In my notes I have the following definition A random variable $T=0,1,2,...,\infty$ is a stopping time , if $\forall n \le \infty$, the event $\{T \le n\}\in \mathscr F_n$
There are several examples (about martingales) where they first say something is a stopping time, so I just report here that initial part
1 The time $T$ of the first Head in a sequence of coinflips is a stopping time, while the time one before the first Head is not
2 A monkey repeatedly types any of the 26 letters of the English alphabet independently with equal chance. Let $T$ be the number of letters that have been typed when the entire word “ABRACADABRA” first appears. $T$ is a stopping time
3 Let $ X_k$ be iid ±1-valued with equal chance. The sum $S_n = \sum_{ k=1}^n X_k$ is called simple symmetric random walk (SSRW). Empty sums are zero, hence $S_0 = 0$. Then $S_n$ is a martingale, and $T : = \inf\{n : S_n = 1\}$ is a stopping time in the natural filtration of the walk
In all of these examples no clear explanation is given on how to prove that T is a stopping time. All they argue is that "using T you don't need to look into the future for stopping the process, so T is a stopping time". I would like to learn to prove this in a more formal way, by using the definition above. Therefore, how can I explicitly see that the event $\{T \le n\}\in \mathscr F_n = \sigma(X_0,X_1, ...,X_n)$ in each case?
These are the notes we are following:
people.maths.bris.ac.uk/~mb13434/mart_thy_notes.pdf
Edit 1 : Following the suggestion for the first case:
Considering $n=1, $ $\mathscr F_1 = \sigma(X_0,X_1)$, $\{T\le 1\}=\{T=0\}\cup \{T=1\}$. $ \{T=0\}$ is composed of the events where I get my first head in the first toss, so $\{T=0\}= \{H\}$ , $ \{T=1\}$ is composed of the events where I get my first head in the second toss so $ \{T=1\}=\{TH\}$ and therefore $\{T\le 1\}=\{H,TH\}$. How to continue from here? What is explicitly $\mathscr F_1$ and what are the values that $X_0$ and $X_1$ tan take?
Edit 2.
Actually I think the correct answer for n=1 of the first question is the following (I just hope someone can confirm the previous one was wrong) Since we are with n=1, $\mathscr F_1 = \sigma(X_0,X_1)$, $\Omega$ should be composed of two-toss events. Therefore: $\Omega=\{HH,HT,TT,TH\}$
$\{T=0\}= \{HH,HT\}$
$ \{T=1\}=\{TH\}$
$ \{T\le 1\}=\{TH,HH,HT\}$ To explicitly write $\mathscr F_1$, Let $X_i(H)=1$, $X_i(T)=0$, $i=1,2 $, and I look at all possible preimages of singletons:
$X_0^{-1}(0)=\{TH,TT\}$
$X_0^{-1}(1)=\{HH,HT\}$
$X_1^{-1}(0)=\{HT,TT\}$
$X_1^{-1}(1)=\{HH,TH\}$
Since the sigma algebra generated by $X_0$ and $X_1$ is the smallest one that make them measurable, I put all these preimages, their complements and unions in the sigma algebra, so:
$ \mathscr F_1= \sigma(X_0,X_1)=\{X_0^{-1}(0),X_0^{-1}(1),X_1^{-1}(0),X_1^{-1}(1),\Omega,\phi,\{HT,TH,TT\}, \{HH,TH,TT\}, \{HH,HT,TT\}, \{HH,HT,TH\}\}$ where I can explicitly see that $ \{T\le 1\}=\{TH,HH,HT\}\in \mathscr F_1$
I see that I still needed to take complements of the three-element sets and maybe even more sets are missing in the sigma-algebra. This rises the question of if there is an algorithmic way for listing them all without missing any? Maybe at the end I always get the power set? Now how to generalize this to arbitrary n?
These examples don't want you to prove anything. They want you to develop intuition.
${\cal F}_n$ is the amount of information available to you at time $n\,.$ Say, ${\cal F}_0=\{\emptyset,\Omega\}\,.$ That means at time $0$ you don't really know if an event happened or not. Later this event $A$ and its complement is in some ${\cal F}_n\,.$ That's when you know if it happened or not.
The time before the first head is clearly not a stopping time because you know if it was before the first head only later and not at that "time".
Typically, every random time that specifies when something happens for the first time is a stopping time.
More formally. Example 3 is the most common: By definition of $$ T=\inf\{n:S_n=1\} $$ we have $$ \{T\le k\}=\bigcup_{n\le k}\{S_n=1\}\,. $$ This shows that $\{T\le k\}\in{\cal F}_k$ and therefore $T$ is a stopping time.
About Example 1.
Let $S_n$ take value in $\{H,T\}$ and define $$ T=\inf\{n:S_n=H\} $$ the time of the first head. We know from the previous considerations that $$ \{T\le k\}=\bigcup_{n\le k}\{S_n=H\} $$ and that $T$ is a stopping time. The time before the first head is $T-1\,.$ We know $$ \{T-1\le k\}=\{T\le k+1\} $$ which is ${\cal F}_{k+1}-$ but not ${\cal F}_k$-measurable.