Probability limit of X’X/n

Summary

• Given: a random sample \(\left( y_i,x_i \right)\) of size \(n\) where \(y_i\) is a scalar and \(x_i={\left( x_{i,1},x_{i,2}, \ldots ,x_{i,k} \right)}'\) is \(k×1\) .
• Result:

\[E\left( \frac{1}{n}X'X \right)=Σ_{xx'}\]

• where \(Σ_{xx'}=E\left( x_ix'_i \right)\)
• Results: The variance of every element in \( \frac{1}{n}X'X\) will go to zero as \(n→∞\) .
• Result:

\[plim \frac{1}{n}X'X=Σ_{xx'}\]