convergence in mean
The sequence of \(X_n\) random variables converges in the \(r\) -th moment to \(X\) if \[ \lim_{n\rightarrow\infty}\mathbb{E}\left[| X_n - X |^r\right] = 0 \]
1. properties
For \(r \geq s \geq 1\), \[ X_n \overset{L^r}{\rightarrow} X \Rightarrow X_n \overset{L^s}{\rightarrow} X \]
2. examples
2.1. convergence in \(L_1\) does not imply convergence in \(L_2\)
Let the outcome space \(\Omega = [0,1]\) (see probability space): \[X_n(\omega) = \begin{cases} n & \text{if } \omega \leq \frac{1}{n^2}\\ 0 & \text{o.w} \end{cases} \] Let \(X=0\). It turns out that \(X_n \overset{L^1}{\rightarrow} X\) but \(X_n \overset{L^2}{\not\rightarrow} X\). We see that, \(\mathbb{E}[X_n] = \frac{1}{n}\) which goes to 0 as \(n\) goes to infinity, but \(\mathbb{E}[X_n^2] = 1\).