Plim-rules

Summary

• \(y_1,y_2,…\) is a sequence of random variables such that \(\textrm{plim } y_n=a\)
• \(z_1,z_2,…\) is a sequence of random variables such that \(\textrm{plim } z_n=b\)
• \(c,d\) are a constant s
• \(g: R→R\) is a function

Then

• \(\textrm{plim } (c+dy_n)=c+d \textrm{plim } (y_n)= c+da\)
• \(\textrm{plim } (g\left( y_n \right))=g\left( \textrm{plim } (y_n) \right)=g\left( a \right)\)
• \(\textrm{plim } (y_n+z_n)=\textrm{plim } (y_n)+\textrm{plim } (z_n)=a+b\)
• \(\textrm{plim } (y_n⋅z_n)=\textrm{plim } (y_n)⋅\textrm{plim } (z_n)=ab\)

Example

• \(x_1,x_2,...,x_n\) is an IID random sample where \(E\left( x_i \right)=μ\) and \(Var\left( x_i \right)=σ^2\) . Then

\[\textrm{plim } {\bar{x}}_n^2=μ^2\]

• even though \(E\left( {\bar{x}}_n^2 \right)≠μ^2\) .