# limsup of a sequence of random variables (definition)

Let $X_n$ be a sequence of random variables.

First, $\limsup X_n=\inf_n\{\sup_{m\ge n}X_m\}$. So,

$$\{\limsup X_n\le c\}=\bigcup_n\bigcap_{m\ge n}\{X_m\le c\}$$

Is it correct to say that if $P\{X_n\le c \text{ i.o.}\}\equiv P\{\limsup\{X_n\le c\}\}=1$ then $\limsup X_n\le c \text{ a.s.}$?

Edit 1:

By the same way

$$\{\liminf X_n\ge c\}=\bigcup_n\bigcap_{m\ge n}\{X_m\ge c\}$$

So, $P\{X_n\ge c \text{ i.o.}\}=1 \Rightarrow \liminf X_n\ge c \text{ a.s.}$

Edit 2:

$$[X_n\geqslant c\ \text{i.o.}]\subseteq[\limsup X_n\geqslant c] \text { and } [X_n\leqslant c\ \text{i.o.}]\subseteq[\liminf X_n\leqslant c]$$
• $P\left\{ X_{n}\geq c\: i.o\right\} \Rightarrow$... How can a value on its own imply something? – drhab Oct 4 '14 at 9:59
• The third line is wrong. In general, $$\{\limsup X_n\le c\}\ne\bigcup_n\bigcap_{m\ge n}\{X_m\le c\}=\liminf\{X_n\le c\}.$$ (I corrected the typo $X_n$ for $X_m$.) It is not even always true that $$\{\limsup X_n\le c\}=\limsup\{X_n\le c\}.$$ For example, if $X_n(\omega)=c+1/n$ for every $n$, then $\limsup X_n(\omega)=\lim X_n(\omega)=c$ hence $\omega$ is in the event $\{\limsup X_n\le c\}$ but $X_n(\omega)\gt c$ for every $n$ hence $\omega$ is neither in the event $\limsup\{X_n\le c\}$ nor in the event $\liminf\{X_n\le c\}$. – Did Oct 4 '14 at 10:36