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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!

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For every $t\geqslant0$, let $N(t)$ denote the value of the counting process at time $t$.

Assume that the counting events occur at the times $X+n$ for $n\geqslant0$ and fix some $t\geqslant0$. Then, for every $s$ in $(0,1)$, $U=N(t+s)-N(t)$ and $V=N(t+1)-N(t+s)$ are not independent since $U$ and $V$ are two Bernoulli random variables such that $U+V=1$ almost surely. Thus the increments are not independent. On the other hand, for every $s$ in $(0,1)$, $N(t+s)-N(t)$ is Bernoulli with probability of success $s$. Likewise, for every $s\geqslant0$, $N(t+s)-N(t)$ is $\lfloor s\rfloor$ plus a Bernoulli random variable with probability of success $s-\lfloor s\rfloor$. Thus the increments are identically distributed.

These arguments follow directly from the definition of stationarity and from the definition of independent increments of a stochastic process. You might want to adapt them to the case when the counting events occur at times $X_n+n$ for $n\geqslant0$, comparing in particular the distributions of $N(t+1)-N(t)$ and $N(t+1.5)-N(t+.5)$ for some integer $t$.

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