On <Emphasis Type="Bold">(c,p)</Emphasis>-pseudostable Random Variables

In (Oleszkiewicz, Lecture Notes in Math. 1807), K. Oleszkiewicz defined a p-pseudostable random variable X as a symmetric random variable for which the following equation holds: where G independent of X has normal distribution N(0,1), X′ denotes independent copy of X, and denotes equality of distributions. In this paper we define and study pseudostable random variables X for which the following equation holds: where c is a quasi-norm on IR, G_p independent of X is symmetric p-stable with the characteristic function e^{−|t|^p}. This is a very natural generalization of the idea of p-pseudostable variables. In this notation X is p-pseudostable iff X is -pseudostable. In the paper we show that if X is (c,p)-pseudostable then there exists r>0, C, D ≥ 0 such that c(a,b)^r=|a|^r+|b|^r and Ee e^itX=exp{− C |t|^p − D |t|^r}.