Re: discrete random variable

zqchen wrote:

On Feb 1, 9:23 am, Niels Diepeveen <n936...@xxxxxxxxxxxx> wrote:
Remember that for a sum of nonnegative numbers (such as probabilities)
convergence implies absolute convergence, so you can calculate the sum in
any convenient order.

could you give an example that sum converges or doesn't if calculated
in different order? thanks

If you search the sci.math archive for "conditionally convergent", you'll
find many interesting discussions of this. However, I didn't see many
simple concrete examples so here are some variations on the standard
example,
log 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 ...

If you reorder this as in exponentially increasing groups of positive and
negative terms, like this

1
- 1/2
+ 1/3 + 1/5
- 1/4 - 1/6
+ 1/7 + 1/9 + 1/11 + 1/13
- 1/8 - 1/10 - 1/12 - 1/14
+ 1/15 + 1/17 + 1/19 + 1/21 + 1/23 + 1/25 + 1/27 + 1/29
- 1/16 - 1/18 - 1/20 - 1/22 - 1/24 - 1/26 - 1/28 - 1/30
....

you can easily see that the absolute value of each group must be greater
than 1/4, because each group has 2^k terms greater than 2^(-k-2).
This is enough to show that the series doesn't converge.
To be more precise, in this order the partial sums still have a liminf of
log 2, but they also have a limsup of 3/2 log 2. If you let the size of the
groups grow even faster, you can get even wilder divergence.

At least as importantly, you can reorder the terms to make the sum converge
to a different limit, for example

(1 - 1/2 - 1/4) + (1/3 - 1/6 - 1/8) + (1/5 - 1/10 - 1/12) + ... =
(1/2 - 1/4) + (1/6 - 1/8) + (1/10 - 1/12) + .... =
(1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ....) / 2 = (log 2) / 2

--
Niels Diepeveen
.

Relevant Pages

  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... this series can be made to sum to anything. ... However, the convergence is not absolute, so we can find ... containing both L and "Square" elements. ... Indeed this is indexed by a different ordinal (2 omega) of the same ...
    (sci.math)
  • Re: 1+2+3+.....
    ... > it "appears" that the sum is zero. ... > You must have convergence and absolute convergence to ... of the sine function. ... > Bernoulli actually went on up through ...
    (sci.math)
  • Re: Riemann Hypothesis
    ... IF the radius of convergence is infinity, then Xi is identical to its ... The sum must be uniform convergent. ... Why not try emailing it to de Branges, ...
    (sci.math)
  • Sum over incomplete Gamma Functions
    ... The result of an integration leads to an infinite sum over incomplete ... and convergence is still far off. ... prv = gsm ...
    (comp.lang.fortran)
  • Re: A mapping from a Banach space to its dual
    ... >Let E be a Banach space and E^* its dual. ... Commonly called weak* convergence, if I'm reading correctly. ... >Does the sum ... for example if E = c_0, the space of all sequences ...
    (sci.math.research)