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The main object of my question is this: if $X$ has a log-normal distribution, $Y = X + Z$ and $Y$ has the same distribution as that of $Z^2$ (in other words, $F_{Z^2} = F_{X+Z}$) and $X, Z$ are independent, then what is the distribution of $Z$? See below a potential example with $X$ being log-normal, and $\log \log Z$ being almost, maybe exactly normal (its CDF pictured below, based on empirical calculations):

enter image description here


Below are details explaining why I came up with this problem, and why I am interested in it. Let's say that $Y = X + Z$, with $X$ and $Z$ being independent, and $Y$ having the same distribution as $Z^2$. In addition, $X \geq 0$ and $Y, Z \geq 1$. The latter condition is needed for reasons that are due to the nature of my problem, and explained below.

If the distribution of $Z$ is known, how do I find the distribution of $X$? Or conversely? Of course one can use the characteristic functions and the convolution theorem. I am interested here in finding the most simplest distributions for $X$ and $Y$, satisfying all the requirements.

The problem is as follows. We are dealing with

$$Z=\sqrt{X_1+\sqrt{X_2+\sqrt{X_3+\cdots}}},$$

where the $X_k$'s are i.i.d. with same distribution as $X$. I am trying to find $X$ such that $Z$ has a simple distribution. The problem seems easier to solve backward: finding $Z$ such that $X$ has a simple distribution.

I tried a log-normal distribution for $X$, more precisely $X=\exp(T)$ with $T \sim$ Normal$(0,1)$. It is pretty obvious that $Z \geq 1$ with probability one. Based on empirical evidence, $\log \log \beta Z$ is pretty well approximated if $\beta = 1$. Is it possible to compute the actual distribution of $Z$? Is my approximation exact? (Update, not, it's just an approximation, not an exact match) If not I am interested in any combination of $X$ and $Z$ where both distributions are simple ones. This is a follow up to my previous question on CV, see here. (Update: a summary about all this research can be found here.)

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    $\begingroup$ Cool question. The convergence for a number of X distributions I tried looks pretty rapid. $\endgroup$ – Glen_b -Reinstate Monica Oct 23 '19 at 3:35
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    $\begingroup$ What is it you're asking me to elaborate more on? I can't tell whether it's the first comment or the second. In the first case I was hoping to avoid the case where someone inadvertently tried to proceed from Y=X+Z by starting with Y and Z (and perhaps implicitly treating them as independent without recognizing that it matters). If you mean the second comment, I am not sure there's much more I can say -- if you try a few common distributions for $X_i$ that fit the conditions, the sequence $Z_i=\sqrt{X_i+Z_{i-1}}$ progresses quite rapidly to what appears to be a stationary distribution ... ctd $\endgroup$ – Glen_b -Reinstate Monica Oct 23 '19 at 4:21
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    $\begingroup$ ctd... some with means that appear tantalizingly close to simple expressions. Be careful writing $Z^2=X+Z$. You don't mean that! You mean instead that $F_{Z^2} = F_{X+Z}$ $\endgroup$ – Glen_b -Reinstate Monica Oct 23 '19 at 4:22
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    $\begingroup$ I believe the answer to your question is that such variables $(X,Y)$ do not exist. That is simply because sums of lognormals are not lognormal, even though squares and square roots of lognormals are lognormal. $\endgroup$ – whuber Oct 24 '19 at 21:18
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    $\begingroup$ The key is "not exactly." Indeed, in the geostatistical literature there is a concept of "conservation of lognormality" that recognizes both (a) sums of lognormals are not lognormal but (b) for many analytical purposes they can be modeled with lognormal distributions. $\endgroup$ – whuber Oct 25 '19 at 12:36

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