# Checking the consistency of the statistic

Let $$X_1, X_2,...,X_n$$ be n random samples from $$N(\theta, \theta^2)$$. We have a statistic $$T = \sum Xi^2$$. I need to prove the consistency of $$\frac{T}{2n}$$ for estimating $$\theta^2$$.

I know that There are two criterion that I need to test here. First that for larger n, we need to prove the statistic as unbiased. Also, its variance should asymptotically converge to 0.

Now, from the $$E(X^2) = 2\theta^2$$, we can clearly see that T is unbiased for $$\theta^2$$. For obtaining variance, I can do:

$$V(\frac{T}{2n}) = \frac{1}{4n^2}V(T) = \frac{1}{4n^2}V(\sum Xi^2) = \frac{n}{4n^2}V(X^2) = \frac{V(X^2)}{4n}$$ which clearly converges to 0. Here, I have not obtained the variance part to show the consistency. Is it the right approach?

As long as $$V(X^2)$$ is bounded, you don't need to explicitly calculate it. And for normal random variables, it is bounded.