Consider $\{X_i\}_{i=1}^{\infty}$ are i.r.v. with $\mathbb{E}(X_i) = \mu$ and $\operatorname{Var}(X_i) = \sigma^2$. Then let $Z(t) = \sup \left\{n : \sum_{i=1}^{n} X_{i} \le t\right\}$.
And we need to show that $\dfrac{Z(t) - t/\mu}{\sigma \sqrt{t/\mu^3}} \xrightarrow[t \to \infty]{d} N(0,1)$.
Have no hot to step it with stochastic processes. I've thought about prove something like law of large numbers for $Z(t)/t$. I guess it should goes to $\frac{1}{\mu}$. But what else ?