See my response to Manfred above. The paper does not define what it means by “expected value”; I am assuming that it means the sum of all possible continuations of the infinite series, multiplied by their probabilities, because that seems consistent and appropriate. It could alternately mean the expectation of the value that an infinite series converges to, which would be a peculiar way of talking about utility calculations (it would be the opposite of time-discounting: the present doesn’t matter, only the infinite future), but would probably also be consistent with the paper.
The paper assumes that you have a universe of possible infinite series, all of which diverge; and proves (not surprisingly) that the sum of an expected value over an infinite number of such infinite series diverges.
If instead of taking the sum x1 + x2 + …, you use time-discounting into the future from the present time t=1:
x1 + f(x2 + f(x3 + f(x4 + …))))
where f(x) = x/c, then you are using exponential time-discounting; and you can find series that meet the particular definition of “unbounded” in the paper, but that are exponentially bounded, and for which the expected value of time-discounted infinite series, multiplied by their probabilities, would converge.
As I just pointed out at that other place in the thread, you are talking about a different paper from the one endoself linked to. The paper we are discussing here does not make the assumption you describe.
See my response to Manfred above. The paper does not define what it means by “expected value”; I am assuming that it means the sum of all possible continuations of the infinite series, multiplied by their probabilities, because that seems consistent and appropriate. It could alternately mean the expectation of the value that an infinite series converges to, which would be a peculiar way of talking about utility calculations (it would be the opposite of time-discounting: the present doesn’t matter, only the infinite future), but would probably also be consistent with the paper.
The paper assumes that you have a universe of possible infinite series, all of which diverge; and proves (not surprisingly) that the sum of an expected value over an infinite number of such infinite series diverges.
If instead of taking the sum x1 + x2 + …, you use time-discounting into the future from the present time t=1:
x1 + f(x2 + f(x3 + f(x4 + …))))
where f(x) = x/c, then you are using exponential time-discounting; and you can find series that meet the particular definition of “unbounded” in the paper, but that are exponentially bounded, and for which the expected value of time-discounted infinite series, multiplied by their probabilities, would converge.
As I just pointed out at that other place in the thread, you are talking about a different paper from the one endoself linked to. The paper we are discussing here does not make the assumption you describe.