Stability of decompositions in the multiplicative scheme (Q920467)

The paper introduces the notion of multiplicative decomposition of a distribution function F(x) and studies the stability of such a decomposition. Let \(X_ 1\), \(X_ 2\) be independent, real random variables and let \(F_ 1(x)\), \(F_ 2(x)\) denote their distribution functions. The M-composition of \(F_ 1\) and \(F_ 2\) is defined to be the distribution function F(x) of the random variable \(X=X_ 1X_ 2.\) Consider a sequence \(F_ n=F_{n_ 1}\circ F_{n_ 2}\) of M- compositions with an appropriately defined Lévy-type distance from F denoted by \(\mu (F_ n,F)\). Then the paper shows that if \(\mu (F_ n,F)\to 0\) as \(n\to \infty\), then \[ \max \inf_{j=1,2}\mu (F_{n_ j},G)\to 0\quad as\quad n\to \infty. \] (Here G denotes a member of the set of all M-components of F(x)). The results can be regarded as analogous to those existing in the literature for the case of additive decompositions.