[Math] Proving consistency for an estimator.

Question

Let $Y_1,Y_2,…,Y_n$ be a random sample of size $n$ from a normal pdf having $\mu=0$. Show that $S_{n}^{2} = \frac{1}{n} \sum_{i=1}^n {Y_{i}^{2}}$ is a consistent estimator for $\sigma^2= Var(Y)$.

Answer 1

Hints:

1. write the definition of the consistent estimator

2. compute the mean and the variance of $S^2_n$

3. apply Chebyshev's inequality to $X = S_n^2$