I have a question on integral rules. Consider the following integral $$ R_J\equiv \int_{-\infty}^{\infty} \Big[\exp\Big(\frac{1}{J}\sum_{k\in \{1,...,2J\}}J\log\Big(G\Big(s-\frac{1}{\sigma_J}(u_{ij}-u_{ik})\Big)\Big)\Big)\Big] \frac{g(s)}{G(s)}ds $$ It is an apparently very complicated object and I don't think that my question requires to explain all the notation. Two are the things to notice
1) the integral is with respect to $s$
2) There as some components in the integrand that depends on $J$ and, indeed, my ultimate goal is to study the behaviour of $R_J$ as $J\rightarrow \infty$.
A book that I am considering proceeds by performing this change of variables (which later turns out to be useful to study the limiting behavior): consider $b_J\in \mathbb{R}$ and $a_J>0$ and impose $$t=\frac{s-b_J}{a_J} \Leftrightarrow s=a_Jt+b_J$$
It follows that $$ R_J=\int_{-\infty}^{\infty} \Big[\exp\Big(\frac{1}{J}\sum_{k\in \{1,...,2J\}}J\log\Big(G\Big(a_Jt+b_J-\frac{1}{\sigma_J}(u_{ij}-u_{ik} \Big)\Big)\Big)\Big] \frac{a_J g(a_Jt+b_J)}{G(a_Jt+b_J)}dt $$
My question is: under which rule/assumptions we can do that? My confusion stems from the fact that we are adding components that depend on $J$. Can we do that?
This is a standard change of variable, of the form
$$R:=\int f(s)\,ds=\int a\,f(at+b)\,dt.$$ or, in your context,
$$R_J:=\int f(s,J)\,ds=\int a_J\,f(a_Jt+b_J,J)\,dt.$$
The fact that $a_J,b_J$ and the function $f$ are depending on the parameter $J$ is irrelevant, because $J$ remains constant "during" the integration.
Now knowing the expression of $R_J$ (as a function of $J$), you can evaluate
$$\lim_{J\to\infty}R_J.$$