Remarks on characterization of normal and stable distributions

Abstract

LetX, Y, Z be independent identically distributed (i.i.d.) random variables. Suppose

$$E\left| {tX + uY + vZ} \right|^p = A(\left| t \right|^q + \left| u \right|^q + \left| v \right|^q )^{{p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} $$

for all realt, u, v, whereq=2 andp≠2m (m=1, 2,...) or 0<p<q<2. It was proved by the author this impliesX, Y, Z have the symmetricq-stable distribution. For two random variables such result is not true. One may suppose that the condition

$$E\left| {tX + uY} \right|^p = A(\left| t \right|^q + \left| u \right|^q )^{{p \mathord{\left/ {\vphantom {p q}} \right. \kern-\nulldelimiterspace} q}} $$

and additional assumption on the behavior ofP{|X|≥x} (x→∞) implyX, Y are stable. In this paper we show it is not valid. The second result is: if the last relation holds for two different exponents andq=2, thenX andY are normal.