What I took CCC as actually meaning was that the differences converge-in-the-usual-sense to zero, or equivalently that one can make them small enough by waiting long enough.
That’s what CCC said in his last post, but it’s not a useful property for showing what was originally claimed by entirelyuseless, who said:
I meant that the changing average of the choices will converge, in the way I expect to happen in the beer case.
This doesn’t follow from shrinking steps, because:
That’s straightforwardly true because if the diameter of your region is D then after n steps your average can’t move by more than D/n per step.
The sum of that over all n is divergent, so the average can move around anywhere if you wait long enough. In fact, for any number of passes of averaging, that will still be true: if the underlying sequence goes somewhere and sticks there long enough, the average will eventually get there. Then the underlying sequence can go somewhere else, and so on.
To put this in more concrete terms, with an example of a property that for many people shows no long-term stability in their lifetime, consider physical location (relative to the geocentric frame). A person may live for years in one place, then years in another town or another continent, and make such moves at various times in their life. For such a person, there is no useful concept of their average location. In the case of the original example, a person’s taste in beer can make just as drastic changes, on top of which, the world changes and new beers are created, the space of the random walk changes as the walk is being made.
entirelyuseless’s original claim:
The random walk doesn’t converge. But the average position does.
I agree. (Did something I said give a contrary impression? I thought I’d said right at the outset that the original claims of both entirelyuseless and CCC are wrong.)
I have already said that I agree that it is mathematically possible to prevent the average from converging, just that this is not likely to happen in real life. In RichardKennaway’s comment, “goes somewhere and sticks there long enough” means progressively longer periods of time, and so is not realistic.
That’s what CCC said in his last post, but it’s not a useful property for showing what was originally claimed by entirelyuseless, who said:
This doesn’t follow from shrinking steps, because:
The sum of that over all n is divergent, so the average can move around anywhere if you wait long enough. In fact, for any number of passes of averaging, that will still be true: if the underlying sequence goes somewhere and sticks there long enough, the average will eventually get there. Then the underlying sequence can go somewhere else, and so on.
To put this in more concrete terms, with an example of a property that for many people shows no long-term stability in their lifetime, consider physical location (relative to the geocentric frame). A person may live for years in one place, then years in another town or another continent, and make such moves at various times in their life. For such a person, there is no useful concept of their average location. In the case of the original example, a person’s taste in beer can make just as drastic changes, on top of which, the world changes and new beers are created, the space of the random walk changes as the walk is being made.
entirelyuseless’s original claim:
cannot be salvaged.
I agree. (Did something I said give a contrary impression? I thought I’d said right at the outset that the original claims of both entirelyuseless and CCC are wrong.)
I have already said that I agree that it is mathematically possible to prevent the average from converging, just that this is not likely to happen in real life. In RichardKennaway’s comment, “goes somewhere and sticks there long enough” means progressively longer periods of time, and so is not realistic.