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Let $S = (S_n)_{n \geq 1}$ be a simple random walk. We denote the hitting time of a point $b$ by $\tau_b = \min \{i \geq 1 : S_i \geq b\}$.

My text says that the events $\displaystyle\{\max_{k \leq n} S_k \geq b\}$ and $\{\tau_b\ \leq n\}$ are identical.

I'm having trouble reading the first set, which seems to be about the maximum value (the values on the $y$-axis of the simple random walk) of $S_k$, for $k \leq n$, but the second set seems to be about "time" (the values on the $x$-axis of the simple random walk). So how can the events be identical? Am I misunderstanding something here?

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If $\tau_b\leqslant n$, then $S_j\geqslant b$ for some $1\leqslant j\leqslant n$, and hence $$\max_{1\leqslant k\leqslant n}S_k\geqslant S_j\geqslant b.$$ If $\max_{1\leqslant k\leqslant n}S_k\geqslant b$, then $\tau_b\leqslant \operatorname{argmax}_{1\leqslant k\leqslant n}S_k\leqslant n$.

It follows that the two sets are equal.

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  • $\begingroup$ What's $\mathrm{argmax}$? It's my first time seeing it. $\endgroup$
    – Irregular User
    Commented May 3, 2016 at 0:39
  • $\begingroup$ Reading through this again, is it correct to understand that it's the argument that "something" takes to get the value of $\max_{1 \leq k \leq n}S_k$? Also, why don't both the sets simply contain one element? $\endgroup$
    – Irregular User
    Commented May 3, 2016 at 0:45
  • $\begingroup$ $\argmax_k S_k$ simply denotes the index $k$ for which $S_k$ is the maximum. As far as how many elements the sets contain, I couldn't tell you, and it isn't relevant. But there could be uncountably many. $\endgroup$
    – Math1000
    Commented May 3, 2016 at 0:56
  • $\begingroup$ what I don't seem to understand is this $\max_{k \leq n} S_k \geq b$... it's just a number, right? Since it gives you the maximum value that $S_k$ hits before $n$, right? $\endgroup$
    – Irregular User
    Commented May 3, 2016 at 18:12
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It turns out that what is trying to be said is that $\mathbb{P}\displaystyle\{\max_{k \leq n} S_k \geq b\}$ and $\mathbb{P}\{\tau_b\ \leq n\}$ are equal, which they indeed are.

The question otherwise makes no sense.

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  • $\begingroup$ It turns out that what is trying to be said is that the events $\{\max_{k \leq n} S_k \geq b\}$ and $\{\tau_b\ \leq n\}$ are equal, which they indeed are. $\endgroup$
    – Did
    Commented May 9, 2016 at 11:20

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