[concreteblock]

The mean converges exponentially to zero with time. It doesn’t grow exponentially. So the theorem you cited also goes in the same direction of my statement.

[kgwgk]

> The mean converges to zero with time. It doesn’t grow exponentially.

  t=0 mean(w) = 1
  t=1 mean(w) = 1/2*1.5 + 1/2*0.6 = 1.05
  t=2 mean(w) = 1/4*1.5*1.5 + 1/2*1.5*0.6 + 1/4*0.6*0.6 = 1.1025
  ....
  t   mean(w) = 1.05^t

Don’t you agree?

[concreteblock]

You’re right. I made a mistake - I thought you were trying to contradict the theorem I stated. I’ve just realized you were saying something orthogonal.

As for the proof of my theorem, By taking logarithms, the process becomes an additive random walk with negative drift (log 1.6 + log 0.5 < 0). This is well known to converge to negative infinity almost surely. After exponentiating to undo the logarithm, this is exactly the statement I made.

It does not matter how many test subjects there are ( as long as there’s finitely many) because, informally speaking , you can just wait for each of them to become irrevocably bankrupt in turn.

[kgwgk]

I think we agree then:

- for a fixed sample size we can find a time large enough that the probability of the sample mean being above $1 is as low as we want

- for a fixed time we can find a sample size large enough that the probability of the sample mean being below $1 is as low as we want

- when both the sample size and the horizon grow without limit which effect dominates will depend on how we make it happen

Adding "almost surely" to "everyone goes bankrupt and will never recover" or "there is a finite time after which no-one ever passes above $0.0000000000000001" is a subtle change but it's enough to allow for someone to go to infinity with infinitesimal probability.

This is why the distribution mean can grow exponentially, it wouldn't be possible if the everyone and no-one in those quotes were strictly true.

[concreteblock]

I agree with the first 3 statements.

Just to confirm, I am using 'almost surely' in the technical sense, which means 'with probability 1.'

Consider the following statement:

If you keep flipping a fair coin every day, it is almost sure that after some day you will have gotten a tails.

This is the same 'almost surely' that I am referring to.

[kgwgk]

We agree!

The point was that you didn't specify "almost surely" previously, that's why I asked for a proof to understand what did you mean exactly when you said that "everyone goes bankrupt and will never recover" and "or "there is a finite time after which no-one ever passes above $0.0000000000000001".

The mean of a random variable that is close to zero is close to zero, the mean of a random variable that is almost surely close to zero can be anything.