Many of the questions I've posted on SE in the last month have been in the goal of helping me solve this particular problem. The questions have all been answered, but I still can't come up with a solution. So, I figured that I should just ask the problem I'm trying to solve directly.

Let $X_n \sim F_n$, where $F_n = (1-(1-F_{n-1})^c)^c$, $F_0 = x$, $c\geq 2$ (integer), and every $F_n$ is a cdf over $(0,1)$.

I want to prove that $\mathbb{E}X_n$ decreases with $n$ for all $c$ (or even, for any particular $c$)! I can show that $F_n$ converges to a Dirac mass at the unique solution to $x_c = (1-(1-x)^c)^c)$ For $c=2$, $x_2 = (3-\sqrt{5})/2 \approx .38$. When looking at a plot of cdfs for increasing $n$'s for the same $c$, all the cdfs cross at $x_n$. The value of $F(x)$ decreases for values of $x$ less than $x_n$ and increases for values of $x$ greater then $x_n$ (as $n$ increases) converging to a vertical line at $x_n$.

Below is a plot of $\mathbb{E}X_n$ for $n = 1$ to $40$ for $c = 2$ to $7$. It is of course a discrete plot, but I have the lines joined for ease of viewing. To generate this plot, I used NIntegrate in Mathematica, though I needed to do it on $1-F^{-1}_n$, as for some reason Mathematica couldn't generate answers on high values of $n$ for the original function. The two should be equivalent, as per Young's theorem, $\int_0^1F(x)\,dx = \int_0^1 1-F^{-1}(x)\,dx$. In my case, $F^{-1}_n(x) = 1-(1-(F^{-1}_{n-1})^{\frac{1}{c}})^{\frac{1}{c}}$, $F^{-1}_n = x$.

enter image description here

As you can see, the $EX_n$ moves very quicky to a minute distance from its fixed point $x_c$. As $c$ increases, the fixed point decreases (eventually will go to 0).

So, it certainly SEEMS to be true that $EX_n$ decreases with $n$ for all $c$. But I can't prove it. Can anyone help me out? (again, I'd be somewhat happy with even just a single $c$) And, if you can't, but you have insight as to why this particular problem may be unsolvable, please share that insight as well.

  • $\begingroup$ Have you considered rewriting so that $Zn = EX_n - EX_{n-1}$? An inductive proof or a contradiction may be readily accessible. $\endgroup$
    – Iterator
    Oct 15 '11 at 16:20
  • $\begingroup$ @Iterator: I have tried (a LOT) but have not been successful. $\endgroup$
    – OctaviaQ
    Oct 15 '11 at 16:35
  • 1
    $\begingroup$ Yes. +1 and deleted my previous comment. $\endgroup$
    – finnw
    Oct 16 '11 at 17:17
  • $\begingroup$ @Jand: I will unfortunately have to retract my proof claim for the moment. I have found a hole that I have yet to be able to patch. Apologies. I should have been more careful before posting something. I checked it several different times, but didn't find the problem until this last time I went through it. $\endgroup$
    – cardinal
    Oct 16 '11 at 23:55
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    $\begingroup$ @Jand: You have a very similar (but slightly different) question on math.SE. Can you clarify whether you are actually interested in both or just one of them and why? $\endgroup$
    – cardinal
    Oct 17 '11 at 23:49

This has been answered on MO by Pietro Majer here.


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