On the law of large numbers for martingales
The author establishes certain constraints under which sufficient conditions in the law of large numbers for a scheme of series of martingales are necessary. In particular, for an array of martingale differences \({\{\xi_{in},\mathcal{F}_{in}\}_{1\leq i\leq k_n, n\geq 1}}\) with \({|\xi_{in}|\leq K}\) for some \(K>0\) and each \(i\) and \(n\), it is obtained that the following conditions (convergence in probability \(\operatorname{P}\)) are equivalent: {\parindent=6mm \begin{itemize} \item[(1)] \({\sum_{i=1}^{k_n}\xi^2_{in}\to 0}\) as \({n\to\infty};\) \item [(2)] \({\sum_{i=1}^{k_n}\operatorname{E}(\xi^2_{in}|\mathcal{F}_{(i-1)n})\to 0}\) as \({n\to\infty};\) \item [(3)] \({\sum_{i=1}^{k_n}\xi_{in}\to0}\) as \({n\to\infty}\) and \({\sqrt{a}\varlimsup_{n\to\infty}\operatorname{P}(\sum_{i=1}^{k_n}\operatorname{E}(\xi^2_{in}|\mathcal{F}_{(i-1)n})>a)=o(1)}\) as \({a\to\infty}.\) \end{itemize}} The author points out in an example that the constraint of the form \[ \sqrt{a}\varlimsup_{n\to\infty}\operatorname{P}(\sum_{i=1}^{k_n}\operatorname{E}(\xi^2_{in}|\mathcal{F}_{(i-1)n})>a)=o(1)\;\;\text{as} \;\;{a\to\infty} \] cannot be omitted.
scientific article; zbMATH DE number 6473892