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Journal Articles Communications in Mathematics Year : 2022

A General Weak Law of Large Numbers for Sequences of $L^{p}$ Random Variables

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Yu-Lin Chou
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Abstract

Without imposing any conditions on dependence structure, we give a seemingly overlooked simple sufficient condition for $L^{p}$ random variables $X_{1}, X_{2}, \dots$ with given $1 \leq p \leq +\infty$ to satisfy \[ \frac{1}{a_{n}}\sum_{i=1}^{b_{n}}(X_{i} - \mathbb{E} X_{i}) \overset{L^{p}}\to 0 \,\,\, \mathrm{as}\, n \to \infty, \] where $(a_{n})_{n \in \mathbb{N}}, (b_{n})_{n \in \mathbb{N}}$ are prespecified unbounded sequences of positive integers. Some unexpected convergences of sample means follow.
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Dates and versions

hal-03842221 , version 1 (07-11-2022)
hal-03842221 , version 2 (11-11-2022)

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Yu-Lin Chou. A General Weak Law of Large Numbers for Sequences of $L^{p}$ Random Variables. Communications in Mathematics, In press, Volume 31 (2023), Issue 1, ⟨10.46298/cm.10292⟩. ⟨hal-03842221v2⟩

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