A probability distribution is of Cauchy type if it is the distribution of $\mu + \sigma X$ where $\mu,\sigma$ are constant (i.e. not random) and the p.d.f. of $X$ is $1/(\pi(1+x^2)).$

Suppose that

[stand by --- I need to edit this paragraph somewhat]

That is proved in "A Characterization of the Cauchy Type" by Frank B. Knight, ''Proc. Amer. Math. Soc.'' 55 (1976), 130–135.

A simpler result is a converse of that: The family of distributions of Cauchy type is closed under that operation.

Which books or papers should be cited to show that simpler proposition, the converse of what Knight's paper says, has been around for a while?