Let $B_{j,n}$ with $j=0,\ldots,n$ be a triangular array of independent Bernoulli random variables such that
$$ \mathbb{P}\left[B_{j,n}=1\right]=p_n,\quad \mathbb{P}\left[B_{j,n}=0\right]=1-p_n $$
with $p_n$ a sequence such that $p_n\rightarrow p$ when $n\rightarrow\infty$. Consider now the random variable
$$ z_{j,n} = \frac{B_{j,n}-p_n}{B_{j-1,n}-p_n},\quad j=1,\ldots,n. $$ Clearly the $z_{j,n}$ are dependent random variables and
$$ \mu_{j,n}=\mathbb{E}[z_{j,n}\mid B_{j-1,n}] = 0. $$
I am looking for a central limit theorem for
$$ S_n=\frac{1}{\sqrt{n}}\,\sum_{j=1}^nz_{j,n} $$ but being the z's dependent I cannot find a simple result. The paper by Dvoretzky (http://digitalassets.lib.berkeley.edu/math/ucb/text/math_s6_v2_article-30.pdf) says that it would be enough that
$$ \sum_{j=1}^n\frac{1}{n}\,\mathbb{E}\left[z_{j,n}^2\mid B_{j-1,n}\right]\stackrel{p}{\longrightarrow} 1,\quad (1) $$ (plus other two conditions that for now I am omitting). Nevertheless using Markov's inequality I am not able to prove (1), any hint? Do someone know a simple result on central limit theorem for triangular arrays of dependent random variables?