Suppose that $X,X_1,X_2,\ldots$ are iid random variables with $\operatorname E|X|^p<\infty$. Does there exist a constant $C>0$ such that $$ P\biggl(\sum_{n=1}^\infty|X_n|I_{\{|X_n|>n^{1/p}\}}<C\biggr)=1, $$ where $I_A$ is the indicator function of an event $A$?
We have that $$ \sum_{n=1}^\infty P(|X|>n^{1/p})\le\operatorname E|X|^P<\infty. $$ Hence, by the Borel-Cantelli lemma, the series contains only a finite number of non-zero elements almost surely. There exists $\Omega_0\subset\Omega$ such that $P(\Omega_0)=1$ and for each $\omega\in\Omega_0$ we have $C(\omega)$ such that $$ \sum_{n=1}^\infty|X_n(\omega)|I_{\{|X_n|>n^{1/p}\}}(\omega)<C(\omega). $$ But this bound depends on $\omega$. If we took $C=\sup_{\omega\in\Omega_0}C(\omega)$, this bound might not be finite, right? So it seems that we cannot bound this series with probability $1$ by a single constant $C>0$. Is that corrrect?
Any help is much appreciated!
You are correct. There is no constant that bounds the sum for all $\omega$. The event \begin{align} |X_n| > n^{1/p} &&\forall n\le N \end{align} has positive probability for all $N\in\mathbb{N}$, lets call it $p_N$. Then for each bound $C$ there exists a $N>C$ for which there is a set of $\omega$'s of size at least $p_N$ such that $$ \sum_{n=1}^{\infty}|X_n|I_{\{X_n>n^{1/p}\}} \ge N > C. $$