In some urn model with parameter $p$, the generating function
$$
f\_p(z) \;=\; \frac{1+p\,z}{1-(1-p)\,z\,(1+p\,z)}
$$
is such that $[z^n]f\_p(z)$ is the probability that an $n$-urn configuration has a particular property.


It is known that this probability tends to zero if $p\gg n^{-1/2}$ and tends to one if $p\ll n^{-1/2}$, so $n^{-1/2}$ is a threshold for the property.


The following approach appears to give the correct probability for the property when $p=c\,n^{-1/2}$:


Let $\rho(p)$ be the dominant singularity of $f\_p$ (a simple pole). Then,
$$
\lim\_{n\to\infty}\rho\big(c\,n^{-1/2}\big)^{-n} \;=\; e^{-c^2} .
$$
Obviously, for fixed $p$, we have $[z^n]f\_p(z)\sim c\_p\,\rho(p)^{-n}$, for some (computable) constant $c\_p$, but substituting $p=c\,n^{-1/2}$ and then taking limits seems rather dubious.


**Question:** Can singularity analysis be used in this way? If so, under what conditions, and where can I find the appropriate justification?


[Of course, the specific details of this urn model and property aren't relevant; it's the general method I'm asking about.]



Using singularity analysis for probability at a threshold?
There are a few things working in this particular question that allow for that to occur. This is a rational function and the denominator is a quadratic in $z$ meaning that we can write $$[z^n] f\_p(z) = c\_1 \rho\_1^{-n} + c\_2 \rho\_2^{-n}$$ where $\rho\_j$ are the roots of the denominator and $c\_j$ are functions of $p$.


In the case at hand, we can substitute $p = c/\sqrt{n}$ and then see if the limit of this exact expression exists. Yes, $\rho\_2$ is larger than $\rho\_1$, but it could *a priori* be the case that $c\_2 \rho\_2^{-n}$ does not tend to $0$ as $n \to \infty$. In this concrete example, you can compute that $c\_2 \to 0$ and $\rho\_2^{-n} \to 0$ for $p = c/\sqrt{n}$ and $n \to \infty$.


Similarly, we need to understand the limiting behavior of $c\_1$. *A priori*, it could be the case that $c\_1 \to 0$ or something strange, but here it can be shown that $c\_1 \to 1$ in the regime listed.


It could be possible that there are more general conditions that guarantee that $c\_1 \not\to 0$ and similarly to deduce behavior on $c\_2 \rho\_2^{-n}$; nothing I've written---or you used in your post---takes into account that this is actually a bivariate rational function, just that it's rational in $z$. Regardless, some info about $c\_1$ has be taken into account, as we could just multiply the generating function by $2$ and alter the asymptotics by a constant factor.


