Reading the book of probability of Achim Klenke I came across the assertion that if $(X_n)$ is a sequence of independent real valued random variables then the limit superior and inferior of the sequence of Cesáro means defined by $S_n:=\frac1n \sum_{k=1}^n X_k$ are almost surely constant, but I cannot see why.
I know that the limit superior and inferior of $(X_n)_n$ are almost surely constant and I already knows that
$$
\liminf X_n \leqslant \liminf S_n\leqslant \limsup S_n\leqslant \limsup X_n
$$
but from here I cannot see why then must be also surely constant $\liminf S_n$ or $\limsup S_n$. The book just assert that but doesn't gives a proof (just gives a proof for the case of $\liminf X_n$). Can someone show me or give me a reference about how to prove this assertion?
Note: at this point of the book there is no central limit theorem neither theorems related to convergence or weak or strong law of large numbers, the major theorem at this point is the Kolmogorov 0-1 law (that is: the tail $\sigma$-algebra of independent $\sigma$-algebras is $P$-trivial).
Best Answer
If $T_n=\frac1{n}\sum_{k=2}^nX_k$ then $|S_n-T_n|\leq\frac1{n}|X_1|$.
From this it follows that for every $\omega\in\Omega$ we will have: $$\limsup S_n(\omega)=\limsup T_n(\omega)$$
In short we have $\limsup S_n=\limsup T_n$ so that we may conclude that $\limsup S_n$ is measurable wrt to $\sigma\{X_2,X_3,\dots\}$.
This can easily be made broader to find that $\limsup S_n$ is measurable wrt to $\sigma\{X_k,X_{k+1},\dots\}$ for every positive integer $k$.
So $\limsup S_n$ is measurable wrt to $\bigcap_{k=1}^{\infty}\sigma\{X_k,X_{k+1},\dots\}$ and it is well known that all sets in that tail $\sigma$-algebra have probability $0$ or $1$.
Consequently random variables that are measurable wrt to it are degenerated.