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

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.