Let $Z_n$ be a stochastic process such that $P(Z_n/a(n) \le x|Z_n>0) \to P(\xi \le x)$, where $a(n) \to \infty$ is a sequence of positive real numbers.
Let $N(t)$ be an integer valued stochastic process independent of $Z_n$ such that $P(N(t)/b(t) \le x) \to P(\eta \le x)$, where $b(t) \to \infty$ is a sequence of positive real numbers.
Consider the superposition $Z_{N(t)}$. Is it true that $P(Z_{N(t)}/a(N(t)) \le x|Z_{N(t)}>0) \to P(\xi \le x)$?
What about $P(Z_{N(t)}/a(b(t)) \le x|Z_{N(t)}>0) \to ???? $