Characterizations of distributions via the stochastic ordering property of random linear forms

We first present a characterization of the normal distribution by the stochastic ordering relationship between a monomial and a random linear form of i.i.d. random variables. This extends a recent result of Oleszkiewicz (1997, Statist. Probab. Lett. 33, 277-280). Secondly, a remarkable characterization of the exponential distribution by geometric compounding is improved. And another characterization of the exponential distribution by the stochastic ordering relationship between a monomial and a linear form with random coefficients is also given. Finally, we investigate the characterization of the Laplace distribution.