Nonstationary law of large numbers for dependent random variables and its application in stochastic optimization (Q1288664)

The following outset is considered for laws of large numbers for dependent random variables. Let \((\Omega,\Sigma,{\mathbb P})\) be a probability space with a nondescending flow of \(\sigma\)-algebras \({\mathcal F}_k\subseteq {\mathcal F}_{k+1}\subseteq\cdots\subseteq \Sigma\), \(k=1,2,\ldots\), and assume that the random variables \(\zeta_k(\omega): \Omega\to \mathbb{R}^n\) are \({\mathcal F}_k\)-measurable, \(k=1,2,\ldots \). Denote the nonstationary conditional expectations by \(z_k(\omega)={\mathbb E}\{\zeta_k(\omega)\mid{\mathcal F}_{k-1}\}\). Consider the sequence of estimates (\(k=0,1,2,\ldots\)) \[ \overline{\zeta}_{k+1}(\omega)=\Pi_Z(\overline{\zeta}_{k}(\omega)-\sigma_k(\overline{\zeta}_{k}(\omega)-\zeta_{k}(\omega))),\quad \overline{\zeta}_1=0, \] and the auxiliary sequence of random variables \[ \overline{z}_{k+1}(\omega)=\Pi_Z(\overline{z}_{k}(\omega)-\sigma_k(\overline{z}_{k}(\omega)-z_{k}(\omega))),\quad \overline{z}_1=0, \] where \(Z\) is a convex set from \(R^n\), \(\Pi_Z\) is the projector onto the set \(Z\), the random variables \(\sigma_k\) are \({\mathcal F}_{k-1}\)-measurable and satisfy the conditions \[ 0\leq\sigma_k\leq 1,\quad \lim_k\sigma_k=0,\quad \sum_{k=1}^\infty\sigma_k=\infty\;\text{ a.s.},\quad \sum_{k=1}^\infty{\mathbb E}\sigma_k^{1+\varepsilon}\|\zeta_k(\omega)-z_k(\omega)\|^{1+\varepsilon}\leq C<\infty \] for some \(\varepsilon\), \(0\leq\varepsilon\leq 1\). The following version of the strong law of large numbers for dependent random variables is proved: Under the above conditions \(\lim_k(\overline{\zeta}_k(\omega)-\overline{z}_k)=0\). In particular, if \(\zeta_k: \Omega\to Z\) and \(\sigma_k=1/k\), then \(\lim_k(1/k)\sum_{s=1}^k(\zeta_s(\omega)-z_s(\omega))=0\) a.s. Under more intricate conditions, other theorems of the same flavor are proved. The motivation of these investigations, the methods used to prove theorems, and their possible applications are related with the stochastic optimization.