Let $0 < r \leq 1/2$ and let $B(n,p)$ be the notation for binomial distribution. Consider the random variable $Z_n$ defined by the recursive equation $$ Z_{n+1} = Z_n + X_n I(Z_{n} \geq r n^2) + Y_n I(Z_{n} < r n^2) $$ and $$ Z_1 \sim B(1,r) $$ where $X_m \sim B(m,p_1)$ and $Y_m \sim B(m,p_2)$ and $I(\cdot)$ is the indicator function.
How to compute the probability distribution of $\lim_{n \rightarrow \infty} \frac{Z_n}{n^2}$
Thanks in advance.