# Distribution of $n^{1/2}\{\hat T_n−T_n(F)\}$ in bootstrap problems

I've read this in a paper, and I don't know how to proof the last statement:

Let $$X_1,...,X_n$$ be independent identically distributed random variables with unkown distribution function F. Suppose the $$\{\hat T_n=\hat T_n(X_1,...,X_n;n≥1\}$$ are real-valued statistics and the $$\{T_n(F);n≥1\}$$ are real-valued functionals such that the asymptotic distribution of $$n^{1/2}\{\hat T_n−T_n(F)\}$$ is normal with mean zero. Let $$H_n(x,F)$$ be the exact distribution function of $$n^{1/2}\{\hat T_n−T_n(F)\}$$. A basic problem in statistics is the estimation of $$H_n(x,F)$$ or functionals of $$H_n(x,F)$$ from the sample. Indeed, the mean and variance of $$H_n(x,F)$$ are, respectively, the bias and variance of $$\hat T_n$$ when $$\hat T_n$$ is regarded as an estimate of $$T_n(F)$$.

Why the variance of $$H_n(x,F)$$ is the same as the one of $$\hat T_n$$ when $$\hat T_n$$ is regarded as an estimate of $$T_n(F)$$?

This is an argument that I've seen in some books dealing with the bootstrap resampling theory, and I can't figure out why that is true.

Thanks