# Why the standard errors of logistic regression are of that form?

In Elements of Statistical Learning, page 125 it is written that the coefficients of a logistic regression converges to $$\mathcal{N}(\beta, (X^T W X)^{-1})$$ with:

• $$X$$ the $$N \times (p+1)$$ matrix of data
• $$W$$ a $$N \times N$$ diagonal matrix of weights with $$i$$th diagonal element $$p(x_i;\beta)(1-p(x_i; \beta))$$ where by definition $$p(x_i;\beta) = \exp(\beta^T x_i)/(1+\exp(\beta^T x_i))$$.

Is there a proof/intuition for the $$(X^T W X)^{-1}$$?