Under what condition the following variables can be independent? Basic independence query. one random variable is defined as $$\gamma_1=\frac{C|h_{sr}|^2}{\sum_{i=1}^{L}D|h_{P_ir}|^2}$$
where $|h_{sr}|$ and all $|h_{P_ir}|$ are independent exponential random variables and $C$ and $D$ are constants. The other random variable is defined as $$\gamma_2=\frac{\min \left[kC|h_{sr}|^2+\sum_{i=1}^{L}D|h_{P_ir}|^2,F\right]|h_{rd}|^2}{\sum_{i=1}^{L}D|h_{P_id}|^2}$$ where $E,k,F$ are constants and $|h_{sr}|^2,|h_{P_ir}|^2,|h_{P_id}|^2, |h_{rd}|^2$ are independent random variables. Is it possible that $\gamma_1$ and $\gamma_2$ are independent?
My Attempt:
In my view both $\gamma_1,\gamma_2$ depend on $h_{sr}$ therefore they can not be independent. But in one of the paper (http://home.iitk.ac.in/~kalamkar/links/Globecom_interference.pdf Title: Interference-Assisted Wireless Energy Harvesting in Cognitive Relay Network with Multiple Primary Transceivers, Read the line above Eq (16) and see Eq (9), Eq (10), Eq (13) and Eq (14) ) it is said that these two variables are independent. Is there something very critical that I am missing? Thanks in advance.    
 A: 
In my view both $\gamma_1,\gamma_2$ depend on $h_{sr}$ therefore they can not be independent.

That's not a valid argument. Two quantities can both depend on a third quantity (among others), yet be mutually independent:
Proposition 1. If $X\sim \text{Gamma}(k_X,\theta_X)$ and $Y\sim \text{Gamma}(k_Y,\theta_Y)$ are independent, then $S=X+Y$ and $R=X/Y$ are independent iff  $\theta_X=\theta_Y$. 
(Proof omitted -- it's a straightforward transformation-of-variables calculation with a simple Jacobian.) 
NB: In my notation, $\text{Exponential}(\theta)= \text{Gamma}(1,\theta)$, and $\text{Gamma}(k,\theta)$ has the probability density function $\frac{1}{\Gamma(k)\,\theta^k}x^{k-1}e^{-\frac{x}{\theta}}[x\ge 0].$ 
There seems to be a typo in your second equation, which I believe should be 
$$\begin{align}\gamma_2
&=\frac{\min \left[k\left(C|h_{sr}|^2+\sum_{i=1}^{L}D|h_{P_ir}|^2\right),F\right]|h_{rd}|^2}{\sum_{i=1}^{L}D|h_{P_id}|^2}\\ \\
&=\frac{\min \left[k\left(\color{blue}{X+Y}\right),F\right]|h_{rd}|^2}{\sum_{i=1}^{L}D|h_{P_id}|^2}\\
\end{align}$$
where $X$ and $Y$ are the numerator and denominator, respectively, of $\gamma_1$:
$$\begin{align}\gamma_1 &= \color{blue}{\frac{X}{Y}}\\ \\
X&=C|h_{sr}|^2\\ \\
Y &= \sum_{i=1}^{L}D|h_{P_ir}|^2.
\end{align}$$
In the paper, $X$ and $Y$ are independent random variables with Exponential and Gamma distributions, respectively. In Appendix B, the author derives the CDF of $X/Y$, but does not mention the independence question that you pose; however, given the assumptions stated in the paper, it's clear that if $X/Y$ and $X+Y$ are independent then so are $\gamma_1$ and $\gamma_2$. (I don't know whether $\theta_X=\theta_Y$ holds in the paper, or whether it may hold in some approximation.)
Another potential line of argument:  If there is a high probability that $X$ is much smaller than $Y$, then, with high probability, 
$$\frac{X}{Y} \approx  \frac{X}{X+Y} $$
so the following might apply:
Proposition 2. If $X\sim \text{Gamma}(k_X,\theta_X)$ and $Y\sim \text{Gamma}(k_Y,\theta_Y)$ are independent, then $S=X+Y$ and $T=\frac{X}{X+Y}$ are independent.
(Again omitting a straightforward proof, but see here.)
