I have a random variable $R_n$ and a constant $w_n$ (which are related to a oriented percolation problem from https://arxiv.org/abs/1610.10018 on section 4.1(ii)) with the following properties:
(Notation: $\mathbb{E}$ is the expected value operator and $\mathbb{P}$ denotes the probability)
Let $R_n^+ = \max\{0,R_n\}$, $\exists$ positive constants $c_1,c_2,c_3$ with $c_1\leq 1$ such that
$c_1 w_n \leq \mathbb{E}(R_n^+)\leq \frac{1}{c_1}w_n$
$\mathbb{P}(R_n\geq 2w_n)\geq c_2$
$\mathbb{P}(R_n\leq w_n)\geq c_3$
The objective is to show that exists another constant $c_4$ such that $$\sqrt{Var(R_n)}\geq c_4 w_n$$
according to the authors this result "is directly implied" but I can't see it.
Here's what I attempted
Using Chebyshev's inequality:
$$\frac{Var(R_n)}{w_n^2}\geq \mathbb{P}(|R_n-\mathbb{E}(R_n)|\geq w_n) = 1-\mathbb{P}(|R_n-\mathbb{E}(R_n)|< w_n)=$$ $$=1 - \mathbb{P}\left[\mathbb{E}(R_n)-w_n<R_n<\mathbb{E}(R_n)+ w_n\right]\geq$$ $$\geq 1 - \mathbb{P}[E(R_n)-w_n<R_n<E(R_n^+)+w_n]\geq 1-\mathbb{P}\left[E(R_n)-w_n<R_n<\frac{w_n}{c_1}+w_n\right]\geq$$ $$\geq 1-\mathbb{P}\left[\mathbb{E}(R_n)-w_n<R_n< \left( \frac{1}{c_1}+1 \right) w_n\right]=$$ $$=\mathbb{P}\left[\left(R_n\leq \mathbb{E}(R_n)-w_n\right)\bigcup \left(R_n\geq \left(\frac{1}{c_1}+1\right)w_n\right)\right]=$$
$$ = \mathbb{P}\left[R_n\leq \mathbb{E}(R_n)-w_n\right]+\mathbb{P}\left[R_n\geq \left(\frac{1}{c_1}+1\right)w_n\right]$$
Now the plan was to somehow find the expressions on the properties to show the existence of $c_4$, but HOW?
Is there any steps I'm not seeing or another simple property that shows this, or maybe another well known inequality involving the variance I could use to try to understand this?
Thanks in advance.