I have these random variable $g_1,\,g_2,\cdots,\,g_K, h_2,\,h_2,\,\cdots,\,h_K$, where all random variables are i.i.d. Suppose $\{g_k\}$ are ordered as follows
$$g_{(1)}\leq g_{(2)}\leq\cdots\leq g_{(K)}$$
and then formed the following random variables $\left\{\theta_k=\frac{g_{(k)}}{h_k}\right\}_{k=1}^K$. In this case, $\{\theta_k\}$ are not independent (right?).
Then I ordered the resulting random variables as
$$\theta_{(1)}\leq \theta_{(2)}\leq\cdots\leq \theta_{(K)}$$
Is it correct then to say that $\text{Pr}\left[\theta_{(K)}\leq x\right]=\prod_{k=1}^K\text{Pr}\left[\theta_{k}\leq x\right]$?