Inverted Distributions

By the inverted distribution of a random variable X we mean the distribution of its reciprocal 1/X.Every distribution G that assigns probability 0 to the value 0 is the inverted distribution of some random variable, namely of X = 1/Y, where Y is distributed according to G.Thus it is impossible to assert any special properties of inverted distributions in general.As we show in Section 2, however, very weak assumptions concerning the distribution of X permit quite strong conclusions concerning G, such as Cauchy tails and multiple modes.Sections 3 and 4 discuss certain approximations and limit results concerning the inverted distributions considered in Section 2.