### On convergence of series of random elements via maximal moment relations with applications to martingale convergence and to convergence of series with $p$-orthogonal summands.

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This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space ${L}^{2}(\Omega ,\mathcal{F},\mathbb{P})$ of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of ${L}^{2}(\Omega ,\mathcal{F},\mathbb{P})$) to be orthogonal to some other sequence in ${L}^{2}(\Omega ,\mathcal{F},\mathbb{P})$. The result obtained is interesting...

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of $m$-linearly negative quadrant dependent random variables ($m=1,2,\cdots $). For a sequence of $m$-linearly negative quadrant dependent random variables $\{{X}_{n},n\ge 1\}$ and $1<p<2$ (resp. $1\le p<2$), conditions are provided under which ${n}^{-1/p}{\sum}_{k=1}^{n}({X}_{k}-E{X}_{k})\to 0$ in ${L}^{1}$ (resp. in ${L}^{p}$). Moreover, for $1\le p<2$, conditions are provided under which ${n}^{-1/p}{\sum}_{k=1}^{n}({X}_{k}-E{X}_{k})$ converges completely to $0$. The current work extends some results of Pyke and Root (1968) and it extends and improves some...

The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$-linearly negative quadrant dependent random variables”.

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