On equivalence of product measures by random translation
If \(X\) is a random element, \(P_X\) denotes the probability measure it induces on its state space. Let now \(\underline{X}\) be a vector with countable i.i.d. random variables as components, and \(\underline {Y}\) a similar vector, with components only assumed independent. Both vectors are taken to be independent. Let \(\underline{Z} = \underline{Y} + \underline{X}\). Kakutani's dichotomy says either \(P_{\underline{Z}} \equiv P_{\underline{X}}\) (equivalent), or \(P_{\underline{Z}} \perp P_{\underline{X}}\) (singular). The paper being reviewed offers a theorem which helps in assessing equivalence, in terms of (a) restricted moments and probabilities of \(\underline {Y}\) (example: \(\sum_k P(|Y_k|\geq \varepsilon) < \infty)\) and (b) conditions on the density of the components of \(\underline{X}\) (example: \(\int^\infty_{-\infty} {[f''(x)]^2\over f(x)} dx < \infty)\). The result extends earlier work with the same concerns. It is also shown that the conditions of the theorem are ``the best possible''.
equivalence with respect to independent identically distributed sequences of random variables
scientific article; zbMATH DE number 775699