Let $\{X_1,...,X_n\}$ iid r.v. We denote $p_X(x)=P_{X_i}(x)$ the PDF of $X_i$ for all $i$. In the section 2.3 page 12 of this paper, we have
Let $p_N(s\mid X_i=x)$ is the probability that given $X_j=x$, there is a r.v. variable $X_k$ ($k\neq j$) s.t. $X_{k}=x+s$ and there is no other r.v. lie between. We can prove that $$p(s\mid X_j=x)=p_X(x+s)[1+F(x)-F(x+s)]^{N-2},$$ where $F$ is the CDF of the $X_i$'s. From this, we can conclude that $$p(s\mid any\ X=x)=\sum_{j=1}^N p_N(s\mid X_j=x)\mathbb P\{X_i=x\}=Np_N(s\mid X_i=x)p_X(x),$$ and finally $$p_N(s)=N\int_\sigma p_N(s\mid X_j=x)p_X(x)dx,$$ where $\sigma $ is the support of the $X_i's$.
Now, we are interested is to compute the spacing distribution function when we have an infinite number of r.v. ,i.e. for $\{X_i\}_{i=1}^\infty$ iid. What they do is that they introduce the substitution $s=\frac{\hat s}{Np_X(x)}$, and says that this distribution is given by $$p(s)=\lim_{n\to \infty}\hat p_N(\hat s)=\lim_{N\to \infty}p\left(s=\frac{\hat s}{Np_X(x)}\right)\frac{ds}{d\hat s}=...=e^{\hat s}.$$
Question : I don't understand the argument here !
1) First, why the scaling $\frac{\hat s}{Np_X(x)}$ ?
2) What does mean $\lim_{N\to \infty}p\left(s=\frac{\hat s}{Np_X(x)}\right)\frac{ds}{d\hat s}$ ? I guess it's $$\lim_{N\to \infty}p_N\left(\frac{\hat s}{Np_N(x)}\right)\frac{d}{d\hat s}\left(\frac{\hat s}{Np_X(x)}\right)=\lim_{N\to \infty}p_N\left(\frac{\hat s}{Np_N(x)}\right)\frac{1}{Np_X(x)},$$
but I've never see such a thing : why introducing the term $\frac{d s}{d\hat s}$ ?