TITLE: Convergence speed of a random dyadic rational generator
QUESTION [5 upvotes]: We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$

two distinct instances $x_r$ and $y_r$ of $M$'s numbers are selected uniformly at random from $M$ (which implies that they cannot be the same instance of any number contained in $M$, viz., $x_r$ is selected and temporarily removed from $M$, thereafter $y_r$ is selected from $M\setminus\{x_r\}$ without removing it, and finally $x_r$ is added back to $M$), and
$z_r=\frac{x_r+y_r}{2}$ is added to $M$.


Question: What is the probability $p_{r,{\epsilon}}$ that we have $\left|z_r-\frac{1}{2}\right|\le\epsilon$ for a given $\epsilon\in\left(0,\frac{1}{2}\right)$?

Edit: I show below the experimental results for different simulations of the random process run with an increasing total number of rounds, and for a single simulation of the random process keeping track of the evolution over time of the total average and the last element added - For the sake of convenience, by writing "rounds", here I am counting the initial insertion of both $0$ and $1$ (simultaneously) as the very first round:
New simul. with 2^1 rounds   -   Avg: 0.5 |   Last added: 0.5
New simul. with 2^2 rounds   -   Avg: 0.5 |   Last added: 0.5
New simul. with 2^3 rounds   -   Avg: 0.472222 |   Last added: 0.5
New simul. with 2^4 rounds   -   Avg: 0.430147 |   Last added: 0.375
New simul. with 2^5 rounds   -   Avg: 0.413826 |   Last added: 0.40625
New simul. with 2^6 rounds   -   Avg: 0.40012 |   Last added: 0.40625
New simul. with 2^7 rounds   -   Avg: 0.38313 |   Last added: 0.46875
New simul. with 2^8 rounds   -   Avg: 0.377516 |   Last added: 0.378906
New simul. with 2^9 rounds   -   Avg: 0.366866 |   Last added: 0.378906
New simul. with 2^10 rounds   -   Avg: 0.362607 |   Last added: 0.342743
New simul. with 2^11 rounds   -   Avg: 0.360595 |   Last added: 0.358353
New simul. with 2^12 rounds   -   Avg: 0.359471 |   Last added: 0.343569
New simul. with 2^13 rounds   -   Avg: 0.364962 |   Last added: 0.336161
New simul. with 2^14 rounds   -   Avg: 0.500135 |   Last added: 0.497771
New simul. with 2^15 rounds   -   Avg: 0.49995 |   Last added: 0.488623
New simul. with 2^16 rounds   -   Avg: 0.602851 |   Last added: 0.590848
New simul. with 2^17 rounds   -   Avg: 0.376087 |   Last added: 0.372888
New simul. with 2^18 rounds   -   Avg: 0.655107 |   Last added: 0.62898
New simul. with 2^19 rounds   -   Avg: 0.182425 |   Last added: 0.201142
New simul. with 2^20 rounds   -   Avg: 0.709139 |   Last added: 0.713385
New simul. with 2^21 rounds   -   Avg: 0.219937 |   Last added: 0.220374
New simul. with 2^22 rounds   -   Avg: 0.112707 |   Last added: 0.112427


Same simul. r=2^1 - Avg: 0.5 | Last added: 0.5
Same simul. r=2^2 - Avg: 0.5 | Last added: 0.75
Same simul. r=2^3 - Avg: 0.545139 | Last added: 0.46875
Same simul. r=2^4 - Avg: 0.625 | Last added: 0.28125
Same simul. r=2^5 - Avg: 0.60393 | Last added: 0.59375
Same simul. r=2^6 - Avg: 0.568329 | Last added: 0.71875
Same simul. r=2^7 - Avg: 0.57769 | Last added: 0.697266
Same simul. r=2^8 - Avg: 0.573474 | Last added: 0.631714
Same simul. r=2^9 - Avg: 0.575036 | Last added: 0.576538
Same simul. r=2^10 - Avg: 0.577153 | Last added: 0.47583
Same simul. r=2^11 - Avg: 0.578355 | Last added: 0.617221
Same simul. r=2^12 - Avg: 0.576684 | Last added: 0.57461
Same simul. r=2^13 - Avg: 0.576757 | Last added: 0.581285
Same simul. r=2^14 - Avg: 0.577305 | Last added: 0.546254
Same simul. r=2^15 - Avg: 0.577683 | Last added: 0.592735
Same simul. r=2^16 - Avg: 0.577662 | Last added: 0.56319
Same simul. r=2^17 - Avg: 0.577692 | Last added: 0.576607
Same simul. r=2^18 - Avg: 0.577675 | Last added: 0.571428
Same simul. r=2^19 - Avg: 0.577657 | Last added: 0.572818
Same simul. r=2^20 - Avg: 0.577655 | Last added: 0.579482
Same simul. r=2^21 - Avg: 0.577652 | Last added: 0.575974
Same simul. r=2^22 - Avg: 0.577654 | Last added: 0.5777
Same simul. r=2^23 - Avg: 0.577659 | Last added: 0.585123
Same simul. r=2^24 - Avg: 0.577657 | Last added: 0.571693
Same simul. r=2^25 - Avg: 0.577659 | Last added: 0.579782
Same simul. r=2^26 - Avg: 0.577659 | Last added: 0.574194
Same simul. r=2^27 - Avg: 0.577659 | Last added: 0.579098

REPLY [3 votes]: This is just a comment, but I wanted to include a graph. There are are $2^r-1$  possibilities for $z_r$. $z_r=\frac{p}{2^r}$ with$1 \le p \le 2^r-1.$ An interesting observation is that the denominator (in lowest terms) is unlikely to be close to $2^r$. If $z_r$ happens to have denominator $2^r$ then it is the only one with that denominator in (the current version of) $M$ and , aside from $\frac01$ and $\frac11$, each of $2,2^2,\cdots, 2^r$ is used as a denominator once. Then the denominator of $z_{r+1}$ is $2^s$ with probability $\frac{2s}{r^2+r}.$ so on average $s \approx \frac{2r}3.$
Here is the probability distribution for $z_6.$ Just under half the time it is $\frac12,\frac38,\frac58,\frac14,$ or $\frac34$ with frequencies roughly $12\%,10\%,10\%,8.5\%,8.5\%. $

That may not have any effect on the average behavior for very large $r$.