Meaning of a liminf and limsup of a sum

Question

Consider a probability space $(\Omega, \mathcal{A}, P)$, a sequence of random variables $(X_n)_{n\in\mathbb{N}}$ and the expressions $$\liminf_p \sum_{k=0}^p X_k(\omega), \quad \limsup_p \sum_{k=0}^p X_k(\omega)$$ In what way does it make sense to interpret them? I thought e.g. the $\limsup$ would simply be $$ \inf_p \sup_{n>p} \sum_{k=0}^n X_k(\omega)$$ but does it make sense? Is the sequence $$\sup_{n>p} \sum_{k=0}^n X_k(\omega)$$ decreasing in $p$ so that it makes sense to take the $\inf$?

Answer 1

Yes, your interpretation is right and the sequence is decreasing in $p$. It works the same way as for a general sequence.

Let $p_2 \geq p_1$. Then

$$ \sup_{n > p_2} \sum_{k = 0}^n X_k(\omega) = \sup \{ \sum_{k = 0}^n X_k(\omega): n > p_2 \} \leq \sup \{ \sum_{k = 0}^n X_k(\omega): n > p_1 \} = \sup_{n > p_1} \sum_{k = 0}^n X_k(\omega) $$

since $$ \{ \sum_{k = 0}^n X_k(\omega): n > p_2 \} \subseteq \{ \sum_{k = 0}^n X_k(\omega): n > p_1 \} $$

In general the $\limsup$ is the $\lim_{p \rightarrow \infty} \sup_{n > p} x_n$, but since $\{\sup_{n > p} x_n\}_{p \in \mathbb{N}}$ is non-increasing, this limit happens to coincide with the $\inf_{p}$