Divergent Stopping Times I have a question regarding a sequence of stopping times in general.
Consider in the context of gambling where $X_n$ denotes your cumulative wealth after the $n^{th}$ gamble.
I recall the intuitive notion of a stopping time with respect to the stochastic process $X$ is a rule that informs you when to quit.
Now, when I think of a sequence of stopping times, it could be some betting strategy that depends on how my wealth evolves over time. For example, I could quit betting when I hit a total wealth of \$0 or \$100. But I could also have a re-enter-the-casino rule as a function of the stopping time. For example, I could set up a re-join rule where I bet on a roulette after 5 cycles once I hit \$100 (e.g. I feel more confident betting after rounds of wins) .
The notion I am a bit unfamiliar is:
$$\lim_{n\rightarrow\infty} \tau_n=\infty.$$
In my example, does this mean as I bet infinitely many times, I will never seize an opportunity to actually stop? If I go bust, this would be true, right? An example would be the gambler's ruin and her stopping time?
$\textbf{My question is why and where is this used?}$
An example I see is when defining something as an approximation in a continuous set-up. Consider the definition of a local martingale.
$X_t$ is a local martingale if there exists a sequence of stopping times $\lim\limits_{n\rightarrow\infty} \tau_n=\infty$ such that $X_{t\wedge\tau_n}$ is a martingale for all $n$.
Are there more intuitive examples where a sequence of stopping times is used to describe something in a continuous set-up to approximate a process? Is this one of the major roles of the sequence?
 A: Your last sentence is basically correct: the sequence of stopping times is often used to approximate a process.  One of the most common applications is to set $\tau_n := \inf_t \{ |X_t| \ge n \}$, i.e. $\tau_n$ is the first time the process $X$ hits $n$.
In the context of betting strategies you discussed, you could consider each $\tau_n$ to be a separate quitting rule that one could follow.  I'm not sure I understand what you mean about "re-entering the casino;" you generally only consider the process stopped at a single $\tau_n$.  In your example, if we consider $X_t$ to be the gambler's wealth at time $t$ (without stopping) and set $\tau_n := \inf_t \{ |X_t| \ge n \}$ then $X^{\tau_n}_t := X_{\tau_n \wedge t}$ is the wealth of a gambler who leaves after reaching a wealth of $n$ dollars (or stays forever if that never happens).  The condition that $\lim_{n \rightarrow \infty} \tau_n = \infty$ basically says that the sequence of stopping times should converge to never stopping, so in this example stopping once your wealth hits $n$ becomes almost the same as never stopping when $n$ is large.
