A general weak law of large numbers for sequences of \(L^p\) random variables (Q6585111)

For some \(1\leq p\leq\infty\), let \(X_1, X_2, \dots\) be a sequence of \(L^p\) random variables. The author shows that if \((a_n)_{n\in\mathbb{N}}\) and \((b_n)_{n\in\mathbb{N}}\) are unbounded sequences of positive integers such that \(a_n^{-1}\sum_{i=1}^{b_n}|X_i|_{L^p}\to0\) as \(n\to\infty\) then\N\[\N\frac{1}{a_n}\sum_{i=1}^{b_n}(X_i-\mathbb{E}X_i)\stackrel{L^p}{\to} 0\N\]\Nas \(n\to\infty\), without assuming any particular dependence structure between the \(X_i\). Several examples and applications are also discussed.