Applying Minkowski's Inequality to Show Convergence of $\overline X_n$ in the $L^p$ spaces I believe that I have been having trouble figuring out the proper first initial step to realizing the following problem. Any suggestions upon how to begin would be very much so appreciated. Here's the problem: 
Assume that $X_1, X_2, \dots$ are an i.i.d. random variables having $E(|X_j|^p) < \infty$ for $1 < p \leq 2$, and $\mu = E(X_j)$. 
Show that $\overline X_n = \frac{X_1 +\dots + X_n}{n} \rightarrow \mu$ in $L^{p}$ as $n \rightarrow \infty$.
 A: This can be solved directly using von Bahr and Esseen's inequality. Here is an alternative approach, by truncation.
Let $R$ be arbitrary but fixed and define $$X'_i:= X_i\mathbf 1\left\{\left|X_i\right|\leqslant R\right\}-\mathbb E\left[X_i\mathbf 1\left\{\left|X_i\right|\leqslant R\right\}\right]\mbox{ and }  $$
$$X''_i:= X_i\mathbf 1\left\{\left|X_i\right|\gt R\right\}-\mathbb E\left[X_i\mathbf 1\left\{\left|X_i\right|\gt R\right\}\right].$$
In this way, we have $X_i-\mu=X'_i+X''_i$ for each $i$. Therefore,
$$\left|\overline{X_n}-\mu\right|\leqslant \frac 1n\left|\sum_{i=1}^nX'_i  \right|+\frac 1n\left|\sum_{i=1}^nX''_i  \right|,$$
and using the elementary inequality $(s+t)^p\leqslant 2^{p-1} \left(s^p+t^p\right)$, we get 
$$\mathbb E\left[ \left|\overline{X_n}-\mu\right|^p\right]\leqslant \frac{2^{p-1}}n\mathbb E\left[ \left|\sum_{i=1}^nX'_i  \right|^p\right]+\frac{2^{p-1}  } n\mathbb E\left[\left|\sum_{i=1}^nX''_i  \right|^p\right].$$
Notice that by Jensen's inequality applied to the function $t\mapsto t^{2/p}$ , 
$$\mathbb E\left[ \left|\sum_{i=1}^nX'_i  \right|^p\right]\leqslant \left(\mathbb E\left[ \left|\sum_{i=1}^nX'_i  \right|^2\right]\right)^{2/p}.  $$Now the remaining tasks are the following:


*

*Show that $$\lim_{n\to +\infty}\frac 1n\left(\mathbb E\left[ \left|\sum_{i=1}^nX'_i  \right|^2\right]\right)^{2/p}=0.$$

*Use Minkowski's inequality to bound $\mathbb E\left[\left|\sum_{i=1}^nX''_i  \right|^p\right]/n$
by a constant independent of $n$ and $R$ times $\mathbb E\left[\left|X_1\right|^p\mathbf 1\left\{\left|X_1\right|\gt R\right\}\right]$.

*Conclude.   

