# Infinite divisiblity of log-normals

What is the law of a piece of a log-normal distribution?

We know that log-normals are infinitely divisible. What would be the law of a root of log-normal?

More specifically, suppose that $$X$$ is a log-normal. Given an integer $$k\geq 2$$, we know that there exist $$X_1,...,X_k$$ i.i.d such that:

$$\mathcal{L}(X) = \mathcal{L}\left(\sum\limits_{i=1}^{k} X_i\right),$$

where $$\mathcal{L}$$ denotes the distribution of a random variable.

Question: Is there a way to simulate directly from $$X_i$$'s law?

• Are lognormals divisible? Is there a theorem or a reference for that? – Matt F. Aug 27 '19 at 15:41
• Maybe relevant tandfonline.com/doi/abs/10.1080/03461238.1977.10405635 (I've only read the free Summary).."In the present paper the author proves that the lognormal distribution is infinitely divisible. This is achieved by showing that the lognormal is the weak limit of a sequence of probability distributions all of which are generalized Γ-convolutions and thus infinitely divisible. It is also proved that the lognormal itself is a generalized Γ-convolution." From practical persepctive, I doubt you'll do better than "directly" simulating for lognormal, unless you have unsual paradigm – Mark L. Stone Aug 27 '19 at 20:19
• The reference you pull up is the right one : they do prove infinite divisibility for log-normal. But i found nothing about the law of the pieces, hence my question. – lrnv Aug 28 '19 at 7:13
• The summary does mention generalized Γ-convolution - but I haven't read the paper, so don't know the details. – Mark L. Stone Aug 28 '19 at 11:27