Let $X_{1}, X_{2}, X_{3}, ...$ be a sequence of iid and integrable random variables with $E[X_{1}] = \mu$. Show that $$ \frac{1}{n} (X_{1}+X_{2}+X_{3}+...+X_{n}) \to \mu $$ in $L^{1}$.

My thoughts so far: Strong law of large numbers implies almost surely convergence. So I still need to get from a.s. convergence to $L^{1}$ convergence. In my lectrure notes I have found two possible ways of doing so:

(a) If $\quad$ $X_{n} \to X \; a.s.$ $\quad$ and $\quad$ $E(|X_{n}|^{p}) \to E(|X|^{p})$ $\quad$ then $\quad$$X_{n} \to X$ in $L^{p}$.

(b) If $\quad$ $X_{n} \to X \;$ in prob. $\quad$ and $\quad$ $\{X_{n} : n \in \mathbb{N}\}$ is UI $\quad$ then $\quad$$X_{n} \to X$ in $L^{1}$.

Which is more promising? I tried both but...it didnt really work out...

Many thanks in advance.

  • $\begingroup$ What didn't work with (b)? $\endgroup$ – kimchi lover Nov 10 '17 at 23:36
  • $\begingroup$ Well i didnt really know how to show that the set of random variables is UI... $\endgroup$ – u49K3df2 Nov 10 '17 at 23:52
  • $\begingroup$ The iid hypothesis makes it trivial. For each $\epsilon>0$ there exists a $K$ such that the nice thing happens for all $X_n$. $\endgroup$ – kimchi lover Nov 11 '17 at 0:01
  • $\begingroup$ Thanks for your help...I am new to this forum and quite new to probability theory...So for now please excuse me if I ask trivial questions... $\endgroup$ – u49K3df2 Nov 12 '17 at 20:26
  • $\begingroup$ I didn't mean to put you down: I just meant, this step is easy. Are you OK with your original question? Did the duplicate existing question & answer satisfy you? $\endgroup$ – kimchi lover Nov 13 '17 at 15:26