Remarks on characterization of normal and stable distributions

Let X, Y, Z be independent identically distributed (i.i.d.) random variables. Suppose {Mathematical expression} for all real t, u, v, where q=2 and p≠2m (m=1, 2,...) or 0<p<q<2. It was proved by the author this implies X, Y, Z have the symmetric q-stable distribution. For two random variables such result is not true. One may suppose that the condition {Mathematical expression} and additional assumption on the behavior of P{|X|≥x} (x→∞) imply X, 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 and q=2, then X and Y are normal.