# Question about the log-normal distribution

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):

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.)

• Cool question. The convergence for a number of X distributions I tried looks pretty rapid. – Glen_b -Reinstate Monica Oct 23 '19 at 3:35
• 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 – Glen_b -Reinstate Monica Oct 23 '19 at 4:21
• 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 $F_{Z^2} = F_{X+Z}$ – Glen_b -Reinstate Monica Oct 23 '19 at 4:22
• 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. – whuber Oct 24 '19 at 21:18
• 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. – whuber Oct 25 '19 at 12:36