Finding the variance of a statistic. $X_1,\cdots,X_n$ are independent random variables from $N(\mu,\sigma^2)$ distribution. Define
$$T=\frac{1}{2(n-1)}\sum_{i=1}^{n-1}(X_{i+1}-X_i)^2$$
I have shown that it is an unbiased estimator of the variance. I need to compare its variance to that of the sample variance. Now how do I find $Var(T)$?
Finding $E(T^2)$ simply by squaring the above expression and then tking expectation is becoming very clumsy!

Simplifying the problem by writing $X_i = \mu + \sigma Y_i$ where $Y_1, Y_2, ...$ are IID $N(0,1)$ variables.
$$T = \frac{\sigma^2}{2(n-1)}\sum_{i=1}^{i=n-1}(Y_{i+1}-Y_i)^2 $$
$$\implies E[T^2] = \frac{\sigma^4}{4(n-1)^2}\{\sum_{i=1}^{i=n-1}E(Y_{i+1}-Y_i)^4+\sum_j\sum_{i \neq j}E(Y_{i+1}-Y_i)^2 E(Y_{j+1}-Y_j)^2\}$$
$$= \frac{\sigma^4}{4(n-1)^2} \{12(n-1)+4(n-1)(n-2)\} = \frac{\sigma^4}{(n-1)} \{n+1\}$$
$$\implies Var(T) = \frac{2\sigma^4}{n-1}$$
