What is the distribution of $\min\limits_{1\le i\le N}\frac{X_i}{Y_i + C}$ as $N \rightarrow \infty$?

Question

Let $Z_i = \frac{X_i}{Y_i + C}$ with $i = 1, 2, \dotsc, N$ denote the sequence of the random variables, where $X_i$ and $Y_i$ are exponentially distributed independent random variables with different means $\lambda_x$ and $\lambda_y$. What will be the distribution of $\min\limits_{i = 1, 2, \dotsc, N}(Z_i)$ as $N \rightarrow \infty$? Will it be a Gumbel distribution? I am not sure.

Answer 1

Note that you do not need to calculate explicitly the distribution of the $Z_i$'s to solve the exercise. Indeed, the $Z_i$'s are iid random variables that take values in $[0,+\infty)$. Denote with $Z$ their common distribution and let $M_N:=\min_{1\le i\le N}\{Z_i\}$. So by a well known result the distribution of $M_N$ is given by $$F_{M_N}(z)=P(M_N\le z)=P\left(\min_{i\le i \le N}\{Z_i\}\le z\right)=\dots=1-\left(1-F_{Z}(z)\right)^N$$ For $z<0$ we have that $F_{M_N}(z)=1-(1-0)^N=0$ for any $N$ and for $z\ge 0$ we have that $$\lim_{N\to+\infty}F_{M_N(z)}=1-\lim_{N\to+\infty}(1-F_Z(z))^N=1-0=1$$ since $1-F_Z(z)<1$. So, if we denote with $M$ the limit distribution of $M_N$ we have by the above that $$F_M(z)=\begin{cases}0, & z<0\\1,&z\ge 0\end{cases}\implies F_M(z)=\mathbf 1_{\{z\ge0\}}$$ which implies that $M$ is a degenerate random variable with $P(M=0)=1$. In other words $$M_N \overset{d}\to 0$$