Dgdrxrt

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: $mathbbE$ is the expected value operator and $mathbbP$ denotes the probability)

Let $R_n^+ = max0,R_n$, $exists$ positive constants $c_1,c_2,c_3$ with $c_1leq 1$ such that

• $c_1 w_n leq mathbbE(R_n^+)leq frac1c_1w_n$




• $mathbbP(R_ngeq 2w_n)geq c_2$




• $mathbbP(R_nleq w_n)geq c_3$

The objective is to show that exists another constant $c_4$ such that $$sqrtVar(R_n)geq c_4 w_n$$

according to the authors this result "is directly implied" but I can't see it.

Here's what I attempted

Using Chebyshev's inequality:

$$fracVar(R_n)w_n^2geq mathbbP(|R_n-mathbbE(R_n)|geq w_n) = 1-mathbbP(|R_n-mathbbE(R_n)|< w_n)=$$
$$=1 - mathbbPleft[mathbbE(R_n)-w_n<R_n<mathbbE(R_n)+ w_nright]geq$$
$$geq 1 - mathbbP[E(R_n)-w_n<R_n<E(R_n^+)+w_n]geq 1-mathbbPleft[E(R_n)-w_n<R_n<fracw_nc_1+w_nright]geq$$
$$geq 1-mathbbPleft[mathbbE(R_n)-w_n<R_n< left( frac1c_1+1 right) w_nright]=$$
$$=mathbbPleft[left(R_nleq mathbbE(R_n)-w_nright)bigcup left(R_ngeq left(frac1c_1+1right)w_nright)right]=$$

$$ = mathbbPleft[R_nleq mathbbE(R_n)-w_nright]+mathbbPleft[R_ngeq left(frac1c_1+1right)w_nright]$$

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.