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Finiteness of a converging random number series

by Loren Booda
Tags: converging, finiteness, number, random, series
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Loren Booda
#1
Feb29-12, 12:54 AM
Loren Booda's Avatar
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1. Imagine a positive point x not equal to zero.

2. Consider a randomly chosen point y with distance to zero less than x.

3. Let y=x. Repeat #2.

4. Is the sum of the y-values finite as y approaches zero?
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TheOtherDave
#2
Feb29-12, 02:11 AM
P: 15
Quote Quote by Loren Booda View Post
1. Imagine a positive point x not equal to zero.

2. Consider a randomly chosen point y with distance to zero less than x.

3. Let y=x. Repeat #2.

4. Is the sum of the y-values finite as y approaches zero?
For step three, it's supposed to be the other way around, right? x is supposed to equal y? Otherwise there's no reason for y to approach zero (or any other number).

I don't know if it always converges, but on average it converges to x (by "average" I mean that for any given random y value, the average of all choices is x/2, so y, on average, equals x/2).
Norwegian
#3
Feb29-12, 05:41 AM
P: 144
Hi, I understand this as follows: denote by ran(x) a random number between 0 and x. Let x1=ran(1), and let xi=ran(xi-1) for x>1.

Let S be the sum Ʃxi.

As noted above, the expected value of S is 1 (does require a very minor argument). The chance of the series not converging is 0. For example the chance of S>N must be less than 1/N, for the average sum to be 1, so the chance of divergence is less than 1/N for any positive N. (A small simulation shows that the chance of the sum exceeding 7 is about 1 in 10 million)

Loren Booda
#4
Feb29-12, 12:24 PM
Loren Booda's Avatar
P: 3,408
Finiteness of a converging random number series

Thanks kindly both of you for your information, which I am attempting to cogitate.


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