Let $X,Y$ be two independent random variables.


The Kac-Bernstein theorem states that if $X+Y,X-Y$ are also independent, then $X,Y$ are Normal.


Are there analogues of this theorem for non-normal, continuous distributions? For instance, what functions $f\_i(X,Y)$ whose mutual independences guarantee that $X,Y$ are exponentially, or Beta distributed?



Analogues of Kac-Bernstein characterisation theorem for non-normal distributions
Such problems were systematically considered in the book *Characterization Problems in Mathematical Statistics* by Kagan, Linnik, and Rao; see e.g. [this review](https://projecteuclid.org/journals/annals-of-statistics/volume-5/issue-3/Review--A-M-Kagan-Yu-V-Linnik-C-Radhakrishna/10.1214/aos/1176343861.full). In particular, see Ch. 8 there on characterization of families of distributions admitting sufficient statistics, and also Section 6.2 on characterization of the gamma distribution. The list of families of distributions characterized in the book includes the normal (of course), exponential, gamma, beta, Cauchy, Wishart, and other distributions.


See also many papers citing this book.


