Remarks on characterization of normal and stable distributions

1572-9230

Distribution ; random variables ; characterization ; function

Springer Online Journal Archives 1860-2000

Mathematics

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.

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