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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?

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

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