Kotlarski's Theorem

Theorem (Kotlarski, 1967): Let $X_1$, $X_2$, and $\theta$ be three independent real-valued random variables and define $Y_1=X_1+\theta$ and $Y_2=X_2+\theta$. If the characteristic function of $(Y_1,Y_2)$ does not vanish, then the joint distribution of $(Y_1,Y_2)$ determines the distributions of $X_1$, $X_2$, and $\theta$ up to a change of the location.

The above theorem was first proved by Kotlarski (1967). See also Prakasa-Rao (1992, section 2.1).

Kotlarski’s theorem appears frequently in the context of identification of econometric models. It can be used, for example, when one observes two error-contaminated measurements of the same variable (when the errors are independent). The distribution of the contaminated variables identifies the distribution of the true variable as well as that of the error(s).

References

• Kotlarski, I. I. (1967). On characterizing the gamma and normal distribution. Pacific Journal of Mathematics 20, 69–76.

• Prakasa-Rao, B. L. S. (1992). Identifiability in Stochastic Models: Characterization of Probability Distributions. Probability and Mathematical Statistics. Academic Press, Boston.