My question:
$\{X_n\}$ is a sequence of random variables. Var$(X_n)\le C\ \ \forall \ n$ and $\rho_{ij}=$Cov$(X_i,X_j)\to 0 $ as $|i-j|\to \infty$ . Show WLLN holds.
In my book there are 3 theorems , and everyone of them includes some kind of expectation calculation or involvement(like finite mean).And moreover none has any association with covariance. How am I to solve?
One can try to show the convergence in $\mathbb L^2$. For $n$ greater than some fixed $R$,
$$\mathbb E\left[\frac 1{n^2}\left(\sum_{i=1}^nX_i\right)^2\right]=\frac 1{n^2}\sum_{i,j=1}^n\rho_{i,j}=\frac 1{n^2}\sum_{i,j=1}^n\rho_{i,j}\mathbf 1\{|i-j|\leqslant R\}+\frac 1{n^2}\sum_{i,j=1}^n\rho_{i,j}\mathbf 1\{|i-j|\gt R\}. $$ The first sum can be estimated in the following way: by Cauchy-Schwarz inequality, $\rho_{i,j}\leqslant C$ for each $i,j$ and the number of $(i,j)$ with $i,j\in\{1,\dots,n\}$ and $|i-j|\leqslant R$ is of order $nR$.
For the second sum, note that $\rho_{i,j}\mathbf 1\{|i-j|\gt R\}\leqslant \sup_{|u-v|\gt R}\rho_{u,v}$ hence the second sum does not exceed $\sup_{|u-v|\gt R}\rho_{u,v}$.