TL;DR:
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is a stopping time some sort of event, or is it a point in discrete time, or something else entirely
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what is an example of something which is not a stopping time?
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is my understanding of the concepts and definitions below correct?
I am having difficulty understanding what a stopping time is.
The definition I am provided with is as follows:
A random time $τ$ is called a stopping time if for any $n$, one can decide whether the event $\{τ ≤ n\}$ (and hence the complementary event $\{τ > n\}$) has occurred by observing the first n variables $X_1, X_2, . . . , X_n$.
We are then given an example:
Time of ruin is a stopping time.
$τ = \min\{n : X_n = 0\}$.
$\{τ > n\} = \{X_1 ≥ 0, X_2 ≥ 0, . . . , X_n > 0\}$.
I don't quite understand what this is supposed to tell us.
random time $τ$ is called a stopping time if for any $n$, one can decide whether the event $\{τ ≤ n\}$ has occurred by observing the first n variables $X_1, X_2, . . . , X_n$
When they say the event $\{ τ \leq n\}$ they are referring to some specific time, are they not? e.g $τ = 1$ or maybe $τ = 4$ as long as $τ \leq n$
Is this correct?
So then if we know the event $ \{τ \leq n\}$ has or has not occurred, we can conclude whether the complementary even $τ > n$ has occurred.
If this is fine so far, then I have issues with the example.
Time of ruin is stopping time,
$τ = \min\{n : X_n = 0\}$
Firstly, time of ruin to me means at a point where you have $0$ or a negative balance of some sort of asset (For a gambler, no more money to gamble with, for a business owner, no more cash to pay expenses or obligations) – is this correct?
In that case Time of ruin should occur when $X_n \leq 0$, correct?
IF that is fine, then continuing, what does
$τ = \min\{n : X_n = 0\}$ mean?
This is not the same as
$τ = \min\{n,X_n\}$, is it? What is it trying to say?
I read it as, the minimum of $n$, such that $X_n = 0$
So it's saying $τ$ is the first point at which we are ruined?
Is my understanding all correct? Can someone provide me with an example of what is NOT a stopping time? Does a "stopping time" refer to a type of event?
Best Answer
You have some event, which you typically don't know when occurs, but that can/will occur some time in the future. The time that this event occurs is random, and it is a stopping time if, at any point in time, you know whether the event has occurred or not.
A few quick examples.
1) Your own (a stopping time): Let $\tau$ denote the time that I'm ruined (i.e. when I have no money left). At any time, I know whether I am ruined or not. For instance, I am not ruined right now. I don't know when ruin occurs, or if it will occur at all, but if it does, I will know.
2) Parking (not a stopping time): Suppose I am driving along a very long road, and that I'm looking for the parking spot which is furthest towards the other end of the road (call this "the last parking spot"). I pass by available spots along the way, but at any time, I never know if I have passed the last free parking spot. Why? I could just have passed some empty spot, but I cannot see if there are more empty spots later on, and I wouldn't know if the spot that I just passed was the last one or not.
3) My birthday this year (a stopping time): This is a deterministic stopping time. At any time, I know whether or not my birthday has occurred this year. In fact, I know exactly when my birthday occurs, which makes this a non-typical stopping time in the sense that it is deterministic.