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I don't think I've seen this notation before $\alpha = \inf \{i > 0 : X_i > X_0 \}$. What does the colon here mean? What does this expression mean in words?

It seems we're assigning the variable $\alpha$ to the RHS, where the RHS is the infimum of a set of random variables $X_1, \ldots, X_{\infty}$ are IID continuous?

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    $\begingroup$ : can mean such that, sometimes written | instead $\endgroup$ Commented Mar 2, 2021 at 3:12
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    $\begingroup$ @J.W.Tanner Ah I see. I'm familiar with the vertical bar notation. $\endgroup$ Commented Mar 2, 2021 at 3:15

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For each $\omega\in\Omega$, $\alpha(\omega)$ is the minimum index $i$ such that $X_i(\omega)>X_0(\omega)$.

For example: If $X_0(\omega_0)=2$ and $X_1(\omega_0)=0.4$, $X_2(\omega_0)=1.3$, $X_3(\omega_0)=2.5$,... Then $\alpha(\omega_0)=3$.

Note: That is wrong defined. $X_i>X_0$ must be changed to $X_i\geq X_0$ (because can be the empty set with $>$)

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  • $\begingroup$ Why is $i$ the minimum index? $\endgroup$ Commented Mar 2, 2021 at 3:15
  • $\begingroup$ @student010101 the set if indexes is closed, therefore the infimum and the minimum are the same.. $\endgroup$ Commented Mar 2, 2021 at 3:18
  • $\begingroup$ There are two $>$s in the definition of $\alpha$. You probably mean the first one, which would be good to note explicitly $\endgroup$ Commented Mar 2, 2021 at 3:19
  • $\begingroup$ I edit for clarification $\endgroup$ Commented Mar 2, 2021 at 3:19
  • $\begingroup$ @CélioAugusto Isn't the set open because there is an infinite number of $X$'s? $\endgroup$ Commented Mar 2, 2021 at 3:24

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