I am reading a dissertation on quantum metrology in which the second chapter deals with the notion of statistical distance. Kullback–Leibler divergence (relative entropy) is introduced as the measure of distance between probability distributions. The author gives a lemma (based on previously given lemmas ad theorems) which asserts that the probability of a type (I think in most references this is referred to as "frequency") $\xi$ generated according to the distribution $p$ can be approximated for large number of trials $n$ as:
$$p(\xi) \approx Ce^{-nD(\xi || p)}$$
in which $D(\xi || p)$ is the relative entropy and $C$ is a constant. In order to define a Riemmanian metric on the space of probabilities the KL-divergence of $\xi=p+\delta p$ and $p$ is calculated which after approximation becomes:
$$D(p+\delta p||p) \approx \frac{1}{2} \sum_a \frac{(\delta p_a)^2}{p_a}$$
Plugging this approximation in the formula for the probability of type $\xi$ we get:
$$p(\xi) \propto exp(-\frac{n}{2}\sum_a \frac{(\delta p_a)^2}{p_a}) \qquad (*)$$
I understand everything up to this point. The author claims that for a neighboring probability vector $\tilde{p}=p+\delta p$ to be distinguishable from $p$, the probability of getting a type $\xi \approx \tilde{p}$ must be low. Hence, the neighboring states will be distinguishable if:
$$\sqrt{n}(\sum_a \frac{(\delta p_a)^2}{p_a})^{\frac{1}{2}}>1$$
and The region of probability space where, according to this equation, probabilities are not distinguishable, is called the region of uncertainty. I don't understand the last equation. We can rewrite the last equation as:
$$\frac{n}{2}\sum_a \frac{(\delta p_a)^2}{p_a}>\frac{1}{2}$$
So basically the exponent in the $(*)$ equation must be greater than $\frac{1}{2}$. I don't understand what is so special about $\frac{1}{2}$. My understanding is that if two probability distributions are distinguishable, their KL-divergence must be greater than zero. In which case the probability given by $(*)$ will be smaller than one. It is very likely that I don't see a very trivial thing here. Any help is appreciated.
P.S. for those of you who have access to ProQuest, the dissertation is titled "Nonlinear quantum metrology" by Sergio Boixo and my question is related to page 17 of this document.