On stability estimations without any conditions of symmetry

Abstract

Let X, X1, X2, ..., Xn be i.i.d. random variables. B. Ramachandran and C.R. Rao have proved that if distributions of sample mean ‾X = ‾X(n) = (X1 + ⋯ + Xn)/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.