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?

Two probability distributions on (Borel subsets of) $\mathbb R$ are of the same "type" if for any random variable $X$ having one of those distributions, the other distribution is that of $\mu + \sigma X$ for some constants $\sigma\ne0, \, \mu.$

A probability distribution is of Cauchy type if it is of the same type as the distribution whose density is $1/(\pi(1+x^2)).$

Suppose that a family of distributions has the property that if the distribution of a random variable $X$ belongs to that family and $a,b,c,d\in \mathbb R$ and $ad-bc\ne0$ then the distribution of $(aX+b)/(cX+d)$ is of the same type. Then that is the family of distributions of Cauchy type. 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?