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

cm:10292 - Communications in Mathematics, November 21, 2022, Volume 31 (2023), Issue 1 - https://doi.org/10.46298/cm.10292
A General Weak Law of Large Numbers for Sequences of $L^{p}$ Random Variables

Authors: Yu-Lin Chou

    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.


    Volume: Volume 31 (2023), Issue 1
    Published on: November 21, 2022
    Accepted on: November 10, 2022
    Submitted on: November 10, 2022
    Keywords: convergence in probability,$L^{p}$-convergence,laws of large numbers,[MATH]Mathematics [math]

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