Let $X$ be a uniform random variable on $(0,1)$ and consider a counting process where events occur at times $X+i$ for $i=0,1,2,...$. The random variable $X$ is generated just once at the beginning of the process. Does the counting process have independent increments? Stationary increments? Why or why not? What if instead, events occur at times $X_i+i$ for $i=0,1,2,...$?
I've looked up definitions for independent and stationary increments but I'm not sure how they can be applied to problems. An explanation of how to use those definitions for this problem would be greatly appreciated. Thanks!