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