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Theorem 5.2.7
Let $ξ_1,ξ_2,...$ be independent and identically distributed with a distribution that is symmetric about zero.Let $Sn=ξ_1+···+ξ_n$. If $a >0$ then $P(\sup_{m≤n}S_m≥a)≤2P(S_n≥a)$

In the proof, $N$ is defined as below

$$N= \inf\{m≤n:S_m> a\}\;\;(\text{ with } \inf\, \emptyset=\infty)$$

and the fact that $\{N\leq n \}\in \mathcal{F}_N$ is used in the proof of the theorem. However, is this fact true?

I thought $\mathcal{F}_N$ is defined when $N$ is fixed. So the reverse should be true. I can't get the concept of $\mathcal{F}_N$ and $\{N\leq n\}$.

Dongri
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    Q1, Q2... –  Apr 16 '21 at 06:01
  • usually $\mathcal{F}_N$ is defined as the $\sigma $-algebra of a stopping time (namely $N$). In the definition of a stopping time $N$ it is assumed that any set of the form ${N\leqslant n}$ is $N$-measurable, i.e., belongs to $\mathcal{F}_N$ – Masacroso Apr 16 '21 at 06:54
  • Ok thanks very much !I think I grasped it – Dongri Apr 16 '21 at 11:18

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