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There is a stochastic process $X(t)$, it is a discrete-time continuous-variable process. I can only know the distribution of $X(t)$ at discrete times $t = \tau, 2\tau, ...$. For such a process, given a constant threshold w, I concern the first passage time of $X(t)$. In other words, I want to know the first time that $X(t)$ exceeds $w$. If I denote the first passage time as $L$, do you think the random variable $L$ is also discretely distributed? Do you think the possible values of $L$ are also $\tau, 2\tau, ...$?

YAN BINGXIN
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  • By definition $L$ is discrete, as the process is discrete-time, but with no information as to the behaviour of $X(t)$, the latter question cannot be answered... – Math1000 Aug 09 '23 at 22:20
  • @Math1000 Thanks for your comment. To clarify, $X(t)$ exhibits independent and identically distributed increments over non-overlapping time increments, denoted as $X(k \tau + \tau) - X(k \tau) \sim F$ for $k=1,2,3,\ldots$, where $ F$ represents any continuous variable distribution. – YAN BINGXIN Aug 10 '23 at 05:06

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