# proof of consistency of sample covariance

• Sep 15th 2012, 01:21 AM
waytogo
proof of consistency of sample covariance
Hi, folks!

Here is what i am dealing with:

Assume $\displaystyle (X_1,Y_1),(X_2,Y_2),...,(X_n,Y_n) - iid$.
I am asked to prove that the sample estimator $\displaystyle \hat{\sigma}_{xy}=\frac{1}{n-1}\sum_{i=1}^n{(X_i-\bar{X})(Y_i-\bar{Y})}$ is consistent, i.e. $\displaystyle \lim_{n\to\infty}P(\mid \hat{\sigma}_{xy}-\sigma_{xy}\mid>\epsilon)=0$.
I am trying to apply Markov's inequality:$\displaystyle P(\mid \hat{\sigma}_{xy}-\sigma_{xy}\mid>\epsilon)<\frac{E\mid\hat{\sigma}_ {xy}-\sigma_{xy}\mid}{\epsilon}$.

So, i get
$\displaystyle E\mid \frac{1}{n-1}\sum_{i=1}^n{(X_i-\bar{X})(Y _i-\bar{Y})}-E(\sum_{i=1}^n{(X_i-EX_i)(Y_i-EY_i)) \mid = E \mid \frac{1}{n-1}\sum_{i=1}^n{(X_iY_i)-E(\sum_{i=1}^n{X_i Y_i})-(\frac{n}{n-1}\bar{X}\bar{Y}-nEX_i EY_i)\mid$.

My intention was to use Law of Large Numbers somewhere, but I can't see how.
Can someone give a hint or if i am wrong until this point show the correct way?