n}(x_2)\cdots X_{H,n}(x_H)\to \prod_{i=1}^H E(X_{i,1}) \qquad\text{as}\quad N\to\infty. \] The main difficulty in the proof of this theorem is that at each stage the null set does not depend on the incoming i.i.d. sequences and associated probability spaces. Further ergodic dynamical systems are considered.
scientific article; zbMATH DE number 1042481