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I'm working on probability, more precisely on Brownian motions and I'm following a book by Peter Mörters and Yuval Peres and they used the following notation ($\vee$): $$\left\{\exists t\in[0,1): \left|\frac{T_k}{n}-t\right|\vee\left|\frac{T_{k-1}}{n}-t\right|\geq \delta\right\}$$ or for example in another context, $$\mathbb P\left\{\sup_{1\leq k\leq n}\frac{(T_k-(k-1))\vee (k-T_{k-1})}{n}\geq \delta\right\}$$ I have never seen this notation ever before and they do not say a word on it in the book.

This does not allow me to understand the proof as I have no idea what they mean.

I would like to know if anyone knows what this means.

(sorry I did not provide more context, but the question is just about the notation)

Thanks in advance.

PNM
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  • Well, my first guess would the inclusive OR. Thus your first expression would say "there is some $t\in [0,1)$ such that at least one of $\frac {T_k}n$, $\frac {T_{k-1}}n$ is within $\delta$ of $t$." I am assuming that $T_i, k, \delta$ have previously been specified or quantified in some way. – lulu Dec 10 '22 at 16:19
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    @lulu or lub? (i.e. here: $\max$), since in the second example it is divided by $n$ – Anne Bauval Dec 10 '22 at 16:21
  • @AnneBauval Maybe, though since there are only finitely many values under discussion it would be a bit odd to not just use "max". – lulu Dec 10 '22 at 16:22
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    @AnneBauval To be sure, there are many authors who appear to seek out complex notation...sometimes because they anticipate generalizations in which you need it, and sometimes because they just like it. – lulu Dec 10 '22 at 16:25

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Have a look at page x of your book ;-) Frequently used notation: "$a\lor b$ the maximum of $a$ and $b$"

Anne Bauval
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