Multiplicative Strong Unimodality

Multiplicative strong unimodality is defined as the preservation of unimodality in products of independent random variables. An Ibragimov type theorem is proved. As an application, preserving unimodality for scale mixtures of gamma distributions is examined. It is also shown that multiplicative strong unimodal probability measures on R appear as images, by the exponential map, of classical strong unimodal ones. The connection to the star order is also established.