On stability estimations without any conditions of symmetry

Abstract

Let X, X₁, X₂, ..., X_n be i.i.d. random variables. B. Ramachandran and C.R. Rao have proved that if distributions of sample mean ‾X = ‾X(n) = (X₁ + ⋯ + X_n)/n and monomial X are coincident at least at two points n = j₁ and n = j₂ such that log j₁/ log j₂ is irrational, then X follows a Cauchy law. Assuming that condition of coincidence of \bar X(n) and X are fulfilled at least for two n values, but only approximately, with some error ε in metric λ, we prove (without any conditions of symmetry) that, in certain sense, characteristic function of X is close to the characteristic function of the Cauchy distribution.