# Simple Econometric question

From LLN, we have $$\text{plim} Y_n=\mu$$. Now you have $$W_n=Y^3_n$$, what is the probability limit of $$W_n$$?

Please help me answer this and thank you!

• Take a look at the continuous mapping theorem. – Matthew Gunn Feb 22 '19 at 23:48
• If you want to only find the probability limit, you can replace "plim" with "lim". Proving it involves probability theory. – chan1142 Feb 23 '19 at 8:35

## 1 Answer

The comments have the answer. Let $$\{X_n\}_{n=1}^{\infty}$$ be a sequence of iid random variables with $$E(X_1)=\mu$$ and $$Var(X_1)=\sigma^2$$ with $$\mu,\sigma^2<\infty$$. Define $$Y_n=\frac{1}{n}\sum_{i=1}^n X_i.$$ By WLLN, $$\text{plim}Y_n=\mu.$$ Since $$f(x)=x^3$$ is a continuous function, by the continuous mapping theorem, $$\text{plim}f(Y_n)=\text{plim}Y_n^3=f(\mu)=\mu^3.$$