Finiteness of a converging random number series

[Loren Booda]

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?

 

[TheOtherDave]

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]

Hi, I understand this as follows: denote by ran(x) a random number between 0 and x. Let x₁=ran(1), and let x_i=ran(x_i-1) for x>1.

Let S be the sum Ʃx_i.

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]

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

 

[blue_raver22]

I would first clarify that the term "converging random number series" refers to a sequence of numbers that approaches a fixed value as the number of terms increases. In this scenario, the sequence is generated by randomly choosing a point y with a distance to zero less than a given positive value x, and then repeating this process with y=x.

Based on this description, it appears that the sequence will converge to zero as y approaches zero. This is because the distance between y and zero decreases with each iteration, eventually reaching a value of zero. Therefore, the sum of the y-values will also approach zero as y approaches zero, making it finite.

However, it is important to note that the finiteness of the sum also depends on the number of iterations or terms in the sequence. If the sequence is infinite, the sum may still be finite, but it could also potentially approach infinity depending on the specific values of x and y. I would recommend further analysis and calculations to determine the exact behavior of the sequence and its sum in this scenario.