Let $X_1, X_2, · · ·$ be sequence of independent nonnegative random variables having a common distribution function $F$ such that $P(X_1=0)<1, E[X_1]=a<\infty$ and $\mathrm{Var}[X_1]=\sigma^2<\infty$. Let's denote $$S_0 = 0, S_n = X_1 + \cdots + X_n$$ for $n = 1, 2, \ldots $ and $$N_t =\sup\left\{n:S_n ≤t\right\}$$ for any $t≥0$.
As $t\rightarrow\infty$ what is the limit distribution of $$\frac{N_t-\frac{t}{a}}{\sqrt{\frac{\sigma^2t}{a^3}}}$$
I always had difficulties getting limit distributions and I get really confused. Can anyone lead me on this please? Thanks.