On new stability estimations in Ramachandran–Rao characterization

B. Ramachandran and C.R. Rao have proved that if X, X1, X2, . . ., Xn are i.i.d. random variables and if distributions of sample mean \bar X = \bar 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 that, in certain sense, characteristic function of X is close to the characteristic function of the Cauchy distribution and construct stability estimation.