Let $YI$ denote the first order statistic of a random sample of size $n$ from a distribution that has the p.d.f. $$f(x) = \begin{cases}e^{-(x - \theta)}&\theta < x < \infty\\ 0&\text{elsewhere}\end{cases}$$ Let $Z_n = n (YI - \theta)$. Investigate the limiting distribution of $Z_n$.
I'm pretty much a new learner on this subject and can't find any source reliable to answer this kind (first order statistic) problem. I only found $n$ or last order statistic and didn't know what is the differences.
Do i need to get the c.d.f first then limit it to the infinity and then use that to find the p.d.f of Zn then turn it into c.d.f again then limit it to infinity?