I am going through this paper on the proof (or better conditions) for convergence of CARLA system by Rodriguez et al (2011). Since I do not come from a pure mathematical background, I need some assistance in understanding parts of the proof or some direction towards understanding the proof, as put forth by the author. I'll try to be brief so as to get to the point but feel free to ask me to add parts that may help you answer the question better. The bold text is from the paper and then italic texts are my comments/observations/questions.
We start from certain conditions which we hereby assume to be correct. Given a pdf $f_t(a)$ at time $t$ defined over the action space $A$ such that $a \in A$. Given, the sequence of pdfs generated over time is a Markovian process and the expected value of $f_{t+1}(a)$ is
\begin{equation} \bar{f}_{t+1}(a) = f_t(a) \left( \bar{\gamma}_t + \dfrac{\alpha \hspace{1mm} \bar{G}_t(a)}{f_t(a)} \right) \end{equation}
where $\bar{G}(a) = \int_{-\infty}^{+\infty} f_t(z) \hspace{1mm} \beta_t(z) \hspace{1mm} e^{\dfrac{1}{2} \left( \dfrac{a - z}{\lambda} \right)} dz$
From the above equation, the following holds for the first derivative of this pdf function
\begin{equation} \begin{matrix} \dfrac{\partial f_t}{\partial t} < 0 & \text{for} \left( \bar{\gamma}_t + \dfrac{\alpha \hspace{1mm} \bar{G}_t(a)}{f_t(a)} \right) < 1 \end{matrix} \end{equation}
\begin{equation} \begin{matrix} \dfrac{\partial f_t}{\partial t} = 0 & \text{for} \left( \bar{\gamma}_t + \dfrac{\alpha \hspace{1mm} \bar{G}_t(a)}{f_t(a)} \right) = 1 \end{matrix} \end{equation}
\begin{equation} \begin{matrix} \dfrac{\partial f_t}{\partial t} > 0 & \text{for} \left( \bar{\gamma}_t + \dfrac{\alpha \hspace{1mm} \bar{G}_t(a)}{f_t(a)} \right) > 1 \end{matrix} \end{equation}
Here,
$\bar{\gamma}_t$ is constant $\forall a \in A$ and
$\int_{-\infty}^{+\infty}f(z)dz = 1$.
This implies
$\exists_{b_1,b_2 \in A}$ s.t. $\dfrac{\bar{G}_t(b_1)}{f_t(b_1)} \neq \dfrac{\bar{G}_t(b_2)}{f_t(b_2)}$
$\implies \exists_{A^+, A^- \subset A, A^+ \cap A^- = 0}, \forall _{a^+ \in A^+,a^- \in A^-}$, s.t.
$\left(\dfrac{\partial f_t(a^+)}{\partial t} > 0 \right) \land \left( \dfrac{\partial f_t(a^-)}{\partial t} < 0 \right)$.**
Here I suppose it means that given the integral of $f_t(a)$ over $\mathbb{R}$ is unity and $\bar{\gamma}_t$ is a constant, the sign of the first derivative of the pdf is dictated by the value of ratio $\dfrac{\bar{G}_t (a)}{f_t (a)}$, as also expressed later. If this ratio is unequal for two points in the action space $b_1, b_2$, then there are indeed regions where the first derivative is both positive and negative. Is this reasoning correct? Is this some kind of inference from mean-value theorem or something else?
The paper continues,
From logical implication from equation above, it can be assured that the sign of $\dfrac{\partial f_t (a)}{\partial t}$ will be determined by the ratio $\dfrac{\bar{G}_t (a)}{f_t (a)}$. Notice subsets $A^+$ and $A^−$ are composed by the elements of $A$ that have not reached their value for the probability density function in equilibrium with $\bar{G}_t (a)$. That is, the $A^+$ subset is composed by all $a \in A$ is too small with respect to $\bar{G}_t (a)$ and vice versa for $A^−$.
What does this last paragraph mean? What does it mean for the elements of $A$ to not have reached their value for pdf in equilibrium with $\bar{G}_t (a)$?
