For all $n\in\mathbb{N}$ let $B_{j,n}$ with $j=1,...,n$ be a triangular array of independent Bernoulli variables such that
$$ \mathbb{P}\left[B_{j,n}=1\right] = p_{B,n},~~\mathbb{P}\left[B_{j,n}=0\right] = 1-p_{B,n}. $$
Besides when $n\rightarrow\infty$ we have $p_{B,n}\rightarrow p_B\in(0,1)$ and $B_{j,n}$ satisfies the Lindeberg's condition, hence
$$ Z_n = \frac{1}{\sqrt{n}}\,\frac{\sum_{j=1}^n(B_{j,n}-p_{F,n})}{s_n}\stackrel{D}{\longrightarrow} \text{N}(0,1), $$ where $s_n^2=\sum_{j=1}^n\text{Var}[B_{j,n}]$. My guess is that
$$ \frac{1}{n^{3/4}}\,\frac{\sum_{j=1}^n(B_{j,n}-p_{F,n})}{s_n}\stackrel{p}{\longrightarrow}0, $$ and this should be a consequence of the fact that $n^{3/4}$ diverges faster than $n^{1/2}$. Nevertheless I didnt find a proper way to prove that (Markov's inequality did not help me).