On (c,p)-pseudostable Random Variables

Autor: G. Mazurkiewicz, J. K. Misiewicz
Rok vydání: 2005
Předmět:
Zdroj: Journal of Theoretical Probability. 18:837-852
ISSN: 1572-9230
0894-9840
DOI: 10.1007/s10959-005-7528-0
Popis: 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: $$\forall a,b \in \ IR \ \exists\ d(a,b)\ aX + bX^{\prime} \mathop{=}^{d} (|a|^p + |b|^p)^{1/p} X + d(a,b) G,$$ where G independent of X has normal distribution N(0,1), X′ denotes independent copy of X, and $$\mathop{=}^{\!\!\!\!d}$$ denotes equality of distributions. In this paper we define and study pseudostable random variables X for which the following equation holds: $$\forall a,b \in \ IR\ \exists d(a,b)\geq 0 \ aX + bX' \mathop{=}^{d} c(a,b) X + d(a,b) G_p,$$ 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 $$(\| \cdot \|_p, 2)$$ -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 }.
Databáze: OpenAIRE
Externí odkaz: