Aug 28, 2015

# Student Puzzle Corner 11: deadline October 20, 2015

Infinite divisibility of Euclidean random variables and vectors has been a core theme in the theory of probability for at least sixty years. The question asked is when can a random variable be decomposed into small independent components. Precisely, a random variable (vector) $X$ is called infinitely divisible if for each $n \geq 1$, $X$ is equal in law to a sum of $n$ iid variables $X_{n1}, \cdots , X_{nn}$, with some common distribution $F_n$. So, obviously any Gaussian or any Poisson or any Gamma or any Cauchy random variable is infinitely divisible. An attractive characterization of infinitely divisible distributions is that the set of all possible weak limits of triangular array partial sums $S_n = X_{n1} + \cdots + X_{nn}$, where $X_{n1}, \cdots , X_{nn}$ are iid with some common distribution $F_n$, coincides with the set of all infinitely divisible distributions. For example, if $F_n$ is a Bernoulli distribution with parameter $\frac{\lambda }{n}$, then the partial sums $S_n$ converge in distribution to a Poisson with mean $\lambda$. So all Poissons arise as weak limits of triangular array partial sums, and hence must be infinitely divisible.

What is not infinitely divisible? Although many special examples are known, what can one say generally? Random variables with a bounded support cannot be infinitely divisible; random variables whose characteristic function $\psi (t)$ ever takes the value zero cannot be infinitely divisible. Thus, Betas cannot be infinitely divisible; interestingly, logarithms of Betas are. On the other hand, although all Gaussians, and in particular the standard normal, are infinitely divisible, fold the standard normal to $(0,\infty )$, and that’s not infinitely divisible. Any non-normal distribution with a normal tail cannot be infinitely divisible. Normal stands at the edge of infinite divisibility in thinness of the tail.

The concept of infinite divisibility has long been extended to non-Euclidean random variables, for example, to groups with some structures, and to random sets. Primary authoritative references include work of Bondesson, Ibragimov, Steutel, Zolotarev, and the all-time classic book of Feller. Infinite divisibility is an enigma; something may be infinitely divisible, and something very, very similar may fail to be so.

Here is the exact problem of this issue:
Fix $\epsilon > 0$. Give an example of two real valued random variables $X, Y$, respectively with densities $f(x), g(x)$, such that $f(x), g(x) > 0$ for all real $x$, $|g(x) – f(x)| \leq \epsilon$ for all real $x$, and $X$ is infinitely divisible, yet $Y$ is not.

Got it? Submit your solution to bulletin@imstat.org, with the subject “Student Puzzle Corner”. Show your workings!

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