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The problem:

We have {$X_n$}, n$\ge$1 - a sequence of independent and equally distributed random variables with finite E|$X_1$|. We need to prove that $E|\frac{S_n}{n} - E$X_1$|\rightarrow 0$, where $S_n = \sum_{k=1}^{n}{X_k}$.

My attempts at solution:

In my understanding, we have convergence in probability and have to probe convergence in mean. I was told to use the definition of expected value and divide the expression into two integrals by multiplying it on I{$\frac{S_n}{n} < n$} + I{$\frac{S_n}{n} \ge n$}. In my understanding, I'll need to use the following property: If $E|X|<\infty$, then $E(|X|I_A)\rightarrow 0$ when $P(A)\rightarrow 0$

What help do I need.

Despite all that, I still feel that I don't understand something about this problem. It feel very easy, but every proof I write is inconclusive. I would be very grateful if someone pointed out what do I missing and how to finally solve this problem, using the definition of expected value.

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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 24 '23 at 16:14
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    Done - hope now it's clearer. –  May 24 '23 at 16:48
  • See here: https://math.stackexchange.com/questions/4620648/let-x-1-x-2-x-3-be-a-sequence-of-independent-and-equally-distributed-ra?rq=1 – Math1000 May 27 '23 at 06:40

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