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Let $(\Omega, \mathbb{P})$ be a probability space and $B\subseteq \Omega$ an event. Let further be $X_1,...,X_n$ random variables.

What does $\tau = \inf \{n\in \mathbb{N} \mid X_n \in B \} $ mean?

For usual the notation $X\in M$ for a set $M$ means $\{w\in \Omega\mid X_n(w) \in M\}$, but that requires that $M$ is part of the image of $X_n(\Omega)$, i.e. $M \subseteq X_n(\Omega)$.

So, for $B$ that notation probably means something different, but what?

Sudix
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    You might be missing a subscript $n$ on the $X$. If ${X_n}_{n \geq 0}$ is a sequence of random variables, $\tau$ is the first time the $X_n$ take on a value in $B$. For instance, if $X_n$ is a random walk and $B = {-4,4}$, then $\tau$ is the first time the random walk hits 4 or -4. One typically uses $T$ or $\tau$ to denote a stopping time, or a random variable such that ${\tau \leq n} \in \sigma(X_1,\dots,X_n)$, i.e. one does not need to look past $n$ seconds to figure out if one needs to stop. Stopping times are central to the theory of martingales and stochastic calculus. – user217285 Dec 14 '17 at 04:13
  • You're right I've deleted the subscript without thinking. But why can $X_n$ even hit a value inside $B$, they're defined on completely different sets? – Sudix Dec 14 '17 at 05:02
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    That's a typo then; $B$ should be a subset of the codomain of $X$ (typically the reals, but $X$ could be a complex Brownian motion for instance), and in this case ${X_n \in B}$ is an (measurable) event for each $n$. – user217285 Dec 14 '17 at 06:33

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