# What does the notation $\alpha = \inf \{i > 0 : X_i > X_0 \}$ mean, where the $\{X_i\}_0^{\infty}$ are IID continuous?

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?

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

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 $$>$$)
• Why is $i$ the minimum index? Commented Mar 2, 2021 at 3:15
• There are two $>$s in the definition of $\alpha$. You probably mean the first one, which would be good to note explicitly Commented Mar 2, 2021 at 3:19
• @CélioAugusto Isn't the set open because there is an infinite number of $X$'s? Commented Mar 2, 2021 at 3:24